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

Average Accuracy: 96.8% → 97.9%

Time: 12.8s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -11.5:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-43}:\\ \;\;\;\;t + \frac{1}{y} \cdot \left(x \cdot \left(z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(z - t\right) \cdot \frac{x}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= y -11.5)
   (fma x (/ (- z t) y) t)
   (if (<= y 5e-43)
     (+ t (* (/ 1.0 y) (* x (- z t))))
     (+ t (* (- z t) (/ x y))))))

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 (y <= -11.5) {
		tmp = fma(x, ((z - t) / y), t);
	} else if (y <= 5e-43) {
		tmp = t + ((1.0 / y) * (x * (z - t)));
	} else {
		tmp = t + ((z - t) * (x / y));
	}
	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 (y <= -11.5)
		tmp = fma(x, Float64(Float64(z - t) / y), t);
	elseif (y <= 5e-43)
		tmp = Float64(t + Float64(Float64(1.0 / y) * Float64(x * Float64(z - t))));
	else
		tmp = Float64(t + Float64(Float64(z - t) * Float64(x / y)));
	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[y, -11.5], N[(x * N[(N[(z - t), $MachinePrecision] / y), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[y, 5e-43], N[(t + N[(N[(1.0 / y), $MachinePrecision] * N[(x * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t + N[(N[(z - t), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	96.8%
Target	96.5%
Herbie	97.9%

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

Derivation

1. Split input into 3 regimes

2. if y < -11.5

   1. Initial program 98.4%

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

   2. Simplified 98.4%

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

      [Start] 98.4	\[ \frac{x}{y} \cdot \left(z - t\right) + t \]
      associate-*l/ [=>] 85.4	\[ \color{blue}{\frac{x \cdot \left(z - t\right)}{y}} + t \]
      associate-*r/ [<=] 98.4	\[ \color{blue}{x \cdot \frac{z - t}{y}} + t \]
      fma-def [=>] 98.4	\[ \color{blue}{\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)} \]

   if -11.5 < y < 5.00000000000000019e-43

   1. Initial program 94.0%

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

   2. Applied egg-rr 94.6%

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

      [Start] 94.0	\[ \frac{x}{y} \cdot \left(z - t\right) + t \]
      associate-*l/ [=>] 97.3	\[ \color{blue}{\frac{x \cdot \left(z - t\right)}{y}} + t \]
      clear-num [=>] 97.2	\[ \color{blue}{\frac{1}{\frac{y}{x \cdot \left(z - t\right)}}} + t \]
      associate-/r* [=>] 94.6	\[ \frac{1}{\color{blue}{\frac{\frac{y}{x}}{z - t}}} + t \]

   3. Applied egg-rr 97.2%

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

      [Start] 94.6	\[ \frac{1}{\frac{\frac{y}{x}}{z - t}} + t \]
      associate-/l/ [=>] 97.2	\[ \frac{1}{\color{blue}{\frac{y}{\left(z - t\right) \cdot x}}} + t \]
      associate-/r/ [=>] 97.2	\[ \color{blue}{\frac{1}{y} \cdot \left(\left(z - t\right) \cdot x\right)} + t \]
      *-commutative [=>] 97.2	\[ \frac{1}{y} \cdot \color{blue}{\left(x \cdot \left(z - t\right)\right)} + t \]

   if 5.00000000000000019e-43 < y

   1. Initial program 98.4%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11.5:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-43}:\\ \;\;\;\;t + \frac{1}{y} \cdot \left(x \cdot \left(z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(z - t\right) \cdot \frac{x}{y}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	64.9%
Cost	1944

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(-\frac{t}{y}\right)\\ \mathbf{elif}\;\frac{x}{y} \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;\frac{x}{y} \leq -2 \cdot 10^{+17}:\\ \;\;\;\;t \cdot \frac{-x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-21}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 20000000000000:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{y}{x}}\\ \end{array} \]

Alternative 2
Accuracy	64.9%
Cost	1424

\[\begin{array}{l} t_1 := \frac{-t}{\frac{y}{x}}\\ \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-21}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 20000000000000:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	64.8%
Cost	1424

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{+17}:\\ \;\;\;\;t \cdot \frac{-x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-21}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 20000000000000:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{y}{x}}\\ \end{array} \]

Alternative 4
Accuracy	92.6%
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -10 \lor \neg \left(\frac{x}{y} \leq 50000000000000\right):\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 5
Accuracy	92.9%
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -10 \lor \neg \left(\frac{x}{y} \leq 50000000\right):\\ \;\;\;\;\frac{x}{\frac{y}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 6
Accuracy	93.0%
Cost	968

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -10:\\ \;\;\;\;\frac{x}{\frac{y}{z - t}}\\ \mathbf{elif}\;\frac{x}{y} \leq 50000000:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y}\\ \end{array} \]

Alternative 7
Accuracy	97.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+41}:\\ \;\;\;\;t + \frac{x}{\frac{y}{z - t}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-43}:\\ \;\;\;\;t + \frac{1}{y} \cdot \left(x \cdot \left(z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(z - t\right) \cdot \frac{x}{y}\\ \end{array} \]

Alternative 8
Accuracy	64.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-39} \lor \neg \left(\frac{x}{y} \leq 10^{-21}\right):\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 9
Accuracy	94.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-188} \lor \neg \left(y \leq 1.15 \cdot 10^{-66}\right):\\ \;\;\;\;t + \left(z - t\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y}\\ \end{array} \]

Alternative 10
Accuracy	96.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-297} \lor \neg \left(t \leq 5.5 \cdot 10^{-117}\right):\\ \;\;\;\;t + \left(z - t\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x}{\frac{y}{z - t}}\\ \end{array} \]

Alternative 11
Accuracy	64.4%
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-21}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \end{array} \]

Alternative 12
Accuracy	63.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-21}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot z}{y}\\ \end{array} \]

Alternative 13
Accuracy	71.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-82} \lor \neg \left(t \leq 1.35 \cdot 10^{-234}\right):\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot z}{y}\\ \end{array} \]

Alternative 14
Accuracy	75.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-32} \lor \neg \left(t \leq 4.85 \cdot 10^{-88}\right):\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \end{array} \]

Alternative 15
Accuracy	49.3%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023138 
(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))