Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2

Percentage Accurate: 86.1% → 99.3%

Time: 10.4s

Precision: binary64

Cost: 7232

• 24.2% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

↓

\[\mathsf{fma}\left(\left(1 + \frac{1}{z}\right) - t, \frac{2}{t}, \frac{x}{y}\right) \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))

↓

(FPCore (x y z t)
 :precision binary64
 (fma (- (+ 1.0 (/ 1.0 z)) t) (/ 2.0 t) (/ x y)))

C

double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
}

↓

double code(double x, double y, double z, double t) {
	return fma(((1.0 + (1.0 / z)) - t), (2.0 / t), (x / y));
}

Julia

function code(x, y, z, t)
	return Float64(Float64(x / y) + Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z)))
end

↓

function code(x, y, z, t)
	return fma(Float64(Float64(1.0 + Float64(1.0 / z)) - t), Float64(2.0 / t), Float64(x / y))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(N[(N[(1.0 + N[(1.0 / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] * N[(2.0 / t), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]

Target

Original	86.1%
Target	98.9%
Herbie	99.3%

\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right) \]

Derivation

1. Initial program 84.6%

   \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

2. Simplified 86.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, 1 - t, 1\right)}{z}, \frac{2}{t}, \frac{x}{y}\right)} \]
   Step-by-step derivation

   [Start] 84.6%	\[ \frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   +-commutative [=>] 84.6%	\[ \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]
   *-commutative [=>] 84.6%	\[ \frac{2 + \color{blue}{\left(1 - t\right) \cdot \left(z \cdot 2\right)}}{t \cdot z} + \frac{x}{y} \]
   associate-*r* [=>] 84.6%	\[ \frac{2 + \color{blue}{\left(\left(1 - t\right) \cdot z\right) \cdot 2}}{t \cdot z} + \frac{x}{y} \]
   distribute-rgt1-in [=>] 84.6%	\[ \frac{\color{blue}{\left(\left(1 - t\right) \cdot z + 1\right) \cdot 2}}{t \cdot z} + \frac{x}{y} \]
   *-commutative [=>] 84.6%	\[ \frac{\left(\left(1 - t\right) \cdot z + 1\right) \cdot 2}{\color{blue}{z \cdot t}} + \frac{x}{y} \]
   times-frac [=>] 86.0%	\[ \color{blue}{\frac{\left(1 - t\right) \cdot z + 1}{z} \cdot \frac{2}{t}} + \frac{x}{y} \]
   fma-def [=>] 86.8%	\[ \color{blue}{\mathsf{fma}\left(\frac{\left(1 - t\right) \cdot z + 1}{z}, \frac{2}{t}, \frac{x}{y}\right)} \]
   *-commutative [<=] 86.8%	\[ \mathsf{fma}\left(\frac{\color{blue}{z \cdot \left(1 - t\right)} + 1}{z}, \frac{2}{t}, \frac{x}{y}\right) \]
   fma-def [=>] 86.8%	\[ \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(z, 1 - t, 1\right)}}{z}, \frac{2}{t}, \frac{x}{y}\right) \]

3. Taylor expanded in z around 0 99.5%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\left(1 + \frac{1}{z}\right) - t}, \frac{2}{t}, \frac{x}{y}\right) \]

4. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(\left(1 + \frac{1}{z}\right) - t, \frac{2}{t}, \frac{x}{y}\right) \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	7232

\[\mathsf{fma}\left(\left(1 + \frac{1}{z}\right) - t, \frac{2}{t}, \frac{x}{y}\right) \]

Alternative 2
Accuracy	64.1%
Cost	1620

\[\begin{array}{l} t_1 := \frac{2}{z \cdot t}\\ t_2 := \frac{2}{t} + -2\\ \mathbf{if}\;\frac{x}{y} \leq -1.12 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 8.5 \cdot 10^{-222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq 2.9 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq 1.55 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Accuracy	64.0%
Cost	1620

\[\begin{array}{l} t_1 := \frac{2}{t} + -2\\ \mathbf{if}\;\frac{x}{y} \leq -3.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 3.65 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 3 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{elif}\;\frac{x}{y} \leq 0.000235:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 2.1 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Accuracy	69.0%
Cost	1240

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ t_2 := \frac{2}{t} + \frac{x}{y}\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+284}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -8.4 \cdot 10^{+229}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -104000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.46 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{2}{t}}{z}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	88.7%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4.7 \cdot 10^{+21} \lor \neg \left(\frac{x}{y} \leq 7.8 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{2}{t} + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(1 + \frac{1}{z}\right) - t}{t}\\ \end{array} \]

Alternative 6
Accuracy	80.2%
Cost	976

\[\begin{array}{l} t_1 := \frac{2 + \frac{2}{z}}{t}\\ t_2 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{t} + \frac{x}{y}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	80.2%
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -0.0028:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-35}:\\ \;\;\;\;2 \cdot \frac{1 + z}{z \cdot t}\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{2}{t} + \frac{x}{y}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	50.5%
Cost	972

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -2.25 \cdot 10^{-284}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 1.42 \cdot 10^{+16}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 9
Accuracy	82.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-91} \lor \neg \left(z \leq 1.1 \cdot 10^{-142}\right):\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t} + -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{z}\\ \end{array} \]

Alternative 10
Accuracy	63.5%
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 1.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{t} + -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 11
Accuracy	98.9%
Cost	832

\[\frac{x}{y} + \left(-2 + \frac{2 + \frac{2}{z}}{t}\right) \]

Alternative 12
Accuracy	36.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -0.033:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 13
Accuracy	20.4%
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2023165 
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"
  :precision binary64

  :herbie-target
  (- (/ (+ (/ 2.0 z) 2.0) t) (- 2.0 (/ x y)))

  (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))