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

Percentage Accurate: 86.9% → 99.4%

Time: 8.0s

Precision: binary64

Cost: 7232

• 22.7% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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

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(x, (1.0 / y), (-2.0 + ((2.0 + (2.0 / z)) / t)));
}

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(x, Float64(1.0 / y), Float64(-2.0 + Float64(Float64(2.0 + Float64(2.0 / z)) / t)))
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[(x * N[(1.0 / y), $MachinePrecision] + N[(-2.0 + N[(N[(2.0 + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	86.9%
Target	99.2%
Herbie	99.4%

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

Derivation

1. Initial program 86.2%

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

2. Simplified 99.2%

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

   [Start] 86.2%	\[ \frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   sub-neg [=>] 86.2%	\[ \frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \color{blue}{\left(1 + \left(-t\right)\right)}}{t \cdot z} \]
   distribute-rgt-in [=>] 86.2%	\[ \frac{x}{y} + \frac{2 + \color{blue}{\left(1 \cdot \left(z \cdot 2\right) + \left(-t\right) \cdot \left(z \cdot 2\right)\right)}}{t \cdot z} \]
   *-lft-identity [=>] 86.2%	\[ \frac{x}{y} + \frac{2 + \left(\color{blue}{z \cdot 2} + \left(-t\right) \cdot \left(z \cdot 2\right)\right)}{t \cdot z} \]
   associate-+r+ [=>] 86.2%	\[ \frac{x}{y} + \frac{\color{blue}{\left(2 + z \cdot 2\right) + \left(-t\right) \cdot \left(z \cdot 2\right)}}{t \cdot z} \]
   cancel-sign-sub-inv [<=] 86.2%	\[ \frac{x}{y} + \frac{\color{blue}{\left(2 + z \cdot 2\right) - t \cdot \left(z \cdot 2\right)}}{t \cdot z} \]
   div-sub [=>] 78.0%	\[ \frac{x}{y} + \color{blue}{\left(\frac{2 + z \cdot 2}{t \cdot z} - \frac{t \cdot \left(z \cdot 2\right)}{t \cdot z}\right)} \]
   associate-*r* [=>] 78.0%	\[ \frac{x}{y} + \left(\frac{2 + z \cdot 2}{t \cdot z} - \frac{\color{blue}{\left(t \cdot z\right) \cdot 2}}{t \cdot z}\right) \]
   associate-*l/ [<=] 78.0%	\[ \frac{x}{y} + \left(\frac{2 + z \cdot 2}{t \cdot z} - \color{blue}{\frac{t \cdot z}{t \cdot z} \cdot 2}\right) \]
   *-inverses [=>] 99.1%	\[ \frac{x}{y} + \left(\frac{2 + z \cdot 2}{t \cdot z} - \color{blue}{1} \cdot 2\right) \]
   metadata-eval [=>] 99.1%	\[ \frac{x}{y} + \left(\frac{2 + z \cdot 2}{t \cdot z} - \color{blue}{2}\right) \]
   sub-neg [=>] 99.1%	\[ \frac{x}{y} + \color{blue}{\left(\frac{2 + z \cdot 2}{t \cdot z} + \left(-2\right)\right)} \]
   metadata-eval [=>] 99.1%	\[ \frac{x}{y} + \left(\frac{2 + z \cdot 2}{t \cdot z} + \color{blue}{-2}\right) \]
   metadata-eval [<=] 99.1%	\[ \frac{x}{y} + \left(\frac{2 + z \cdot 2}{t \cdot z} + \color{blue}{2 \cdot -1}\right) \]
   +-commutative [<=] 99.1%	\[ \frac{x}{y} + \color{blue}{\left(2 \cdot -1 + \frac{2 + z \cdot 2}{t \cdot z}\right)} \]
   metadata-eval [=>] 99.1%	\[ \frac{x}{y} + \left(\color{blue}{-2} + \frac{2 + z \cdot 2}{t \cdot z}\right) \]
   associate-/l/ [<=] 99.2%	\[ \frac{x}{y} + \left(-2 + \color{blue}{\frac{\frac{2 + z \cdot 2}{z}}{t}}\right) \]

3. Applied egg-rr 99.4%

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

   [Start] 99.2%	\[ \frac{x}{y} + \left(-2 + \frac{2 + \frac{2}{z}}{t}\right) \]
   div-inv [=>] 99.0%	\[ \color{blue}{x \cdot \frac{1}{y}} + \left(-2 + \frac{2 + \frac{2}{z}}{t}\right) \]
   fma-def [=>] 99.4%	\[ \color{blue}{\mathsf{fma}\left(x, \frac{1}{y}, -2 + \frac{2 + \frac{2}{z}}{t}\right)} \]

4. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	7232

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

Alternative 2
Accuracy	70.5%
Cost	1488

\[\begin{array}{l} t_1 := -2 + \frac{2}{z \cdot t}\\ \mathbf{if}\;\frac{x}{y} \leq -3.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 1.02 \cdot 10^{-286}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 9.5 \cdot 10^{-141}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - 2\\ \end{array} \]

Alternative 3
Accuracy	98.4%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -17000000 \lor \neg \left(z \leq 0.0009\right):\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{\frac{2}{z}}{t}\right)\\ \end{array} \]

Alternative 4
Accuracy	65.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -8 \cdot 10^{-7} \lor \neg \left(\frac{x}{y} \leq 1.65 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{x}{y} - 2\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \end{array} \]

Alternative 5
Accuracy	78.7%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-194} \lor \neg \left(t \leq 2.3 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \end{array} \]

Alternative 6
Accuracy	65.3%
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 8200000:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7
Accuracy	99.2%
Cost	832

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

Alternative 8
Accuracy	61.5%
Cost	716

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -1.85 \cdot 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-238}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;t \leq 0.0021:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	80.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-11} \lor \neg \left(t \leq 0.0011\right):\\ \;\;\;\;\frac{x}{y} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \end{array} \]

Alternative 10
Accuracy	46.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.1 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 4.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 11
Accuracy	19.2%
Cost	192

\[\frac{2}{t} \]

Reproduce

herbie shell --seed 2023171 
(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))))