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

Average Accuracy: 85.0% → 99.9%

Time: 15.1s

Precision: binary64

Cost: 960

Math

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

↓

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

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

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x / y) + ((2.0d0 + ((z * 2.0d0) * (1.0d0 - t))) / (t * z))
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x / y) + (2.0d0 * ((1.0d0 + (1.0d0 / z)) / t))) - 2.0d0
end function

Java

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

↓

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

Python

def code(x, y, z, t):
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z))

↓

def code(x, y, z, t):
	return ((x / y) + (2.0 * ((1.0 + (1.0 / z)) / t))) - 2.0

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 Float64(Float64(Float64(x / y) + Float64(2.0 * Float64(Float64(1.0 + Float64(1.0 / z)) / t))) - 2.0)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
end

↓

function tmp = code(x, y, z, t)
	tmp = ((x / y) + (2.0 * ((1.0 + (1.0 / z)) / t))) - 2.0;
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[(x / y), $MachinePrecision] + N[(2.0 * N[(N[(1.0 + N[(1.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]

Target

Original	85.0%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Initial program 85.0%

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

2. Simplified 85.0%

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

   [Start] 85.0	\[ \frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   +-commutative [=>] 85.0	\[ \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]
   *-commutative [=>] 85.0	\[ \frac{2 + \color{blue}{\left(1 - t\right) \cdot \left(z \cdot 2\right)}}{t \cdot z} + \frac{x}{y} \]
   associate-*r* [=>] 85.0	\[ \frac{2 + \color{blue}{\left(\left(1 - t\right) \cdot z\right) \cdot 2}}{t \cdot z} + \frac{x}{y} \]
   distribute-rgt1-in [=>] 85.0	\[ \frac{\color{blue}{\left(\left(1 - t\right) \cdot z + 1\right) \cdot 2}}{t \cdot z} + \frac{x}{y} \]
   *-commutative [=>] 85.0	\[ \frac{\left(\left(1 - t\right) \cdot z + 1\right) \cdot 2}{\color{blue}{z \cdot t}} + \frac{x}{y} \]
   times-frac [=>] 85.0	\[ \color{blue}{\frac{\left(1 - t\right) \cdot z + 1}{z} \cdot \frac{2}{t}} + \frac{x}{y} \]
   fma-def [=>] 85.0	\[ \color{blue}{\mathsf{fma}\left(\frac{\left(1 - t\right) \cdot z + 1}{z}, \frac{2}{t}, \frac{x}{y}\right)} \]
   *-commutative [<=] 85.0	\[ \mathsf{fma}\left(\frac{\color{blue}{z \cdot \left(1 - t\right)} + 1}{z}, \frac{2}{t}, \frac{x}{y}\right) \]
   fma-def [=>] 85.0	\[ \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(z, 1 - t, 1\right)}}{z}, \frac{2}{t}, \frac{x}{y}\right) \]

3. Taylor expanded in t around inf 99.9%

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

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	90.7%
Cost	1104

\[\begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{z \cdot t}\\ t_2 := \frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-11}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-69}:\\ \;\;\;\;-2 + \frac{\frac{2}{z}}{t}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	91.5%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+16} \lor \neg \left(\frac{x}{y} \leq 10^{+30}\right):\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{\frac{2}{z} - -2}{t}\\ \end{array} \]

Alternative 3
Accuracy	69.1%
Cost	976

\[\begin{array}{l} t_1 := -2 - \frac{\frac{-2}{t}}{z}\\ t_2 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{+124}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	69.1%
Cost	976

\[\begin{array}{l} t_1 := \frac{2}{z \cdot t} - 2\\ t_2 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{+125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	52.5%
Cost	972

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -9.6 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 2:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6
Accuracy	98.9%
Cost	969

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

Alternative 7
Accuracy	80.8%
Cost	844

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -5 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1100:\\ \;\;\;\;\frac{2}{z \cdot t} - 2\\ \mathbf{elif}\;t \leq 0.00086:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	87.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-6} \lor \neg \left(z \leq 9 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{\frac{-2}{t}}{z}\\ \end{array} \]

Alternative 9
Accuracy	99.9%
Cost	832

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

Alternative 10
Accuracy	69.2%
Cost	716

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -52000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	68.8%
Cost	716

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-29}:\\ \;\;\;\;-2 + \frac{\frac{2}{z}}{t}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Accuracy	69.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-25} \lor \neg \left(t \leq 5.2 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{x}{y} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t}\\ \end{array} \]

Alternative 13
Accuracy	46.3%
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-26}:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq 1.05:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 14
Accuracy	25.3%
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2023147 
(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))))