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

Average Error: 9.4 → 0.1

Time: 11.8s

Precision: binary64

Cost: 832

Math

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

↓

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

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

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) + ((2.0d0 + (2.0d0 / z)) / t))
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 + ((2.0 + (2.0 / z)) / t));
}

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

Target

Original	9.4
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 9.4

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

2. Simplified 0.1

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

   [Start] 9.4	\[ \frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   +-rgt-identity [<=] 9.4	\[ \color{blue}{\left(\frac{x}{y} + 0\right)} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   mul0-lft [<=] 9.4	\[ \left(\frac{x}{y} + \color{blue}{0 \cdot \frac{2}{t \cdot z}}\right) + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   associate-+r+ [<=] 9.4	\[ \color{blue}{\frac{x}{y} + \left(0 \cdot \frac{2}{t \cdot z} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\right)} \]
   mul0-lft [=>] 9.4	\[ \frac{x}{y} + \left(\color{blue}{0} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\right) \]
   +-lft-identity [=>] 9.4	\[ \frac{x}{y} + \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} \]
   sub-neg [=>] 9.4	\[ \frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \color{blue}{\left(1 + \left(-t\right)\right)}}{t \cdot z} \]
   distribute-rgt-in [=>] 9.4	\[ \frac{x}{y} + \frac{2 + \color{blue}{\left(1 \cdot \left(z \cdot 2\right) + \left(-t\right) \cdot \left(z \cdot 2\right)\right)}}{t \cdot z} \]
   associate-+r+ [=>] 9.4	\[ \frac{x}{y} + \frac{\color{blue}{\left(2 + 1 \cdot \left(z \cdot 2\right)\right) + \left(-t\right) \cdot \left(z \cdot 2\right)}}{t \cdot z} \]
   cancel-sign-sub-inv [<=] 9.4	\[ \frac{x}{y} + \frac{\color{blue}{\left(2 + 1 \cdot \left(z \cdot 2\right)\right) - t \cdot \left(z \cdot 2\right)}}{t \cdot z} \]
   div-sub [=>] 9.4	\[ \frac{x}{y} + \color{blue}{\left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} - \frac{t \cdot \left(z \cdot 2\right)}{t \cdot z}\right)} \]
   associate-*r* [=>] 9.4	\[ \frac{x}{y} + \left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} - \frac{\color{blue}{\left(t \cdot z\right) \cdot 2}}{t \cdot z}\right) \]
   associate-*l/ [<=] 9.4	\[ \frac{x}{y} + \left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} - \color{blue}{\frac{t \cdot z}{t \cdot z} \cdot 2}\right) \]
   *-inverses [=>] 0.1	\[ \frac{x}{y} + \left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} - \color{blue}{1} \cdot 2\right) \]
   metadata-eval [=>] 0.1	\[ \frac{x}{y} + \left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} - \color{blue}{2}\right) \]
   sub-neg [=>] 0.1	\[ \frac{x}{y} + \color{blue}{\left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} + \left(-2\right)\right)} \]
   metadata-eval [=>] 0.1	\[ \frac{x}{y} + \left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} + \color{blue}{-2}\right) \]
   metadata-eval [<=] 0.1	\[ \frac{x}{y} + \left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} + \color{blue}{2 \cdot -1}\right) \]
   +-commutative [<=] 0.1	\[ \frac{x}{y} + \color{blue}{\left(2 \cdot -1 + \frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z}\right)} \]
   metadata-eval [=>] 0.1	\[ \frac{x}{y} + \left(\color{blue}{-2} + \frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z}\right) \]
   associate-*r* [=>] 0.1	\[ \frac{x}{y} + \left(-2 + \frac{2 + \color{blue}{\left(1 \cdot z\right) \cdot 2}}{t \cdot z}\right) \]
   distribute-rgt1-in [=>] 0.1	\[ \frac{x}{y} + \left(-2 + \frac{\color{blue}{\left(1 \cdot z + 1\right) \cdot 2}}{t \cdot z}\right) \]
   times-frac [=>] 9.6	\[ \frac{x}{y} + \left(-2 + \color{blue}{\frac{1 \cdot z + 1}{t} \cdot \frac{2}{z}}\right) \]
   *-commutative [<=] 9.6	\[ \frac{x}{y} + \left(-2 + \color{blue}{\frac{2}{z} \cdot \frac{1 \cdot z + 1}{t}}\right) \]
   associate-*r/ [=>] 0.1	\[ \frac{x}{y} + \left(-2 + \color{blue}{\frac{\frac{2}{z} \cdot \left(1 \cdot z + 1\right)}{t}}\right) \]
   *-lft-identity [=>] 0.1	\[ \frac{x}{y} + \left(-2 + \frac{\frac{2}{z} \cdot \left(\color{blue}{z} + 1\right)}{t}\right) \]
   distribute-rgt-in [=>] 0.1	\[ \frac{x}{y} + \left(-2 + \frac{\color{blue}{z \cdot \frac{2}{z} + 1 \cdot \frac{2}{z}}}{t}\right) \]
   /-rgt-identity [<=] 0.1	\[ \frac{x}{y} + \left(-2 + \frac{\color{blue}{\frac{z}{1}} \cdot \frac{2}{z} + 1 \cdot \frac{2}{z}}{t}\right) \]
   times-frac [<=] 0.1	\[ \frac{x}{y} + \left(-2 + \frac{\color{blue}{\frac{z \cdot 2}{1 \cdot z}} + 1 \cdot \frac{2}{z}}{t}\right) \]
   *-lft-identity [=>] 0.1	\[ \frac{x}{y} + \left(-2 + \frac{\frac{z \cdot 2}{\color{blue}{z}} + 1 \cdot \frac{2}{z}}{t}\right) \]
   *-commutative [=>] 0.1	\[ \frac{x}{y} + \left(-2 + \frac{\frac{\color{blue}{2 \cdot z}}{z} + 1 \cdot \frac{2}{z}}{t}\right) \]
   associate-/l* [=>] 0.1	\[ \frac{x}{y} + \left(-2 + \frac{\color{blue}{\frac{2}{\frac{z}{z}}} + 1 \cdot \frac{2}{z}}{t}\right) \]
   *-inverses [=>] 0.1	\[ \frac{x}{y} + \left(-2 + \frac{\frac{2}{\color{blue}{1}} + 1 \cdot \frac{2}{z}}{t}\right) \]
   metadata-eval [=>] 0.1	\[ \frac{x}{y} + \left(-2 + \frac{\color{blue}{2} + 1 \cdot \frac{2}{z}}{t}\right) \]
   *-lft-identity [=>] 0.1	\[ \frac{x}{y} + \left(-2 + \frac{2 + \color{blue}{\frac{2}{z}}}{t}\right) \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	30.8
Cost	1232

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -75000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -4.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 9.5 \cdot 10^{-47}:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{x}{y} \leq 1150000000:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Error	15.9
Cost	1228

\[\begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{t}\\ \mathbf{if}\;\frac{x}{y} \leq -0.0085:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq -1 \cdot 10^{-314}:\\ \;\;\;\;-2 + \frac{\frac{2}{t}}{z}\\ \mathbf{elif}\;\frac{x}{y} \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	5.0
Cost	1097

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -360000 \lor \neg \left(\frac{x}{y} \leq 11500\right):\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{2 + \frac{2}{z}}{t}\\ \end{array} \]

Alternative 4
Error	4.9
Cost	1097

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.45 \cdot 10^{-7} \lor \neg \left(\frac{x}{y} \leq 0.0019\right):\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{2 + \frac{2}{z}}{t}\\ \end{array} \]

Alternative 5
Error	0.7
Cost	969

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

Alternative 6
Error	20.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6900000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 35000000000:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7
Error	12.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-18} \lor \neg \left(t \leq 105000\right):\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \end{array} \]

Alternative 8
Error	20.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-33} \lor \neg \left(t \leq 1.1 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t}\\ \end{array} \]

Alternative 9
Error	34.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 10
Error	48.1
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2022364 
(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))))