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

Average Error: 9.7 → 0.1

Time: 34.7s

Precision: binary64

Cost: 1088

Math

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

↓

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

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 t) (- (* 2.0 (/ (/ 1.0 t) z)) 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 / t) + ((2.0 * ((1.0 / t) / z)) - 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 / t) + ((2.0d0 * ((1.0d0 / t) / z)) - 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 / t) + ((2.0 * ((1.0 / t) / z)) - 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 / t) + ((2.0 * ((1.0 / t) / z)) - 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(x / y) + Float64(Float64(2.0 / t) + Float64(Float64(2.0 * Float64(Float64(1.0 / t) / z)) - 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 / t) + ((2.0 * ((1.0 / t) / z)) - 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[(x / y), $MachinePrecision] + N[(N[(2.0 / t), $MachinePrecision] + N[(N[(2.0 * N[(N[(1.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	9.7
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 9.7

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

2. Taylor expanded in t around 0 0.1

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

3. Simplified 0.1

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

   [Start] 0.1	\[ \frac{x}{y} + \left(\left(2 \cdot \frac{1}{t \cdot z} + \frac{2}{t}\right) - 2\right) \]
   rational.json-simplify-48 [=>] 0.1	\[ \frac{x}{y} + \color{blue}{\left(\frac{2}{t} + \left(2 \cdot \frac{1}{t \cdot z} - 2\right)\right)} \]
   rational.json-simplify-46 [=>] 0.1	\[ \frac{x}{y} + \left(\frac{2}{t} + \left(2 \cdot \color{blue}{\frac{\frac{1}{t}}{z}} - 2\right)\right) \]

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	5.0
Cost	1616

\[\begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{z \cdot t} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \end{array} \]

Alternative 2
Error	1.1
Cost	1352

\[\begin{array}{l} t_1 := \frac{x}{y} + \frac{2 \cdot z + 2}{t \cdot z}\\ \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 0.001:\\ \;\;\;\;\frac{2}{z \cdot t} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	34.5
Cost	1116

\[\begin{array}{l} \mathbf{if}\;t \leq -216000000000:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;t \leq -9.6 \cdot 10^{-129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 10^{+230}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Error	23.4
Cost	1108

\[\begin{array}{l} t_1 := \frac{0.25}{t} \cdot 8 + -2\\ t_2 := \frac{x}{y} - 2\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+250}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-153}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{2}{t}}{z}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+59}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	23.2
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{t}\\ \mathbf{if}\;z \leq -3100000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{x}{y} - 2\\ \mathbf{elif}\;z \leq 0.0225:\\ \;\;\;\;\frac{\frac{2}{t}}{z}\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{t} \cdot 8 + -2\\ \end{array} \]

Alternative 6
Error	19.9
Cost	848

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	19.6
Cost	848

\[\begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ t_2 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{-15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	19.6
Cost	848

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{2}{t}}{z}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	11.9
Cost	840

\[\begin{array}{l} t_1 := \frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	6.3
Cost	840

\[\begin{array}{l} t_1 := \frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	6.3
Cost	840

\[\begin{array}{l} t_1 := \frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{if}\;z \leq -1.96 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{2}{t}}{z} + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	12.4
Cost	712

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	33.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -3400:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 14
Error	47.0
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2023067 
(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))))