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

Average Error: 9.3 → 0.1

Time: 19.4s

Precision: binary64

Cost: 1088

Math

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

↓

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

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

Target

Original	9.3
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 9.3

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

2. Taylor expanded in t around 0 0.1

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	5.6
Cost	1744

\[\begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ t_2 := \frac{x}{y} + 2 \cdot \frac{1 - t}{t}\\ \mathbf{if}\;\frac{x}{y} \leq -1.8 \cdot 10^{+49}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq 0.022:\\ \;\;\;\;\left(t_1 + \frac{2}{t}\right) - 2\\ \mathbf{elif}\;\frac{x}{y} \leq 4.5 \cdot 10^{+141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq 4.6 \cdot 10^{+200}:\\ \;\;\;\;\frac{x}{y} + t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1352

\[\begin{array}{l} t_1 := \frac{x}{y} + 2 \cdot \frac{1 - t}{t}\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	17.0
Cost	1104

\[\begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{t \cdot z}\\ t_2 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -3.05 \cdot 10^{+141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-199}:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	6.0
Cost	1100

\[\begin{array}{l} t_1 := \frac{x}{y} + 2 \cdot \frac{1 - t}{t}\\ \mathbf{if}\;z \leq -0.00041:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{z \cdot t} - 2\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	20.0
Cost	976

\[\begin{array}{l} t_1 := \frac{2}{z \cdot t} - 2\\ t_2 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -41000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	11.5
Cost	972

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -41000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-27}:\\ \;\;\;\;\frac{2 + 2 \cdot \frac{1}{z}}{t}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	19.8
Cost	848

\[\begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ t_2 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -2.25 \cdot 10^{-36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	20.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -8.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 6.1 \cdot 10^{+26}:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 9
Error	19.7
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 14.5:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	34.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 11
Error	48.7
Cost	192

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

Reproduce

herbie shell --seed 2023077 
(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))))