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

Average Error: 9.3 → 0.1

Time: 19.8s

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	9.3
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

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

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	31.0
Cost	1492

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 3.9 \cdot 10^{-243}:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 2.45 \cdot 10^{-14}:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{x}{y} \leq 7500000000:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Error	6.9
Cost	1236

\[\begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{t \cdot z}\\ t_2 := \frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.0152:\\ \;\;\;\;\frac{2 + 2 \cdot \frac{1}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	6.9
Cost	1236

\[\begin{array}{l} t_1 := \frac{x}{y} + \left(-\frac{\frac{-2}{t}}{z}\right)\\ t_2 := \frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.0152:\\ \;\;\;\;\frac{2 + 2 \cdot \frac{1}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	23.1
Cost	1112

\[\begin{array}{l} t_1 := \frac{2}{t} - 2\\ t_2 := \frac{x}{y} - 2\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+274}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	23.1
Cost	1112

\[\begin{array}{l} t_1 := \frac{2}{t} - 2\\ t_2 := \frac{x}{y} - 2\\ \mathbf{if}\;z \leq -3 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.34 \cdot 10^{-70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{2}{t}}{z}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+276}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	0.9
Cost	1096

\[\begin{array}{l} t_1 := \frac{x}{y} + \left(\frac{2}{z \cdot t} - 2\right)\\ \mathbf{if}\;t \leq -17000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{y} + \frac{2 \cdot z + 2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	15.7
Cost	968

\[\begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{t}\\ \mathbf{if}\;\frac{x}{y} \leq -2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 2800000000:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	0.8
Cost	968

\[\begin{array}{l} t_1 := \frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{if}\;z \leq -68000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.39:\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{z \cdot t} - 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	20.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -250:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 85000000000:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 10
Error	19.8
Cost	840

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;\frac{x}{y} \leq -0.00078:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 7500000000:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	11.4
Cost	840

\[\begin{array}{l} t_1 := \frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	6.9
Cost	840

\[\begin{array}{l} t_1 := \frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	33.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -17000000:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 14
Error	47.2
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2023074 
(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))))