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

Average Accuracy: 85.1% → 99.9%

Time: 14.8s

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	85.1%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Initial program 85.1%

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

2. Simplified 99.9%

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

   [Start] 85.1	\[ \frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   +-rgt-identity [<=] 85.1	\[ \color{blue}{\left(\frac{x}{y} + 0\right)} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   mul0-lft [<=] 85.1	\[ \left(\frac{x}{y} + \color{blue}{0 \cdot \frac{2}{t \cdot z}}\right) + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   associate-+r+ [<=] 85.1	\[ \color{blue}{\frac{x}{y} + \left(0 \cdot \frac{2}{t \cdot z} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\right)} \]
   mul0-lft [=>] 85.1	\[ \frac{x}{y} + \left(\color{blue}{0} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\right) \]
   +-lft-identity [=>] 85.1	\[ \frac{x}{y} + \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} \]
   sub-neg [=>] 85.1	\[ \frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \color{blue}{\left(1 + \left(-t\right)\right)}}{t \cdot z} \]
   distribute-rgt-in [=>] 85.1	\[ \frac{x}{y} + \frac{2 + \color{blue}{\left(1 \cdot \left(z \cdot 2\right) + \left(-t\right) \cdot \left(z \cdot 2\right)\right)}}{t \cdot z} \]
   associate-+r+ [=>] 85.1	\[ \frac{x}{y} + \frac{\color{blue}{\left(2 + 1 \cdot \left(z \cdot 2\right)\right) + \left(-t\right) \cdot \left(z \cdot 2\right)}}{t \cdot z} \]
   cancel-sign-sub-inv [<=] 85.1	\[ \frac{x}{y} + \frac{\color{blue}{\left(2 + 1 \cdot \left(z \cdot 2\right)\right) - t \cdot \left(z \cdot 2\right)}}{t \cdot z} \]
   div-sub [=>] 85.2	\[ \frac{x}{y} + \color{blue}{\left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} - \frac{t \cdot \left(z \cdot 2\right)}{t \cdot z}\right)} \]
   associate-*r* [=>] 85.2	\[ \frac{x}{y} + \left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} - \frac{\color{blue}{\left(t \cdot z\right) \cdot 2}}{t \cdot z}\right) \]
   associate-*l/ [<=] 85.2	\[ \frac{x}{y} + \left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} - \color{blue}{\frac{t \cdot z}{t \cdot z} \cdot 2}\right) \]
   *-inverses [=>] 99.8	\[ \frac{x}{y} + \left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} - \color{blue}{1} \cdot 2\right) \]
   metadata-eval [=>] 99.8	\[ \frac{x}{y} + \left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} - \color{blue}{2}\right) \]
   sub-neg [=>] 99.8	\[ \frac{x}{y} + \color{blue}{\left(\frac{2 + 1 \cdot \left(z \cdot 2\right)}{t \cdot z} + \left(-2\right)\right)} \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	68.0%
Cost	2400

\[\begin{array}{l} t_1 := -2 + \frac{2}{t}\\ t_2 := -2 + \frac{2}{z \cdot t}\\ \mathbf{if}\;\frac{x}{y} \leq -9 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -380000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq -3.1 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 1.42 \cdot 10^{-257}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq 5.2 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 0.22:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq 2.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 2.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Accuracy	68.0%
Cost	2400

\[\begin{array}{l} t_1 := -2 + \frac{2}{t}\\ t_2 := \frac{\frac{2}{t}}{z}\\ t_3 := -2 + t_2\\ \mathbf{if}\;\frac{x}{y} \leq -9 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -560000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{x}{y} \leq -3.8 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 4 \cdot 10^{-260}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{x}{y} \leq 5.8 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 0.125:\\ \;\;\;\;-2 + \frac{2}{z \cdot t}\\ \mathbf{elif}\;\frac{x}{y} \leq 5.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 2.2 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Accuracy	69.7%
Cost	2268

\[\begin{array}{l} t_1 := -2 + \frac{2}{t}\\ t_2 := -2 + \frac{\frac{2}{t}}{z}\\ t_3 := \frac{2 + \frac{2}{z}}{t}\\ \mathbf{if}\;\frac{x}{y} \leq -6.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -4.6 \cdot 10^{-10}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{x}{y} \leq -9.2 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 9.5 \cdot 10^{-264}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq 4.05 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 2.2 \cdot 10^{-38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq 1.32 \cdot 10^{+48}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Accuracy	89.4%
Cost	1236

\[\begin{array}{l} t_1 := -2 + \frac{\frac{2}{t}}{z}\\ t_2 := \frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ t_3 := \frac{x}{y} + \frac{2}{z \cdot t}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-183}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-36}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	89.4%
Cost	1236

\[\begin{array}{l} t_1 := -2 + \frac{2 + \frac{2}{z}}{t}\\ t_2 := \frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ t_3 := \frac{x}{y} + \frac{2}{z \cdot t}\\ \mathbf{if}\;z \leq -4 \cdot 10^{-36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.9 \cdot 10^{-169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-180}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-39}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Accuracy	67.9%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.12 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -140000000000:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \mathbf{elif}\;\frac{x}{y} \leq 1.85 \cdot 10^{+41}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7
Accuracy	67.9%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.12 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -290000:\\ \;\;\;\;\frac{\frac{2}{t}}{z}\\ \mathbf{elif}\;\frac{x}{y} \leq 1.85 \cdot 10^{+41}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 8
Accuracy	67.9%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.12 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -16600000000:\\ \;\;\;\;\frac{\frac{2}{z}}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 1.85 \cdot 10^{+41}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 9
Accuracy	98.2%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \lor \neg \left(z \leq 9.5 \cdot 10^{-36}\right):\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{z \cdot t}\right)\\ \end{array} \]

Alternative 10
Accuracy	87.2%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-28} \lor \neg \left(z \leq 1.5 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{\frac{2}{t}}{z}\\ \end{array} \]

Alternative 11
Accuracy	68.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -8.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 1.85 \cdot 10^{+41}:\\ \;\;\;\;-2 + \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 12
Accuracy	44.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -3.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 1.6 \cdot 10^{+34}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 13
Accuracy	24.5%
Cost	192

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

Reproduce

herbie shell --seed 2023137 
(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))))