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

Average Accuracy: 84.9% → 99.9%

Time: 16.5s

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	84.9%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Initial program 84.9%

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

2. Simplified 99.9%

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

   [Start] 84.9	\[ \frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   sub-neg [=>] 84.9	\[ \frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \color{blue}{\left(1 + \left(-t\right)\right)}}{t \cdot z} \]
   distribute-rgt-in [=>] 84.9	\[ \frac{x}{y} + \frac{2 + \color{blue}{\left(1 \cdot \left(z \cdot 2\right) + \left(-t\right) \cdot \left(z \cdot 2\right)\right)}}{t \cdot z} \]
   *-lft-identity [=>] 84.9	\[ \frac{x}{y} + \frac{2 + \left(\color{blue}{z \cdot 2} + \left(-t\right) \cdot \left(z \cdot 2\right)\right)}{t \cdot z} \]
   associate-+r+ [=>] 84.9	\[ \frac{x}{y} + \frac{\color{blue}{\left(2 + z \cdot 2\right) + \left(-t\right) \cdot \left(z \cdot 2\right)}}{t \cdot z} \]
   cancel-sign-sub-inv [<=] 84.9	\[ \frac{x}{y} + \frac{\color{blue}{\left(2 + z \cdot 2\right) - t \cdot \left(z \cdot 2\right)}}{t \cdot z} \]
   div-sub [=>] 84.9	\[ \frac{x}{y} + \color{blue}{\left(\frac{2 + z \cdot 2}{t \cdot z} - \frac{t \cdot \left(z \cdot 2\right)}{t \cdot z}\right)} \]
   associate-*r* [=>] 84.9	\[ \frac{x}{y} + \left(\frac{2 + z \cdot 2}{t \cdot z} - \frac{\color{blue}{\left(t \cdot z\right) \cdot 2}}{t \cdot z}\right) \]
   associate-*l/ [<=] 84.9	\[ \frac{x}{y} + \left(\frac{2 + z \cdot 2}{t \cdot z} - \color{blue}{\frac{t \cdot z}{t \cdot z} \cdot 2}\right) \]
   *-inverses [=>] 99.8	\[ \frac{x}{y} + \left(\frac{2 + z \cdot 2}{t \cdot z} - \color{blue}{1} \cdot 2\right) \]
   metadata-eval [=>] 99.8	\[ \frac{x}{y} + \left(\frac{2 + z \cdot 2}{t \cdot z} - \color{blue}{2}\right) \]
   sub-neg [=>] 99.8	\[ \frac{x}{y} + \color{blue}{\left(\frac{2 + z \cdot 2}{t \cdot z} + \left(-2\right)\right)} \]
   metadata-eval [=>] 99.8	\[ \frac{x}{y} + \left(\frac{2 + z \cdot 2}{t \cdot z} + \color{blue}{-2}\right) \]
   metadata-eval [<=] 99.8	\[ \frac{x}{y} + \left(\frac{2 + z \cdot 2}{t \cdot z} + \color{blue}{2 \cdot -1}\right) \]
   +-commutative [<=] 99.8	\[ \frac{x}{y} + \color{blue}{\left(2 \cdot -1 + \frac{2 + z \cdot 2}{t \cdot z}\right)} \]
   metadata-eval [=>] 99.8	\[ \frac{x}{y} + \left(\color{blue}{-2} + \frac{2 + z \cdot 2}{t \cdot z}\right) \]
   associate-/l/ [<=] 99.8	\[ \frac{x}{y} + \left(-2 + \color{blue}{\frac{\frac{2 + z \cdot 2}{z}}{t}}\right) \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	92.4%
Cost	1485

\[\begin{array}{l} t_1 := \frac{2}{z \cdot t}\\ \mathbf{if}\;\frac{x}{y} \leq -1.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y} + t_1\\ \mathbf{elif}\;\frac{x}{y} \leq -0.00047 \lor \neg \left(\frac{x}{y} \leq 64\right):\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} + \left(t_1 - 2\right)\\ \end{array} \]

Alternative 2
Accuracy	92.4%
Cost	1357

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{z \cdot t}\\ \mathbf{elif}\;\frac{x}{y} \leq -0.00037 \lor \neg \left(\frac{x}{y} \leq 60\right):\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{-2 - \frac{2}{z}}{t}\\ \end{array} \]

Alternative 3
Accuracy	52.9%
Cost	1232

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2.7 \cdot 10^{-289}:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{x}{y} \leq 1.9 \cdot 10^{-198}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{elif}\;\frac{x}{y} \leq 0.000155:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Accuracy	67.0%
Cost	1113

\[\begin{array}{l} t_1 := \frac{2}{z \cdot t}\\ t_2 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -2.25 \cdot 10^{+36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-160}:\\ \;\;\;\;-2 - \frac{-2}{t}\\ \mathbf{elif}\;t \leq 0.0076 \lor \neg \left(t \leq 6 \cdot 10^{+69}\right) \land t \leq 2.9 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	79.3%
Cost	1106

\[\begin{array}{l} \mathbf{if}\;t \leq -3.05 \cdot 10^{+35} \lor \neg \left(t \leq 0.055\right) \land \left(t \leq 6 \cdot 10^{+69} \lor \neg \left(t \leq 2.9 \cdot 10^{+100}\right)\right):\\ \;\;\;\;\frac{x}{y} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} + \frac{2}{z \cdot t}\\ \end{array} \]

Alternative 6
Accuracy	80.8%
Cost	1104

\[\begin{array}{l} t_1 := \frac{2}{z \cdot t}\\ t_2 := \frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{t} + t_1\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{y} - 2\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	97.7%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -2900 \lor \neg \left(z \leq 3.1 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(-2 + \frac{2}{z \cdot t}\right)\\ \end{array} \]

Alternative 8
Accuracy	79.3%
Cost	849

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.0031:\\ \;\;\;\;\frac{2 + \frac{2}{z}}{t}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+69} \lor \neg \left(t \leq 2.9 \cdot 10^{+100}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z \cdot t}\\ \end{array} \]

Alternative 9
Accuracy	90.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2900 \lor \neg \left(z \leq 1.8 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{z \cdot t}\\ \end{array} \]

Alternative 10
Accuracy	68.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -3.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 0.000155:\\ \;\;\;\;-2 - \frac{-2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 11
Accuracy	69.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -4100000 \lor \neg \left(t \leq 10800\right):\\ \;\;\;\;\frac{x}{y} - 2\\ \mathbf{else}:\\ \;\;\;\;-2 - \frac{-2}{t}\\ \end{array} \]

Alternative 12
Accuracy	46.9%
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-14}:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq 3.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 13
Accuracy	25.7%
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2023151 
(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))))