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

Average Error: 9.2 → 0.1

Time: 1.5min

Precision: binary64

Cost: 832

Math

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

↓

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

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

Target

Original	9.2
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 9.2

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

2. Applied egg-rr 9.1

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

3. Taylor expanded in t around inf 0.1

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

4. Simplified 0.1

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

Alternatives

Alternative 1
Error	24.5
Cost	1112

\[\begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ t_2 := \frac{2}{t} - 2\\ t_3 := \frac{x}{y} + -2\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-145}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.7 \cdot 10^{-217}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+36}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	24.5
Cost	1112

\[\begin{array}{l} t_1 := \frac{2}{t} - 2\\ t_2 := \frac{x}{y} + -2\\ \mathbf{if}\;z \leq -1 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.15 \cdot 10^{-145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{2}{z}}{t}\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+34}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	12.1
Cost	1104

\[\begin{array}{l} t_1 := \frac{x}{y} + \left(\frac{2}{t} + -2\right)\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{2}{z}}{t}\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-217}:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	13.5
Cost	976

\[\begin{array}{l} t_1 := \frac{2 + \frac{2}{z}}{t}\\ t_2 := \frac{x}{y} + -2\\ \mathbf{if}\;t \leq -245000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	0.8
Cost	968

\[\begin{array}{l} t_1 := \frac{x}{y} + \left(\frac{2}{t} + -2\right)\\ \mathbf{if}\;z \leq -115000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t \cdot z} + -2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	20.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2.8:\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{elif}\;\frac{x}{y} \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7
Error	5.9
Cost	840

\[\begin{array}{l} t_1 := \frac{x}{y} + \left(\frac{2}{t} + -2\right)\\ \mathbf{if}\;z \leq -3 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	5.9
Cost	840

\[\begin{array}{l} t_1 := \frac{x}{y} + \left(\frac{2}{t} + -2\right)\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{y} + \frac{\frac{2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	16.5
Cost	716

\[\begin{array}{l} t_1 := \frac{x}{y} + -2\\ \mathbf{if}\;t \leq -0.37:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	34.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -7800:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 7.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 11
Error	20.2
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{y} + -2\\ \mathbf{if}\;t \leq -0.000155:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	48.2
Cost	192

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

Reproduce

herbie shell --seed 2023033 
(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))))