xlohi (overflows)

Average Error: 62.0 → 51.5

Time: 10.0s

Precision: binary64

Cost: 448

\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]

Math

\[\frac{x - lo}{hi - lo} \]

↓

\[\frac{hi}{lo} \cdot \frac{hi}{lo} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))

↓

(FPCore (lo hi x) :precision binary64 (* (/ hi lo) (/ hi lo)))

C

double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

↓

double code(double lo, double hi, double x) {
	return (hi / lo) * (hi / lo);
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (x - lo) / (hi - lo)
end function

↓

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (hi / lo) * (hi / lo)
end function

Java

public static double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

↓

public static double code(double lo, double hi, double x) {
	return (hi / lo) * (hi / lo);
}

Python

def code(lo, hi, x):
	return (x - lo) / (hi - lo)

↓

def code(lo, hi, x):
	return (hi / lo) * (hi / lo)

Julia

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end

↓

function code(lo, hi, x)
	return Float64(Float64(hi / lo) * Float64(hi / lo))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x - lo) / (hi - lo);
end

↓

function tmp = code(lo, hi, x)
	tmp = (hi / lo) * (hi / lo);
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]

↓

code[lo_, hi_, x_] := N[(N[(hi / lo), $MachinePrecision] * N[(hi / lo), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around inf 64.0

   \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{lo} + \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right)\right) - -1 \cdot \frac{hi}{lo}} \]

3. Simplified 51.9

   \[\leadsto \color{blue}{\left(1 - \frac{x - hi}{lo} \cdot \frac{hi}{lo}\right) - \frac{x - hi}{lo}} \]
   Proof

   [Start] 64.0	\[ \left(-1 \cdot \frac{x}{lo} + \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right)\right) - -1 \cdot \frac{hi}{lo} \]
   sub-neg [=>] 64.0	\[ \color{blue}{\left(-1 \cdot \frac{x}{lo} + \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right)\right) + \left(--1 \cdot \frac{hi}{lo}\right)} \]
   +-commutative [=>] 64.0	\[ \color{blue}{\left(\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) + -1 \cdot \frac{x}{lo}\right)} + \left(--1 \cdot \frac{hi}{lo}\right) \]
   mul-1-neg [=>] 64.0	\[ \left(\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) + \color{blue}{\left(-\frac{x}{lo}\right)}\right) + \left(--1 \cdot \frac{hi}{lo}\right) \]
   unsub-neg [=>] 64.0	\[ \color{blue}{\left(\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) - \frac{x}{lo}\right)} + \left(--1 \cdot \frac{hi}{lo}\right) \]
   associate-+l- [=>] 64.0	\[ \color{blue}{\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) - \left(\frac{x}{lo} - \left(--1 \cdot \frac{hi}{lo}\right)\right)} \]
   mul-1-neg [=>] 64.0	\[ \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) - \left(\frac{x}{lo} - \left(-\color{blue}{\left(-\frac{hi}{lo}\right)}\right)\right) \]
   remove-double-neg [=>] 64.0	\[ \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) - \left(\frac{x}{lo} - \color{blue}{\frac{hi}{lo}}\right) \]
   div-sub [<=] 64.0	\[ \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) - \color{blue}{\frac{x - hi}{lo}} \]

4. Taylor expanded in lo around 0 64.0

   \[\leadsto \color{blue}{-1 \cdot \frac{\left(x - hi\right) \cdot hi}{{lo}^{2}}} \]

5. Simplified 51.6

   \[\leadsto \color{blue}{\frac{hi}{lo} \cdot \frac{-\left(x - hi\right)}{lo}} \]
   Proof

   [Start] 64.0	\[ -1 \cdot \frac{\left(x - hi\right) \cdot hi}{{lo}^{2}} \]
   *-commutative [=>] 64.0	\[ \color{blue}{\frac{\left(x - hi\right) \cdot hi}{{lo}^{2}} \cdot -1} \]
   *-commutative [<=] 64.0	\[ \frac{\color{blue}{hi \cdot \left(x - hi\right)}}{{lo}^{2}} \cdot -1 \]
   unpow2 [=>] 64.0	\[ \frac{hi \cdot \left(x - hi\right)}{\color{blue}{lo \cdot lo}} \cdot -1 \]
   times-frac [=>] 51.6	\[ \color{blue}{\left(\frac{hi}{lo} \cdot \frac{x - hi}{lo}\right)} \cdot -1 \]
   associate-*l* [=>] 51.6	\[ \color{blue}{\frac{hi}{lo} \cdot \left(\frac{x - hi}{lo} \cdot -1\right)} \]
   *-commutative [<=] 51.6	\[ \frac{hi}{lo} \cdot \color{blue}{\left(-1 \cdot \frac{x - hi}{lo}\right)} \]
   mul-1-neg [=>] 51.6	\[ \frac{hi}{lo} \cdot \color{blue}{\left(-\frac{x - hi}{lo}\right)} \]
   distribute-frac-neg [<=] 51.6	\[ \frac{hi}{lo} \cdot \color{blue}{\frac{-\left(x - hi\right)}{lo}} \]

6. Taylor expanded in hi around inf 64.0

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]

7. Simplified 51.5

   \[\leadsto \color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}} \]
   Proof

   [Start] 64.0	\[ \frac{{hi}^{2}}{{lo}^{2}} \]
   unpow2 [=>] 64.0	\[ \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} \]
   unpow2 [=>] 64.0	\[ \frac{hi \cdot hi}{\color{blue}{lo \cdot lo}} \]
   times-frac [=>] 51.5	\[ \color{blue}{\frac{hi}{lo} \cdot \frac{hi}{lo}} \]

8. Final simplification 51.5

   \[\leadsto \frac{hi}{lo} \cdot \frac{hi}{lo} \]

Alternatives

Alternative 1
Error	52.1
Cost	320

\[\frac{lo - x}{lo} \]

Alternative 2
Error	52.0
Cost	320

\[\frac{x - lo}{hi} \]

Alternative 3
Error	52.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023046 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))