xlohi (overflows)

Average Error: 3.1% → 99.2%

Time: 11.2s

Precision: binary64

Cost: 7232.00

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

Math

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

↓

\[\mathsf{fma}\left(\frac{hi - x}{lo}, \frac{1}{1 - \frac{hi}{lo}}, 1\right) \]

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (fma (/ (- hi x) lo) (/ 1.0 (- 1.0 (/ hi lo))) 1.0))

C

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

↓

double code(double lo, double hi, double x) {
	return fma(((hi - x) / lo), (1.0 / (1.0 - (hi / lo))), 1.0);
}

Julia

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

↓

function code(lo, hi, x)
	return fma(Float64(Float64(hi - x) / lo), Float64(1.0 / Float64(1.0 - Float64(hi / lo))), 1.0)
end

Wolfram

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

↓

code[lo_, hi_, x_] := N[(N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 3.1

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

2. Taylor expanded in lo around inf 0.0

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

3. Simplified 18.9

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

   [Start] 0.0	\[ \left(-1 \cdot \frac{x}{lo} + \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right)\right) - -1 \cdot \frac{hi}{lo} \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) + -1 \cdot \frac{x}{lo}\right)} - -1 \cdot \frac{hi}{lo} \]
   associate--l+ [=>] 0.0	\[ \color{blue}{\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) + \left(-1 \cdot \frac{x}{lo} - -1 \cdot \frac{hi}{lo}\right)} \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)} + \left(-1 \cdot \frac{x}{lo} - -1 \cdot \frac{hi}{lo}\right) \]
   associate-*r/ [=>] 0.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \left(\color{blue}{\frac{-1 \cdot x}{lo}} - -1 \cdot \frac{hi}{lo}\right) \]
   associate-*r/ [=>] 0.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \left(\frac{-1 \cdot x}{lo} - \color{blue}{\frac{-1 \cdot hi}{lo}}\right) \]
   div-sub [<=] 0.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \color{blue}{\frac{-1 \cdot x - -1 \cdot hi}{lo}} \]
   distribute-lft-out-- [=>] 0.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \frac{\color{blue}{-1 \cdot \left(x - hi\right)}}{lo} \]
   associate-*r/ [<=] 0.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \color{blue}{-1 \cdot \frac{x - hi}{lo}} \]
   associate-+r+ [<=] 0.0	\[ \color{blue}{1 + \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + -1 \cdot \frac{x - hi}{lo}\right)} \]

4. Applied egg-rr 18.9

   \[\leadsto 1 + \color{blue}{\frac{\frac{hi - x}{lo}}{\frac{1}{1 + \frac{hi}{lo}}}} \]

5. Taylor expanded in hi around 0 99.2

   \[\leadsto 1 + \frac{\frac{hi - x}{lo}}{\color{blue}{1 + -1 \cdot \frac{hi}{lo}}} \]

6. Simplified 99.2

   \[\leadsto 1 + \frac{\frac{hi - x}{lo}}{\color{blue}{1 - \frac{hi}{lo}}} \]
   Proof

   [Start] 99.2	\[ 1 + \frac{\frac{hi - x}{lo}}{1 + -1 \cdot \frac{hi}{lo}} \]
   mul-1-neg [=>] 99.2	\[ 1 + \frac{\frac{hi - x}{lo}}{1 + \color{blue}{\left(-\frac{hi}{lo}\right)}} \]
   sub-neg [<=] 99.2	\[ 1 + \frac{\frac{hi - x}{lo}}{\color{blue}{1 - \frac{hi}{lo}}} \]

7. Applied egg-rr 99.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{hi - x}{lo}, \frac{1}{1 - \frac{hi}{lo}}, 1\right)} \]

8. Final simplification 99.2

   \[\leadsto \mathsf{fma}\left(\frac{hi - x}{lo}, \frac{1}{1 - \frac{hi}{lo}}, 1\right) \]

Alternatives

Alternative 1
Error	99.2%
Cost	832.00

\[1 + \frac{\frac{hi - x}{lo}}{1 - \frac{hi}{lo}} \]

Alternative 2
Error	18.9%
Cost	704.00

\[1 + \frac{hi}{lo} \cdot \left(1 + \frac{hi}{lo}\right) \]

Alternative 3
Error	98.2%
Cost	704.00

\[1 + \frac{\frac{hi}{lo}}{1 - \frac{hi}{lo}} \]

Alternative 4
Error	18.8%
Cost	320.00

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

Alternative 5
Error	18.7%
Cost	64.00

\[1 \]

Reproduce

herbie shell --seed 2023097 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))