xlohi (overflows)

Average Error: 62.0 → 1.0

Time: 18.8s

Precision: binary64

Cost: 15040

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x - hi}{lo}\\ \frac{1 - \mathsf{fma}\left(t_0, \frac{hi}{lo}, t_0\right) \cdot \frac{x}{lo}}{t_0 + \mathsf{fma}\left(hi, \frac{x - hi}{lo \cdot lo}, 1\right)} \end{array} \]

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x hi) lo)))
   (/
    (- 1.0 (* (fma t_0 (/ hi lo) t_0) (/ x lo)))
    (+ t_0 (fma hi (/ (- x hi) (* lo 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) {
	double t_0 = (x - hi) / lo;
	return (1.0 - (fma(t_0, (hi / lo), t_0) * (x / lo))) / (t_0 + fma(hi, ((x - hi) / (lo * lo)), 1.0));
}

Julia

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

↓

function code(lo, hi, x)
	t_0 = Float64(Float64(x - hi) / lo)
	return Float64(Float64(1.0 - Float64(fma(t_0, Float64(hi / lo), t_0) * Float64(x / lo))) / Float64(t_0 + fma(hi, Float64(Float64(x - hi) / Float64(lo * lo)), 1.0)))
end

Wolfram

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

↓

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]}, N[(N[(1.0 - N[(N[(t$95$0 * N[(hi / lo), $MachinePrecision] + t$95$0), $MachinePrecision] * N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(hi * N[(N[(x - hi), $MachinePrecision] / N[(lo * lo), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $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. Applied egg-rr 51.9

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

5. Simplified 51.9

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

   [Start] 51.9	\[ \left(1 - \frac{-1}{\frac{lo}{hi} \cdot \left(-\frac{lo}{x - hi}\right)}\right) - \frac{x - hi}{lo} \]
   distribute-rgt-neg-out [=>] 51.9	\[ \left(1 - \frac{-1}{\color{blue}{-\frac{lo}{hi} \cdot \frac{lo}{x - hi}}}\right) - \frac{x - hi}{lo} \]
   *-commutative [<=] 51.9	\[ \left(1 - \frac{-1}{-\color{blue}{\frac{lo}{x - hi} \cdot \frac{lo}{hi}}}\right) - \frac{x - hi}{lo} \]
   distribute-rgt-neg-out [<=] 51.9	\[ \left(1 - \frac{-1}{\color{blue}{\frac{lo}{x - hi} \cdot \left(-\frac{lo}{hi}\right)}}\right) - \frac{x - hi}{lo} \]
   associate-/r* [=>] 51.9	\[ \left(1 - \color{blue}{\frac{\frac{-1}{\frac{lo}{x - hi}}}{-\frac{lo}{hi}}}\right) - \frac{x - hi}{lo} \]
   distribute-neg-frac [=>] 51.9	\[ \left(1 - \frac{\frac{-1}{\frac{lo}{x - hi}}}{\color{blue}{\frac{-lo}{hi}}}\right) - \frac{x - hi}{lo} \]

6. Applied egg-rr 51.9

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

7. Simplified 46.8

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

   [Start] 51.9	\[ \frac{1 - \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right) \cdot \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right)}{1 + \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right)} \]
   fma-udef [=>] 51.9	\[ \frac{1 - \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right) \cdot \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right)}{1 + \color{blue}{\left(\frac{x - hi}{lo} \cdot \frac{hi}{lo} + \frac{x - hi}{lo}\right)}} \]
   associate-*r/ [=>] 54.4	\[ \frac{1 - \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right) \cdot \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right)}{1 + \left(\color{blue}{\frac{\frac{x - hi}{lo} \cdot hi}{lo}} + \frac{x - hi}{lo}\right)} \]
   associate-*l/ [<=] 51.9	\[ \frac{1 - \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right) \cdot \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right)}{1 + \left(\color{blue}{\frac{\frac{x - hi}{lo}}{lo} \cdot hi} + \frac{x - hi}{lo}\right)} \]
   *-commutative [<=] 51.9	\[ \frac{1 - \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right) \cdot \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right)}{1 + \left(\color{blue}{hi \cdot \frac{\frac{x - hi}{lo}}{lo}} + \frac{x - hi}{lo}\right)} \]
   associate-+l+ [<=] 51.9	\[ \frac{1 - \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right) \cdot \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right)}{\color{blue}{\left(1 + hi \cdot \frac{\frac{x - hi}{lo}}{lo}\right) + \frac{x - hi}{lo}}} \]
   +-commutative [<=] 51.9	\[ \frac{1 - \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right) \cdot \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right)}{\color{blue}{\frac{x - hi}{lo} + \left(1 + hi \cdot \frac{\frac{x - hi}{lo}}{lo}\right)}} \]
   +-commutative [=>] 51.9	\[ \frac{1 - \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right) \cdot \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right)}{\frac{x - hi}{lo} + \color{blue}{\left(hi \cdot \frac{\frac{x - hi}{lo}}{lo} + 1\right)}} \]
   fma-def [=>] 51.9	\[ \frac{1 - \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right) \cdot \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right)}{\frac{x - hi}{lo} + \color{blue}{\mathsf{fma}\left(hi, \frac{\frac{x - hi}{lo}}{lo}, 1\right)}} \]
   associate-/l/ [=>] 46.8	\[ \frac{1 - \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right) \cdot \mathsf{fma}\left(\frac{x - hi}{lo}, \frac{hi}{lo}, \frac{x - hi}{lo}\right)}{\frac{x - hi}{lo} + \mathsf{fma}\left(hi, \color{blue}{\frac{x - hi}{lo \cdot lo}}, 1\right)} \]

8. Taylor expanded in hi around 0 1.0

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

9. Final simplification 1.0

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

Alternatives

Alternative 1
Error	1.0
Cost	8000

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

Alternative 2
Error	51.5
Cost	448

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

Alternative 3
Error	52.0
Cost	320

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

Alternative 4
Error	52.0
Cost	256

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

Alternative 5
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023038 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))