xlohi (overflows)

Average Error: 62.0 → 0.3

Time: 11.2s

Precision: binary64

Cost: 7616

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

Math

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

↓

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

FPCore

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

↓

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

Julia

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

↓

function code(lo, hi, x)
	t_0 = Float64(-1.0 + Float64(lo / hi))
	return fma(-1.0, Float64(Float64(x / hi) / t_0), Float64(Float64(lo / hi) / t_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[(-1.0 + N[(lo / hi), $MachinePrecision]), $MachinePrecision]}, N[(-1.0 * N[(N[(x / hi), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(lo / hi), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.0

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

2. Taylor expanded in hi around inf 64.0

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

3. Simplified 57.9

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

   [Start] 64.0	\[ \left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi} \]
   +-commutative [=>] 64.0	\[ \color{blue}{\left(\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \frac{x}{hi}\right)} - \frac{lo}{hi} \]
   associate--l+ [=>] 64.0	\[ \color{blue}{\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)} \]
   *-commutative [=>] 64.0	\[ \frac{\color{blue}{\left(x - lo\right) \cdot lo}}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   unpow2 [=>] 64.0	\[ \frac{\left(x - lo\right) \cdot lo}{\color{blue}{hi \cdot hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   times-frac [=>] 57.9	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 57.9	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

4. Applied egg-rr 0.5

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

5. Taylor expanded in hi around inf 64.0

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

6. Simplified 0.5

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

   [Start] 64.0	\[ \frac{-1 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   associate-*r/ [=>] 64.0	\[ \frac{\color{blue}{\frac{-1 \cdot {\left(x - lo\right)}^{2}}{{hi}^{2}}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   unpow2 [=>] 64.0	\[ \frac{\frac{-1 \cdot {\left(x - lo\right)}^{2}}{\color{blue}{hi \cdot hi}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   associate-*r/ [<=] 64.0	\[ \frac{\color{blue}{-1 \cdot \frac{{\left(x - lo\right)}^{2}}{hi \cdot hi}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   associate-/r* [=>] 64.0	\[ \frac{-1 \cdot \color{blue}{\frac{\frac{{\left(x - lo\right)}^{2}}{hi}}{hi}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   unpow2 [=>] 64.0	\[ \frac{-1 \cdot \frac{\frac{\color{blue}{\left(x - lo\right) \cdot \left(x - lo\right)}}{hi}}{hi}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   associate-*r/ [<=] 17.9	\[ \frac{-1 \cdot \frac{\color{blue}{\left(x - lo\right) \cdot \frac{x - lo}{hi}}}{hi}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   associate-*l/ [<=] 0.5	\[ \frac{-1 \cdot \color{blue}{\left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}\right)}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   unpow2 [<=] 0.5	\[ \frac{-1 \cdot \color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]
   neg-mul-1 [<=] 0.5	\[ \frac{\color{blue}{-{\left(\frac{x - lo}{hi}\right)}^{2}}}{\frac{x - lo}{hi} \cdot \left(\frac{lo}{hi} - 1\right)} \]

7. Taylor expanded in x around 0 62.0

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

8. Simplified 0.3

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

   [Start] 62.0	\[ -1 \cdot \frac{x}{hi \cdot \left(\frac{lo}{hi} - 1\right)} + \frac{lo}{hi \cdot \left(\frac{lo}{hi} - 1\right)} \]
   fma-def [=>] 62.0	\[ \color{blue}{\mathsf{fma}\left(-1, \frac{x}{hi \cdot \left(\frac{lo}{hi} - 1\right)}, \frac{lo}{hi \cdot \left(\frac{lo}{hi} - 1\right)}\right)} \]
   associate-/r* [=>] 61.8	\[ \mathsf{fma}\left(-1, \color{blue}{\frac{\frac{x}{hi}}{\frac{lo}{hi} - 1}}, \frac{lo}{hi \cdot \left(\frac{lo}{hi} - 1\right)}\right) \]
   sub-neg [=>] 61.8	\[ \mathsf{fma}\left(-1, \frac{\frac{x}{hi}}{\color{blue}{\frac{lo}{hi} + \left(-1\right)}}, \frac{lo}{hi \cdot \left(\frac{lo}{hi} - 1\right)}\right) \]
   metadata-eval [=>] 61.8	\[ \mathsf{fma}\left(-1, \frac{\frac{x}{hi}}{\frac{lo}{hi} + \color{blue}{-1}}, \frac{lo}{hi \cdot \left(\frac{lo}{hi} - 1\right)}\right) \]
   associate-/r* [=>] 0.3	\[ \mathsf{fma}\left(-1, \frac{\frac{x}{hi}}{\frac{lo}{hi} + -1}, \color{blue}{\frac{\frac{lo}{hi}}{\frac{lo}{hi} - 1}}\right) \]
   sub-neg [=>] 0.3	\[ \mathsf{fma}\left(-1, \frac{\frac{x}{hi}}{\frac{lo}{hi} + -1}, \frac{\frac{lo}{hi}}{\color{blue}{\frac{lo}{hi} + \left(-1\right)}}\right) \]
   metadata-eval [=>] 0.3	\[ \mathsf{fma}\left(-1, \frac{\frac{x}{hi}}{\frac{lo}{hi} + -1}, \frac{\frac{lo}{hi}}{\frac{lo}{hi} + \color{blue}{-1}}\right) \]

9. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.5
Cost	1472

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

Alternative 2
Error	0.5
Cost	960

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

Alternative 3
Error	1.0
Cost	576

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

Alternative 4
Error	51.5
Cost	448

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

Alternative 5
Error	52.0
Cost	256

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

Alternative 6
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023018 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))