xlohi (overflows)

Average Error: 62.0 → 0.5

Time: 11.9s

Precision: binary64

Cost: 48064

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{hi - x}{lo}\\ t_1 := {t_0}^{2}\\ 1 + \frac{\mathsf{fma}\left(t_1, t_0, {\left(\left(hi - x\right) \cdot \left(hi \cdot {lo}^{-2}\right)\right)}^{3}\right)}{{\left(\left(hi - x\right) \cdot \frac{hi}{lo \cdot lo}\right)}^{2} - \left(t_1 \cdot \frac{hi}{lo} - t_1\right)} \end{array} \]

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- hi x) lo)) (t_1 (pow t_0 2.0)))
   (+
    1.0
    (/
     (fma t_1 t_0 (pow (* (- hi x) (* hi (pow lo -2.0))) 3.0))
     (- (pow (* (- hi x) (/ hi (* lo lo))) 2.0) (- (* t_1 (/ hi lo)) t_1))))))

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 = (hi - x) / lo;
	double t_1 = pow(t_0, 2.0);
	return 1.0 + (fma(t_1, t_0, pow(((hi - x) * (hi * pow(lo, -2.0))), 3.0)) / (pow(((hi - x) * (hi / (lo * lo))), 2.0) - ((t_1 * (hi / lo)) - t_1)));
}

Julia

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

↓

function code(lo, hi, x)
	t_0 = Float64(Float64(hi - x) / lo)
	t_1 = t_0 ^ 2.0
	return Float64(1.0 + Float64(fma(t_1, t_0, (Float64(Float64(hi - x) * Float64(hi * (lo ^ -2.0))) ^ 3.0)) / Float64((Float64(Float64(hi - x) * Float64(hi / Float64(lo * lo))) ^ 2.0) - Float64(Float64(t_1 * Float64(hi / lo)) - t_1))))
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[(hi - x), $MachinePrecision] / lo), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, N[(1.0 + N[(N[(t$95$1 * t$95$0 + N[Power[N[(N[(hi - x), $MachinePrecision] * N[(hi * N[Power[lo, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[(hi - x), $MachinePrecision] * N[(hi / N[(lo * lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[(t$95$1 * N[(hi / lo), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $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

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

   [Start] 64.0	\[ \left(-1 \cdot \frac{x}{lo} + \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right)\right) - -1 \cdot \frac{hi}{lo} \]
   +-commutative [=>] 64.0	\[ \color{blue}{\left(\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) + -1 \cdot \frac{x}{lo}\right)} - -1 \cdot \frac{hi}{lo} \]
   associate--l+ [=>] 64.0	\[ \color{blue}{\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) + \left(-1 \cdot \frac{x}{lo} - -1 \cdot \frac{hi}{lo}\right)} \]
   +-commutative [=>] 64.0	\[ \color{blue}{\left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right)} + \left(-1 \cdot \frac{x}{lo} - -1 \cdot \frac{hi}{lo}\right) \]
   associate-*r/ [=>] 64.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \left(\color{blue}{\frac{-1 \cdot x}{lo}} - -1 \cdot \frac{hi}{lo}\right) \]
   associate-*r/ [=>] 64.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \left(\frac{-1 \cdot x}{lo} - \color{blue}{\frac{-1 \cdot hi}{lo}}\right) \]
   div-sub [<=] 64.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \color{blue}{\frac{-1 \cdot x - -1 \cdot hi}{lo}} \]
   distribute-lft-out-- [=>] 64.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \frac{\color{blue}{-1 \cdot \left(x - hi\right)}}{lo} \]
   associate-*r/ [<=] 64.0	\[ \left(1 + \frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}}\right) + \color{blue}{-1 \cdot \frac{x - hi}{lo}} \]
   associate-+r+ [<=] 64.0	\[ \color{blue}{1 + \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + -1 \cdot \frac{x - hi}{lo}\right)} \]

4. Applied egg-rr 51.9

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

5. Applied egg-rr 57.9

   \[\leadsto 1 + \color{blue}{\frac{{\left(\frac{hi - x}{lo}\right)}^{3} + {\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{3}}{{\left(\frac{hi - x}{lo}\right)}^{2} + \left({\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{2} - \frac{hi - x}{lo} \cdot \frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}} \]

6. Simplified 0.6

   \[\leadsto 1 + \color{blue}{\frac{{\left(\left(hi - x\right) \cdot \frac{hi}{lo \cdot lo}\right)}^{3} + {\left(\frac{hi - x}{lo}\right)}^{3}}{{\left(\left(hi - x\right) \cdot \frac{hi}{lo \cdot lo}\right)}^{2} - \left(\frac{hi}{lo} \cdot {\left(\frac{hi - x}{lo}\right)}^{2} - {\left(\frac{hi - x}{lo}\right)}^{2}\right)}} \]
   Proof

   [Start] 57.9	\[ 1 + \frac{{\left(\frac{hi - x}{lo}\right)}^{3} + {\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{3}}{{\left(\frac{hi - x}{lo}\right)}^{2} + \left({\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{2} - \frac{hi - x}{lo} \cdot \frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)} \]
   +-commutative [=>] 57.9	\[ 1 + \frac{\color{blue}{{\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{3} + {\left(\frac{hi - x}{lo}\right)}^{3}}}{{\left(\frac{hi - x}{lo}\right)}^{2} + \left({\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{2} - \frac{hi - x}{lo} \cdot \frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)} \]
   associate-/l* [<=] 64.0	\[ 1 + \frac{{\color{blue}{\left(\frac{\left(hi - x\right) \cdot hi}{lo \cdot lo}\right)}}^{3} + {\left(\frac{hi - x}{lo}\right)}^{3}}{{\left(\frac{hi - x}{lo}\right)}^{2} + \left({\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{2} - \frac{hi - x}{lo} \cdot \frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)} \]
   *-commutative [<=] 64.0	\[ 1 + \frac{{\left(\frac{\color{blue}{hi \cdot \left(hi - x\right)}}{lo \cdot lo}\right)}^{3} + {\left(\frac{hi - x}{lo}\right)}^{3}}{{\left(\frac{hi - x}{lo}\right)}^{2} + \left({\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{2} - \frac{hi - x}{lo} \cdot \frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)} \]
   associate-/r* [=>] 64.0	\[ 1 + \frac{{\color{blue}{\left(\frac{\frac{hi \cdot \left(hi - x\right)}{lo}}{lo}\right)}}^{3} + {\left(\frac{hi - x}{lo}\right)}^{3}}{{\left(\frac{hi - x}{lo}\right)}^{2} + \left({\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{2} - \frac{hi - x}{lo} \cdot \frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)} \]
   associate-/l* [=>] 54.3	\[ 1 + \frac{{\left(\frac{\color{blue}{\frac{hi}{\frac{lo}{hi - x}}}}{lo}\right)}^{3} + {\left(\frac{hi - x}{lo}\right)}^{3}}{{\left(\frac{hi - x}{lo}\right)}^{2} + \left({\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{2} - \frac{hi - x}{lo} \cdot \frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)} \]
   associate-/r* [<=] 52.3	\[ 1 + \frac{{\color{blue}{\left(\frac{hi}{\frac{lo}{hi - x} \cdot lo}\right)}}^{3} + {\left(\frac{hi - x}{lo}\right)}^{3}}{{\left(\frac{hi - x}{lo}\right)}^{2} + \left({\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{2} - \frac{hi - x}{lo} \cdot \frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)} \]
   associate-/l/ [<=] 51.4	\[ 1 + \frac{{\color{blue}{\left(\frac{\frac{hi}{lo}}{\frac{lo}{hi - x}}\right)}}^{3} + {\left(\frac{hi - x}{lo}\right)}^{3}}{{\left(\frac{hi - x}{lo}\right)}^{2} + \left({\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{2} - \frac{hi - x}{lo} \cdot \frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)} \]
   associate-/r/ [=>] 51.4	\[ 1 + \frac{{\color{blue}{\left(\frac{\frac{hi}{lo}}{lo} \cdot \left(hi - x\right)\right)}}^{3} + {\left(\frac{hi - x}{lo}\right)}^{3}}{{\left(\frac{hi - x}{lo}\right)}^{2} + \left({\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{2} - \frac{hi - x}{lo} \cdot \frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)} \]
   *-commutative [=>] 51.4	\[ 1 + \frac{{\color{blue}{\left(\left(hi - x\right) \cdot \frac{\frac{hi}{lo}}{lo}\right)}}^{3} + {\left(\frac{hi - x}{lo}\right)}^{3}}{{\left(\frac{hi - x}{lo}\right)}^{2} + \left({\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{2} - \frac{hi - x}{lo} \cdot \frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)} \]
   associate-/r* [<=] 57.9	\[ 1 + \frac{{\left(\left(hi - x\right) \cdot \color{blue}{\frac{hi}{lo \cdot lo}}\right)}^{3} + {\left(\frac{hi - x}{lo}\right)}^{3}}{{\left(\frac{hi - x}{lo}\right)}^{2} + \left({\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{2} - \frac{hi - x}{lo} \cdot \frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)} \]
   +-commutative [=>] 57.9	\[ 1 + \frac{{\left(\left(hi - x\right) \cdot \frac{hi}{lo \cdot lo}\right)}^{3} + {\left(\frac{hi - x}{lo}\right)}^{3}}{\color{blue}{\left({\left(\frac{hi - x}{\frac{lo \cdot lo}{hi}}\right)}^{2} - \frac{hi - x}{lo} \cdot \frac{hi - x}{\frac{lo \cdot lo}{hi}}\right) + {\left(\frac{hi - x}{lo}\right)}^{2}}} \]

7. Applied egg-rr 0.5

   \[\leadsto 1 + \frac{\color{blue}{\mathsf{fma}\left({\left(\frac{hi - x}{lo}\right)}^{2}, \frac{hi - x}{lo}, {\left(\left(hi - x\right) \cdot \left(hi \cdot {lo}^{-2}\right)\right)}^{3}\right)}}{{\left(\left(hi - x\right) \cdot \frac{hi}{lo \cdot lo}\right)}^{2} - \left(\frac{hi}{lo} \cdot {\left(\frac{hi - x}{lo}\right)}^{2} - {\left(\frac{hi - x}{lo}\right)}^{2}\right)} \]

8. Final simplification 0.5

   \[\leadsto 1 + \frac{\mathsf{fma}\left({\left(\frac{hi - x}{lo}\right)}^{2}, \frac{hi - x}{lo}, {\left(\left(hi - x\right) \cdot \left(hi \cdot {lo}^{-2}\right)\right)}^{3}\right)}{{\left(\left(hi - x\right) \cdot \frac{hi}{lo \cdot lo}\right)}^{2} - \left({\left(\frac{hi - x}{lo}\right)}^{2} \cdot \frac{hi}{lo} - {\left(\frac{hi - x}{lo}\right)}^{2}\right)} \]

Alternatives

Alternative 1
Error	0.6
Cost	27968

\[\begin{array}{l} t_0 := \frac{hi - x}{lo}\\ t_1 := {t_0}^{2}\\ 1 + \frac{{t_0}^{3}}{{\left(\left(hi - x\right) \cdot \frac{hi}{lo \cdot lo}\right)}^{2} - \left(t_1 \cdot \frac{hi}{lo} - t_1\right)} \end{array} \]

Alternative 2
Error	50.4
Cost	20736

\[\begin{array}{l} t_0 := 1 - \frac{x}{hi}\\ \frac{x}{hi} - \mathsf{log1p}\left(\mathsf{fma}\left(t_0, \frac{lo}{hi}, 0.5 \cdot {\left(t_0 \cdot \frac{lo}{hi}\right)}^{2}\right)\right) \end{array} \]

Alternative 3
Error	51.9
Cost	7360

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

Alternative 4
Error	51.9
Cost	832

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

Alternative 5
Error	51.9
Cost	704

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

Alternative 6
Error	52.0
Cost	320

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

Alternative 7
Error	52.0
Cost	256

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

Alternative 8
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023054 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))