xlohi (overflows)

Average Accuracy: 3.1% → 99.4%

Time: 14.9s

Precision: binary64

Cost: 41472

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

Math

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

↓

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

FPCore

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

↓

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

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

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around inf 0.0%

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

3. Simplified 9.3%

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

   [Start] 0.0	\[ \left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi} \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \frac{x}{hi}\right)} - \frac{lo}{hi} \]
   associate--l+ [=>] 0.0	\[ \color{blue}{\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)} \]
   *-commutative [=>] 0.0	\[ \frac{\color{blue}{\left(x - lo\right) \cdot lo}}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   unpow2 [=>] 0.0	\[ \frac{\left(x - lo\right) \cdot lo}{\color{blue}{hi \cdot hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   times-frac [=>] 9.3	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 9.3	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

4. Applied egg-rr 99.2%

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

   [Start] 9.3	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi} \]
   flip3-+ [=>] 9.3	\[ \color{blue}{\frac{{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \frac{x - lo}{hi}\right)}} \]
   associate-*l/ [=>] 8.8	\[ \frac{{\color{blue}{\left(\frac{\left(x - lo\right) \cdot \frac{lo}{hi}}{hi}\right)}}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \frac{x - lo}{hi}\right)} \]
   *-un-lft-identity [=>] 8.8	\[ \frac{{\left(\frac{\left(x - lo\right) \cdot \frac{lo}{hi}}{\color{blue}{1 \cdot hi}}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \frac{x - lo}{hi}\right)} \]
   times-frac [=>] 9.3	\[ \frac{{\color{blue}{\left(\frac{x - lo}{1} \cdot \frac{\frac{lo}{hi}}{hi}\right)}}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \frac{x - lo}{hi}\right)} \]
   flip-- [=>] 0.0	\[ \frac{{\left(\frac{\color{blue}{\frac{x \cdot x - lo \cdot lo}{x + lo}}}{1} \cdot \frac{\frac{lo}{hi}}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \frac{x - lo}{hi}\right)} \]
   associate-/l/ [=>] 0.0	\[ \frac{{\left(\color{blue}{\frac{x \cdot x - lo \cdot lo}{1 \cdot \left(x + lo\right)}} \cdot \frac{\frac{lo}{hi}}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \frac{x - lo}{hi}\right)} \]
   *-un-lft-identity [<=] 0.0	\[ \frac{{\left(\frac{x \cdot x - lo \cdot lo}{\color{blue}{x + lo}} \cdot \frac{\frac{lo}{hi}}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \frac{x - lo}{hi}\right)} \]
   flip-- [<=] 9.3	\[ \frac{{\left(\color{blue}{\left(x - lo\right)} \cdot \frac{\frac{lo}{hi}}{hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \frac{x - lo}{hi}\right)} \]
   associate-/l/ [=>] 19.8	\[ \frac{{\left(\left(x - lo\right) \cdot \color{blue}{\frac{lo}{hi \cdot hi}}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) + \left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi} - \left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right) \cdot \frac{x - lo}{hi}\right)} \]

5. Applied egg-rr 99.4%

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

   [Start] 99.2	\[ \frac{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{3} + {\left(\frac{x - lo}{hi}\right)}^{3}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   +-commutative [=>] 99.2	\[ \frac{\color{blue}{{\left(\frac{x - lo}{hi}\right)}^{3} + {\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{3}}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   unpow3 [=>] 99.4	\[ \frac{\color{blue}{\left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}\right) \cdot \frac{x - lo}{hi}} + {\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{3}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   fma-def [=>] 99.4	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}, \frac{x - lo}{hi}, {\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{3}\right)}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   pow2 [=>] 99.4	\[ \frac{\mathsf{fma}\left(\color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2}}, \frac{x - lo}{hi}, {\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{3}\right)}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   *-commutative [=>] 99.4	\[ \frac{\mathsf{fma}\left({\left(\frac{x - lo}{hi}\right)}^{2}, \frac{x - lo}{hi}, {\color{blue}{\left(\frac{lo}{hi \cdot hi} \cdot \left(x - lo\right)\right)}}^{3}\right)}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   div-inv [=>] 99.4	\[ \frac{\mathsf{fma}\left({\left(\frac{x - lo}{hi}\right)}^{2}, \frac{x - lo}{hi}, {\left(\color{blue}{\left(lo \cdot \frac{1}{hi \cdot hi}\right)} \cdot \left(x - lo\right)\right)}^{3}\right)}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   associate-*l* [=>] 99.4	\[ \frac{\mathsf{fma}\left({\left(\frac{x - lo}{hi}\right)}^{2}, \frac{x - lo}{hi}, {\color{blue}{\left(lo \cdot \left(\frac{1}{hi \cdot hi} \cdot \left(x - lo\right)\right)\right)}}^{3}\right)}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   pow2 [=>] 99.4	\[ \frac{\mathsf{fma}\left({\left(\frac{x - lo}{hi}\right)}^{2}, \frac{x - lo}{hi}, {\left(lo \cdot \left(\frac{1}{\color{blue}{{hi}^{2}}} \cdot \left(x - lo\right)\right)\right)}^{3}\right)}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   pow-flip [=>] 99.4	\[ \frac{\mathsf{fma}\left({\left(\frac{x - lo}{hi}\right)}^{2}, \frac{x - lo}{hi}, {\left(lo \cdot \left(\color{blue}{{hi}^{\left(-2\right)}} \cdot \left(x - lo\right)\right)\right)}^{3}\right)}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{\mathsf{fma}\left({\left(\frac{x - lo}{hi}\right)}^{2}, \frac{x - lo}{hi}, {\left(lo \cdot \left({hi}^{\color{blue}{-2}} \cdot \left(x - lo\right)\right)\right)}^{3}\right)}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]

6. Taylor expanded in x around inf 99.4%

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

7. Simplified 99.4%

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

   [Start] 99.4	\[ \frac{\mathsf{fma}\left({\left(\frac{x - lo}{hi}\right)}^{2}, \frac{x - lo}{hi}, {\left(lo \cdot \frac{x}{{hi}^{2}}\right)}^{3}\right)}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   unpow2 [=>] 99.4	\[ \frac{\mathsf{fma}\left({\left(\frac{x - lo}{hi}\right)}^{2}, \frac{x - lo}{hi}, {\left(lo \cdot \frac{x}{\color{blue}{hi \cdot hi}}\right)}^{3}\right)}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]

8. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.2%
Cost	27840

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

Alternative 2
Accuracy	99.2%
Cost	14656

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

Alternative 3
Accuracy	19.5%
Cost	576

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

Alternative 4
Accuracy	19.5%
Cost	448

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

Alternative 5
Accuracy	18.8%
Cost	320

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

Alternative 6
Accuracy	18.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023138 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))