xlohi (overflows)

Average Error: 62.0 → 0.4

Time: 13.9s

Precision: binary64

Cost: 47936

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

Math

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

↓

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

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 lo) (pow hi -2.0))) 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 - lo) * pow(hi, -2.0))), 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(Float64(x - lo) * (hi ^ -2.0))) ^ 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[(N[(x - lo), $MachinePrecision] * N[Power[hi, -2.0], $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 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 58.0

   \[\leadsto \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi} + \frac{x - lo}{hi}} \]
   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 [=>] 58.0	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 58.0	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

4. Applied egg-rr 0.5

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

5. Applied egg-rr 0.4

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

6. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.5
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
Error	0.5
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
Error	51.5
Cost	7360

\[\left(2 + \left|1 + \frac{hi}{lo}\right| \cdot \frac{hi - x}{lo}\right) + -1 \]

Alternative 4
Error	51.5
Cost	448

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

Alternative 5
Error	52.0
Cost	320

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

Alternative 6
Error	52.0
Cost	256

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

Alternative 7
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023083 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))