xlohi (overflows)

Average Accuracy: 3.1% → 21.8%

Time: 16.5s

Precision: binary64

Cost: 33856

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

Math

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

↓

\[\mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \frac{x}{hi} \cdot \frac{lo}{hi}, \mathsf{fma}\left(-0.5, {\left(\frac{lo}{hi}\right)}^{2}, \frac{-lo}{hi}\right)\right)\right)\right) \]

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (expm1
  (+
   (/ x hi)
   (fma
    -0.5
    (* (/ x hi) (/ x hi))
    (fma
     2.0
     (* (/ x hi) (/ lo hi))
     (fma -0.5 (pow (/ lo hi) 2.0) (/ (- lo) hi)))))))

C

double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

↓

double code(double lo, double hi, double x) {
	return expm1(((x / hi) + fma(-0.5, ((x / hi) * (x / hi)), fma(2.0, ((x / hi) * (lo / hi)), fma(-0.5, pow((lo / hi), 2.0), (-lo / hi))))));
}

Julia

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

↓

function code(lo, hi, x)
	return expm1(Float64(Float64(x / hi) + fma(-0.5, Float64(Float64(x / hi) * Float64(x / hi)), fma(2.0, Float64(Float64(x / hi) * Float64(lo / hi)), fma(-0.5, (Float64(lo / hi) ^ 2.0), Float64(Float64(-lo) / hi))))))
end

Wolfram

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

↓

code[lo_, hi_, x_] := N[(Exp[N[(N[(x / hi), $MachinePrecision] + N[(-0.5 * N[(N[(x / hi), $MachinePrecision] * N[(x / hi), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(x / hi), $MachinePrecision] * N[(lo / hi), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Power[N[(lo / hi), $MachinePrecision], 2.0], $MachinePrecision] + N[((-lo) / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around 0 18.8%

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

3. Simplified 18.8%

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

   [Start] 18.8	\[ \frac{x}{hi} + -1 \cdot \left(lo \cdot \left(\frac{1}{hi} + -1 \cdot \frac{x}{{hi}^{2}}\right)\right) \]
   mul-1-neg [=>] 18.8	\[ \frac{x}{hi} + \color{blue}{\left(-lo \cdot \left(\frac{1}{hi} + -1 \cdot \frac{x}{{hi}^{2}}\right)\right)} \]
   unsub-neg [=>] 18.8	\[ \color{blue}{\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} + -1 \cdot \frac{x}{{hi}^{2}}\right)} \]
   mul-1-neg [=>] 18.8	\[ \frac{x}{hi} - lo \cdot \left(\frac{1}{hi} + \color{blue}{\left(-\frac{x}{{hi}^{2}}\right)}\right) \]
   unsub-neg [=>] 18.8	\[ \frac{x}{hi} - lo \cdot \color{blue}{\left(\frac{1}{hi} - \frac{x}{{hi}^{2}}\right)} \]
   unpow2 [=>] 18.8	\[ \frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - \frac{x}{\color{blue}{hi \cdot hi}}\right) \]

4. Applied egg-rr 18.8%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{hi} - lo \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)} \]
   Step-by-step derivation

   [Start] 18.8	\[ \frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right) \]
   expm1-log1p-u [=>] 18.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - \frac{x}{hi \cdot hi}\right)\right)\right)} \]
   associate-/r* [=>] 18.8	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{hi} - lo \cdot \left(\frac{1}{hi} - \color{blue}{\frac{\frac{x}{hi}}{hi}}\right)\right)\right) \]
   sub-div [=>] 18.8	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{hi} - lo \cdot \color{blue}{\frac{1 - \frac{x}{hi}}{hi}}\right)\right) \]

5. Taylor expanded in lo around 0 0.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{-0.5 \cdot \frac{{\left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right)}^{2} \cdot {lo}^{2}}{{\left(\frac{x}{hi} + 1\right)}^{2}} + \left(\log \left(\frac{x}{hi} + 1\right) + lo \cdot \left(\frac{x}{\left(\frac{x}{hi} + 1\right) \cdot {hi}^{2}} - \frac{1}{\left(\frac{x}{hi} + 1\right) \cdot hi}\right)\right)}\right) \]

6. Simplified 21.8%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(\left(1 - \frac{x}{hi}\right) \cdot \frac{lo}{hi}\right)}^{2}}{{\left(1 + \frac{x}{hi}\right)}^{2}}, \mathsf{fma}\left(lo, \frac{x}{\left(hi \cdot hi\right) \cdot \left(1 + \frac{x}{hi}\right)} + \frac{-1}{hi + hi \cdot \frac{x}{hi}}, \mathsf{log1p}\left(\frac{x}{hi}\right)\right)\right)}\right) \]
   Step-by-step derivation

   [Start] 0.0	\[ \mathsf{expm1}\left(-0.5 \cdot \frac{{\left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right)}^{2} \cdot {lo}^{2}}{{\left(\frac{x}{hi} + 1\right)}^{2}} + \left(\log \left(\frac{x}{hi} + 1\right) + lo \cdot \left(\frac{x}{\left(\frac{x}{hi} + 1\right) \cdot {hi}^{2}} - \frac{1}{\left(\frac{x}{hi} + 1\right) \cdot hi}\right)\right)\right) \]
   fma-def [=>] 0.0	\[ \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right)}^{2} \cdot {lo}^{2}}{{\left(\frac{x}{hi} + 1\right)}^{2}}, \log \left(\frac{x}{hi} + 1\right) + lo \cdot \left(\frac{x}{\left(\frac{x}{hi} + 1\right) \cdot {hi}^{2}} - \frac{1}{\left(\frac{x}{hi} + 1\right) \cdot hi}\right)\right)}\right) \]

7. Taylor expanded in hi around inf 0.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\frac{x}{hi} + \left(-0.5 \cdot \frac{{x}^{2}}{{hi}^{2}} + \left(2 \cdot \frac{lo \cdot x}{{hi}^{2}} + \left(-1 \cdot \frac{lo}{hi} + -0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}}\right)\right)\right)}\right) \]

8. Simplified 21.8%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \frac{x}{hi} \cdot \frac{lo}{hi}, \mathsf{fma}\left(-0.5, {\left(\frac{lo}{hi}\right)}^{2}, \frac{-lo}{hi}\right)\right)\right)}\right) \]
   Step-by-step derivation

   [Start] 0.0	\[ \mathsf{expm1}\left(\frac{x}{hi} + \left(-0.5 \cdot \frac{{x}^{2}}{{hi}^{2}} + \left(2 \cdot \frac{lo \cdot x}{{hi}^{2}} + \left(-1 \cdot \frac{lo}{hi} + -0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}}\right)\right)\right)\right) \]
   fma-def [=>] 0.0	\[ \mathsf{expm1}\left(\frac{x}{hi} + \color{blue}{\mathsf{fma}\left(-0.5, \frac{{x}^{2}}{{hi}^{2}}, 2 \cdot \frac{lo \cdot x}{{hi}^{2}} + \left(-1 \cdot \frac{lo}{hi} + -0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}}\right)\right)}\right) \]
   unpow2 [=>] 0.0	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{\color{blue}{x \cdot x}}{{hi}^{2}}, 2 \cdot \frac{lo \cdot x}{{hi}^{2}} + \left(-1 \cdot \frac{lo}{hi} + -0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}}\right)\right)\right) \]
   unpow2 [=>] 0.0	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x \cdot x}{\color{blue}{hi \cdot hi}}, 2 \cdot \frac{lo \cdot x}{{hi}^{2}} + \left(-1 \cdot \frac{lo}{hi} + -0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}}\right)\right)\right) \]
   times-frac [=>] 0.0	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \color{blue}{\frac{x}{hi} \cdot \frac{x}{hi}}, 2 \cdot \frac{lo \cdot x}{{hi}^{2}} + \left(-1 \cdot \frac{lo}{hi} + -0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}}\right)\right)\right) \]
   fma-def [=>] 0.0	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \color{blue}{\mathsf{fma}\left(2, \frac{lo \cdot x}{{hi}^{2}}, -1 \cdot \frac{lo}{hi} + -0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}}\right)}\right)\right) \]
   *-commutative [=>] 0.0	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \frac{\color{blue}{x \cdot lo}}{{hi}^{2}}, -1 \cdot \frac{lo}{hi} + -0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}}\right)\right)\right) \]
   unpow2 [=>] 0.0	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \frac{x \cdot lo}{\color{blue}{hi \cdot hi}}, -1 \cdot \frac{lo}{hi} + -0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}}\right)\right)\right) \]
   times-frac [=>] 0.0	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \color{blue}{\frac{x}{hi} \cdot \frac{lo}{hi}}, -1 \cdot \frac{lo}{hi} + -0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}}\right)\right)\right) \]
   +-commutative [=>] 0.0	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \frac{x}{hi} \cdot \frac{lo}{hi}, \color{blue}{-0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}} + -1 \cdot \frac{lo}{hi}}\right)\right)\right) \]
   fma-def [=>] 0.0	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \frac{x}{hi} \cdot \frac{lo}{hi}, \color{blue}{\mathsf{fma}\left(-0.5, \frac{{lo}^{2}}{{hi}^{2}}, -1 \cdot \frac{lo}{hi}\right)}\right)\right)\right) \]
   unpow2 [=>] 0.0	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \frac{x}{hi} \cdot \frac{lo}{hi}, \mathsf{fma}\left(-0.5, \frac{\color{blue}{lo \cdot lo}}{{hi}^{2}}, -1 \cdot \frac{lo}{hi}\right)\right)\right)\right) \]
   unpow2 [=>] 0.0	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \frac{x}{hi} \cdot \frac{lo}{hi}, \mathsf{fma}\left(-0.5, \frac{lo \cdot lo}{\color{blue}{hi \cdot hi}}, -1 \cdot \frac{lo}{hi}\right)\right)\right)\right) \]
   times-frac [=>] 21.8	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \frac{x}{hi} \cdot \frac{lo}{hi}, \mathsf{fma}\left(-0.5, \color{blue}{\frac{lo}{hi} \cdot \frac{lo}{hi}}, -1 \cdot \frac{lo}{hi}\right)\right)\right)\right) \]
   unpow2 [<=] 21.8	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \frac{x}{hi} \cdot \frac{lo}{hi}, \mathsf{fma}\left(-0.5, \color{blue}{{\left(\frac{lo}{hi}\right)}^{2}}, -1 \cdot \frac{lo}{hi}\right)\right)\right)\right) \]
   associate-*r/ [=>] 21.8	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \frac{x}{hi} \cdot \frac{lo}{hi}, \mathsf{fma}\left(-0.5, {\left(\frac{lo}{hi}\right)}^{2}, \color{blue}{\frac{-1 \cdot lo}{hi}}\right)\right)\right)\right) \]
   mul-1-neg [=>] 21.8	\[ \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \frac{x}{hi} \cdot \frac{lo}{hi}, \mathsf{fma}\left(-0.5, {\left(\frac{lo}{hi}\right)}^{2}, \frac{\color{blue}{-lo}}{hi}\right)\right)\right)\right) \]

9. Final simplification 21.8%

   \[\leadsto \mathsf{expm1}\left(\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{x}{hi} \cdot \frac{x}{hi}, \mathsf{fma}\left(2, \frac{x}{hi} \cdot \frac{lo}{hi}, \mathsf{fma}\left(-0.5, {\left(\frac{lo}{hi}\right)}^{2}, \frac{-lo}{hi}\right)\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy	21.8%
Cost	19776

\[\mathsf{expm1}\left(\mathsf{fma}\left(-0.5, {\left(\frac{lo}{hi}\right)}^{2}, \frac{-lo}{hi}\right)\right) \]

Alternative 2
Accuracy	18.8%
Cost	1216

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

Alternative 3
Accuracy	18.8%
Cost	832

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

Alternative 4
Accuracy	18.8%
Cost	576

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

Alternative 5
Accuracy	18.8%
Cost	256

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

Alternative 6
Accuracy	18.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023164 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))