xlohi (overflows)

Percentage Accurate: 3.1% → 18.8%
Time: 7.6s
Alternatives: 3
Speedup: N/A×

Specification

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

Your Program's Arguments

Results

Enter valid numbers for all inputs

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 3 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Alternative 1: 18.8% accurate, 0.8× speedup?

\[\frac{\frac{lo}{hi} \cdot x - lo}{hi} \]
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. Step-by-step derivation

    1. mul-1-neg18.8%

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

    2. unsub-neg18.8%

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

    3. mul-1-neg18.8%

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

    4. unsub-neg18.8%

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

    5. unpow218.8%

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

  4. Simplified18.8%

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

  5. Taylor expanded in lo around -inf 18.8%

    \[\leadsto \color{blue}{-1 \cdot \left(\left(\frac{1}{hi} - \frac{x}{{hi}^{2}}\right) \cdot lo\right)} \]
  6. Step-by-step derivation

    1. mul-1-neg18.8%

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

    2. *-commutative18.8%

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

    3. unpow-118.8%

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

    4. sub-neg18.8%

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

    5. mul-1-neg18.8%

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

    6. distribute-rgt-in18.8%

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

    7. unpow-118.8%

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

    8. associate-*l/18.8%

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

    9. *-lft-identity18.8%

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

    10. associate-*r/18.8%

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

    11. associate-*l/10.0%

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

    12. associate-*r*10.0%

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

    13. *-commutative10.0%

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

    14. mul-1-neg10.0%

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

    15. distribute-neg-frac10.0%

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

    16. unsub-neg10.0%

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

    17. unpow210.0%

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

    18. times-frac18.8%

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

  7. Simplified18.8%

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

    1. associate-*r/18.8%

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

    2. sub-div18.8%

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

  9. Applied egg-rr18.8%

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

  10. Final simplification18.8%

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

Alternative 2: 18.8% accurate, 1.8× speedup?

\[\frac{-lo}{hi} \]
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. Step-by-step derivation

    1. mul-1-neg18.8%

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

    2. unsub-neg18.8%

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

    3. mul-1-neg18.8%

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

    4. unsub-neg18.8%

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

    5. unpow218.8%

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

  4. Simplified18.8%

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

  5. Taylor expanded in x around 0 18.8%

    \[\leadsto \color{blue}{-1 \cdot \frac{lo}{hi}} \]
  6. Step-by-step derivation

    1. neg-mul-118.8%

      \[\leadsto \color{blue}{-\frac{lo}{hi}} \]

    2. distribute-neg-frac18.8%

      \[\leadsto \color{blue}{\frac{-lo}{hi}} \]

  7. Simplified18.8%

    \[\leadsto \color{blue}{\frac{-lo}{hi}} \]

  8. Final simplification18.8%

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

Alternative 3: 18.7% accurate, 7.0× speedup?

\[1 \]
Derivation

  1. Initial program 3.1%

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

  2. Taylor expanded in lo around inf 18.7%

    \[\leadsto \color{blue}{1} \]

  3. Final simplification18.7%

    \[\leadsto 1 \]

Reproduce

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herbie shell --seed 2023167 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))