Average Error: 62.0 → 52.0
Time: 19.3s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo}\]
↓
\[\frac{\sqrt[3]{\frac{x}{hi} \cdot \frac{x}{hi} + \left(\frac{lo}{hi} \cdot \left(\frac{x}{hi} + -1\right)\right) \cdot \left(\frac{lo}{hi} \cdot \left(1 - \frac{x}{hi}\right)\right)} \cdot \sqrt[3]{\frac{x}{hi} \cdot \frac{x}{hi} + \left(\frac{lo}{hi} \cdot \left(\frac{x}{hi} + -1\right)\right) \cdot \left(\frac{lo}{hi} \cdot \left(1 - \frac{x}{hi}\right)\right)}}{\sqrt[3]{\frac{x}{hi} - \frac{lo}{\frac{hi}{\frac{x}{hi} + -1}}} \cdot \sqrt[3]{\frac{x}{hi} - \frac{lo}{\frac{hi}{\frac{x}{hi} + -1}}}} \cdot \sqrt[3]{\frac{x}{hi} - \frac{lo}{hi} \cdot \left(1 - \frac{x}{hi}\right)}\]
\frac{x - lo}{hi - lo}
↓
\frac{\sqrt[3]{\frac{x}{hi} \cdot \frac{x}{hi} + \left(\frac{lo}{hi} \cdot \left(\frac{x}{hi} + -1\right)\right) \cdot \left(\frac{lo}{hi} \cdot \left(1 - \frac{x}{hi}\right)\right)} \cdot \sqrt[3]{\frac{x}{hi} \cdot \frac{x}{hi} + \left(\frac{lo}{hi} \cdot \left(\frac{x}{hi} + -1\right)\right) \cdot \left(\frac{lo}{hi} \cdot \left(1 - \frac{x}{hi}\right)\right)}}{\sqrt[3]{\frac{x}{hi} - \frac{lo}{\frac{hi}{\frac{x}{hi} + -1}}} \cdot \sqrt[3]{\frac{x}{hi} - \frac{lo}{\frac{hi}{\frac{x}{hi} + -1}}}} \cdot \sqrt[3]{\frac{x}{hi} - \frac{lo}{hi} \cdot \left(1 - \frac{x}{hi}\right)}
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x)
 :precision binary64
 (*
  (/
   (*
    (cbrt
     (+
      (* (/ x hi) (/ x hi))
      (* (* (/ lo hi) (+ (/ x hi) -1.0)) (* (/ lo hi) (- 1.0 (/ x hi))))))
    (cbrt
     (+
      (* (/ x hi) (/ x hi))
      (* (* (/ lo hi) (+ (/ x hi) -1.0)) (* (/ lo hi) (- 1.0 (/ x hi)))))))
   (*
    (cbrt (- (/ x hi) (/ lo (/ hi (+ (/ x hi) -1.0)))))
    (cbrt (- (/ x hi) (/ lo (/ hi (+ (/ x hi) -1.0)))))))
  (cbrt (- (/ x hi) (* (/ lo hi) (- 1.0 (/ x hi)))))))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return ((cbrt(((x / hi) * (x / hi)) + (((lo / hi) * ((x / hi) + -1.0)) * ((lo / hi) * (1.0 - (x / hi))))) * cbrt(((x / hi) * (x / hi)) + (((lo / hi) * ((x / hi) + -1.0)) * ((lo / hi) * (1.0 - (x / hi)))))) / (cbrt((x / hi) - (lo / (hi / ((x / hi) + -1.0)))) * cbrt((x / hi) - (lo / (hi / ((x / hi) + -1.0)))))) * cbrt((x / hi) - ((lo / hi) * (1.0 - (x / hi))));
}

Error

Bits error versus lo

Bits error versus hi

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 62.0

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

  2. Taylor expanded around 0 57.9

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

  3. Simplified52.0

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

  4. Taylor expanded around 0 52.0

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

  5. Simplified52.0

    \[\leadsto \frac{x}{hi} + \color{blue}{\frac{lo}{\frac{hi}{\frac{x}{hi} - 1}}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt_binary64_113652.0

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

  8. Simplified52.0

    \[\leadsto \left(\sqrt[3]{\frac{x}{hi} + \frac{lo}{\frac{hi}{\frac{x}{hi} - 1}}} \cdot \sqrt[3]{\frac{x}{hi} + \frac{lo}{\frac{hi}{\frac{x}{hi} - 1}}}\right) \cdot \color{blue}{\sqrt[3]{\frac{x}{hi} + \frac{lo}{hi} \cdot \left(\frac{x}{hi} - 1\right)}}\]
  9. Using strategy rm

  10. Applied flip-+_binary64_107552.0

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

  11. Applied cbrt-div_binary64_113352.0

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

  12. Applied flip-+_binary64_107552.0

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

  13. Applied cbrt-div_binary64_113352.0

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

  14. Applied frac-times_binary64_111152.0

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

  15. Simplified52.0

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

  16. Final simplification52.0

    \[\leadsto \frac{\sqrt[3]{\frac{x}{hi} \cdot \frac{x}{hi} + \left(\frac{lo}{hi} \cdot \left(\frac{x}{hi} + -1\right)\right) \cdot \left(\frac{lo}{hi} \cdot \left(1 - \frac{x}{hi}\right)\right)} \cdot \sqrt[3]{\frac{x}{hi} \cdot \frac{x}{hi} + \left(\frac{lo}{hi} \cdot \left(\frac{x}{hi} + -1\right)\right) \cdot \left(\frac{lo}{hi} \cdot \left(1 - \frac{x}{hi}\right)\right)}}{\sqrt[3]{\frac{x}{hi} - \frac{lo}{\frac{hi}{\frac{x}{hi} + -1}}} \cdot \sqrt[3]{\frac{x}{hi} - \frac{lo}{\frac{hi}{\frac{x}{hi} + -1}}}} \cdot \sqrt[3]{\frac{x}{hi} - \frac{lo}{hi} \cdot \left(1 - \frac{x}{hi}\right)}\]

Reproduce

herbie shell --seed 2021176 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))