Average Error: 62.0 → 51.6
Time: 4.6s
Precision: binary64
\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]
\[\frac{x - lo}{hi - lo} \]
↓
\[\sqrt{{\left(\frac{lo}{hi}\right)}^{6}} \]
(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))
↓
(FPCore (lo hi x) :precision binary64 (sqrt (pow (/ lo hi) 6.0)))
double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
double code(double lo, double hi, double x) {
	return sqrt(pow((lo / hi), 6.0));
}

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (x - lo) / (hi - lo)
end function
↓
real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = sqrt(((lo / hi) ** 6.0d0))
end function

public static double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}
↓
public static double code(double lo, double hi, double x) {
	return Math.sqrt(Math.pow((lo / hi), 6.0));
}

def code(lo, hi, x):
	return (x - lo) / (hi - lo)
↓
def code(lo, hi, x):
	return math.sqrt(math.pow((lo / hi), 6.0))

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end
↓
function code(lo, hi, x)
	return sqrt((Float64(lo / hi) ^ 6.0))
end

function tmp = code(lo, hi, x)
	tmp = (x - lo) / (hi - lo);
end
↓
function tmp = code(lo, hi, x)
	tmp = sqrt(((lo / hi) ^ 6.0));
end

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]
↓
code[lo_, hi_, x_] := N[Sqrt[N[Power[N[(lo / hi), $MachinePrecision], 6.0], $MachinePrecision]], $MachinePrecision]

\frac{x - lo}{hi - lo}
↓
\sqrt{{\left(\frac{lo}{hi}\right)}^{6}}

Error

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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 in hi around inf 64.0

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

  3. Simplified51.9

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

  4. Applied egg-rr51.9

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

  5. Taylor expanded in lo around -inf 51.6

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

  6. Simplified51.6

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

  7. Applied egg-rr51.6

    \[\leadsto \color{blue}{\sqrt{{\left(\frac{lo}{hi}\right)}^{6}}} \]

  8. Final simplification51.6

    \[\leadsto \sqrt{{\left(\frac{lo}{hi}\right)}^{6}} \]

Reproduce

herbie shell --seed 2022192 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))