(/ (- x lo) (- hi lo))

Average Error: 62.0 → 51.7

Time: 7.3s

Precision: binary64

Cost: 13376

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

Math

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

↓

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

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (* (/ (/ lo hi) (sqrt hi)) (/ lo (sqrt hi))))

C

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

↓

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

Fortran

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 = ((lo / hi) / sqrt(hi)) * (lo / sqrt(hi))
end function

Java

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 ((lo / hi) / Math.sqrt(hi)) * (lo / Math.sqrt(hi));
}

Python

def code(lo, hi, x):
	return (x - lo) / (hi - lo)

↓

def code(lo, hi, x):
	return ((lo / hi) / math.sqrt(hi)) * (lo / math.sqrt(hi))

Julia

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

↓

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

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x - lo) / (hi - lo);
end

↓

function tmp = code(lo, hi, x)
	tmp = ((lo / hi) / sqrt(hi)) * (lo / sqrt(hi));
end

Wolfram

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

↓

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

   \[\leadsto \color{blue}{\left(\frac{lo}{hi} + 1\right) \cdot \frac{x - lo}{hi}} \]
   Proof
   (*.f64 (+.f64 (/.f64 lo hi) 1) (/.f64 (-.f64 x lo) hi)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 (/.f64 lo hi) (/.f64 (-.f64 x lo) hi)) (/.f64 (-.f64 x lo) hi))): 101 points increase in error, 99 points decrease in error
   (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 lo (-.f64 x lo)) (*.f64 hi hi))) (/.f64 (-.f64 x lo) hi)): 256 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 (*.f64 lo (-.f64 x lo)) (Rewrite<= unpow2_binary64 (pow.f64 hi 2))) (/.f64 (-.f64 x lo) hi)): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 (*.f64 lo (-.f64 x lo)) (pow.f64 hi 2)) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 x hi) (/.f64 lo hi)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (*.f64 lo (-.f64 x lo)) (pow.f64 hi 2)) (/.f64 x hi)) (/.f64 lo hi))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 x hi) (/.f64 (*.f64 lo (-.f64 x lo)) (pow.f64 hi 2)))) (/.f64 lo hi)): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 52.5

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

5. Taylor expanded in lo around inf 64.0

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

6. Simplified 51.7

   \[\leadsto \sqrt{{\color{blue}{\left(\frac{lo}{hi} \cdot \frac{-lo}{hi}\right)}}^{2}} \]
   Proof
   (*.f64 (/.f64 lo hi) (/.f64 (neg.f64 lo) hi)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 lo hi) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 lo hi)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 lo hi) (/.f64 lo hi)))): 0 points increase in error, 0 points decrease in error
   (neg.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 lo lo) (*.f64 hi hi)))): 256 points increase in error, 0 points decrease in error
   (neg.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 lo 2)) (*.f64 hi hi))): 0 points increase in error, 0 points decrease in error
   (neg.f64 (/.f64 (pow.f64 lo 2) (Rewrite<= unpow2_binary64 (pow.f64 hi 2)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 lo 2) (pow.f64 hi 2)))): 0 points increase in error, 0 points decrease in error

7. Applied egg-rr 51.7

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

8. Final simplification 51.7

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

Alternatives

Alternative 1
Error	51.5
Cost	448

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

Alternative 2
Error	51.7
Cost	448

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

Alternative 3
Error	52.0
Cost	256

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

Alternative 4
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022294 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))