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

Average Error: 62.0 → 51.7

Time: 8.5s

Precision: binary64

Cost: 26304

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

Math

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

↓

\[\sqrt{{\left(\frac{lo}{hi}\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{x - lo}{hi}\right)}^{3}} \]

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (* (sqrt (pow (/ lo hi) 2.0)) (cbrt (pow (/ (- x lo) hi) 3.0))))

C

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), 2.0)) * cbrt(pow(((x - lo) / hi), 3.0));
}

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 Math.sqrt(Math.pow((lo / hi), 2.0)) * Math.cbrt(Math.pow(((x - lo) / hi), 3.0));
}

Julia

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

↓

function code(lo, hi, x)
	return Float64(sqrt((Float64(lo / hi) ^ 2.0)) * cbrt((Float64(Float64(x - lo) / hi) ^ 3.0)))
end

Wolfram

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

↓

code[lo_, hi_, x_] := N[(N[Sqrt[N[Power[N[(lo / hi), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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))): 91 points increase in error, 114 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

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

5. Taylor expanded in lo around inf 51.7

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

6. Applied egg-rr 51.7

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

7. Final simplification 51.7

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

Alternatives

Alternative 1
Error	51.7
Cost	13568

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

Alternative 2
Error	51.7
Cost	576

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

Alternative 3
Error	52.0
Cost	320

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

Alternative 4
Error	52.0
Cost	256

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

Alternative 5
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022308 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))