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

Average Error: 62.0 → 51.5

Time: 5.4s

Precision: binary64

Cost: 448

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = (hi / lo) * (hi / lo)
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 (hi / lo) * (hi / lo);
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 62.0

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

2. Taylor expanded in lo around inf 64.0

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

3. Simplified 51.9

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

4. Applied egg-rr 51.9

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

5. Taylor expanded in hi around inf 64.0

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]

6. Simplified 51.5

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

7. Final simplification 51.5

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

Alternatives

Alternative 1
Error	52.0
Cost	256

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

Alternative 2
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022217 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))