xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 16.3s

Alternatives: 5

Speedup: 18.0×

Specification

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

Math

\[\begin{array}{l} \\ \frac{x - lo}{hi - lo} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return (x - 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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - lo}{hi - lo} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return (x - 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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 18.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{\mathsf{fma}\left(\frac{x - hi}{lo}, hi, -hi\right)}{lo} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return 1.0 - (fma(((x - hi) / lo), hi, -hi) / lo);
}

Julia

function code(lo, hi, x)
	return Float64(1.0 - Float64(fma(Float64(Float64(x - hi) / lo), hi, Float64(-hi)) / lo))
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[(N[(N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] * hi + (-hi)), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around -inf

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{\left(x + \frac{hi \cdot \left(x - hi\right)}{lo}\right) - hi}{lo}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 18.8

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 - \frac{\mathsf{fma}\left(\frac{x - hi}{lo}, hi, -1 \cdot hi\right)}{lo} \]
7. Step-by-step derivation

8. Applied rewrites 18.8%

   \[\leadsto 1 - \frac{\mathsf{fma}\left(\frac{x - hi}{lo}, hi, -hi\right)}{lo} \]
9. Add Preprocessing

Alternative 2: 18.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{x}{hi} - 1, lo, x\right)}{hi} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ (fma (- (/ x hi) 1.0) lo x) hi))

C

double code(double lo, double hi, double x) {
	return fma(((x / hi) - 1.0), lo, x) / hi;
}

Julia

function code(lo, hi, x)
	return Float64(fma(Float64(Float64(x / hi) - 1.0), lo, x) / hi)
end

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(N[(x / hi), $MachinePrecision] - 1.0), $MachinePrecision] * lo + x), $MachinePrecision] / hi), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around inf

   \[\leadsto \color{blue}{\frac{\left(x + \frac{{lo}^{2} \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \left(lo + -1 \cdot \frac{lo \cdot \left(x - lo\right)}{hi}\right)}{hi}} \]
4. Step-by-step derivation

   1. remove-double-neg N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. associate--r+ N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 14.6%

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

6. Taylor expanded in lo around 0

   \[\leadsto \frac{x + lo \cdot \left(\frac{x}{hi} - 1\right)}{hi} \]
7. Step-by-step derivation

8. Applied rewrites 18.8%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{hi} - 1, lo, x\right)}{hi} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))