xlohi (overflows)

Percentage Accurate: 3.1% → 99.4%

Time: 19.7s

Alternatives: 7

Speedup: 7.0×

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: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x - lo}{1 - \frac{lo}{hi}}}{hi} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return ((x - lo) / (1.0 - (lo / hi))) / 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) / (1.0d0 - (lo / hi))) / hi
end function

Java

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

Python

def code(lo, hi, x):
	return ((x - lo) / (1.0 - (lo / hi))) / hi

Julia

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

MATLAB

function tmp = code(lo, hi, x)
	tmp = ((x - lo) / (1.0 - (lo / hi))) / hi;
end

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. distribute-lft-out N/A

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

   7. distribute-neg-frac N/A

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

   8. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\frac{-1 \cdot \left(x - lo\right) + -1 \cdot \frac{lo \cdot \left(x - lo\right)}{hi}}{hi}} \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{-1 \cdot \left(x - lo\right) + -1 \cdot \frac{lo \cdot \left(x - lo\right)}{hi}}{hi}\right)}\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(-1 \cdot \left(x - lo\right) + -1 \cdot \frac{lo \cdot \left(x - lo\right)}{hi}\right), \color{blue}{hi}\right)\right) \]

5. Simplified 10.3%

   \[\leadsto \color{blue}{0 - \frac{\left(x - lo\right) \cdot \left(-1 - \frac{lo}{hi}\right)}{hi}} \]
6. Step-by-step derivation

   1. flip-- N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \left(\frac{-1 \cdot -1 - \frac{lo}{hi} \cdot \frac{lo}{hi}}{-1 + \frac{lo}{hi}}\right)\right), hi\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\left(-1 \cdot -1 - \frac{lo}{hi} \cdot \frac{lo}{hi}\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\left(1 - \frac{lo}{hi} \cdot \frac{lo}{hi}\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo}{hi} \cdot \frac{lo}{hi}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo}{hi} \cdot \frac{1}{\frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   6. frac-times N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo \cdot 1}{hi \cdot \frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo \cdot \frac{1}{1}}{hi \cdot \frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   8. div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{lo}{1}}{hi \cdot \frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   9. /-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo}{hi \cdot \frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \left(hi \cdot \frac{hi}{lo}\right)\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo}\right)\right)\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \left(\frac{lo}{hi}\right)\right)\right)\right), hi\right)\right) \]

   14. /-lowering-/.f64 31.2%

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(lo, hi\right)\right)\right)\right), hi\right)\right) \]

7. Applied egg-rr 31.2%

   \[\leadsto 0 - \frac{\left(x - lo\right) \cdot \color{blue}{\frac{1 - \frac{lo}{hi \cdot \frac{hi}{lo}}}{-1 + \frac{lo}{hi}}}}{hi} \]

8. Taylor expanded in lo around 0

   \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(lo, hi\right)\right)\right)\right), hi\right)\right) \]
9. Step-by-step derivation

10. Simplified 99.3%

    \[\leadsto 0 - \frac{\left(x - lo\right) \cdot \frac{\color{blue}{1}}{-1 + \frac{lo}{hi}}}{hi} \]
11. Step-by-step derivation

    1. sub0-neg N/A

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

    2. distribute-neg-frac N/A

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

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(x - lo\right) \cdot \frac{1}{-1 + \frac{lo}{hi}}\right)\right), \color{blue}{hi}\right) \]

    4. un-div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{x - lo}{-1 + \frac{lo}{hi}}\right)\right), hi\right) \]

    5. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{x - lo}{\mathsf{neg}\left(\left(-1 + \frac{lo}{hi}\right)\right)}\right), hi\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - lo\right), \left(\mathsf{neg}\left(\left(-1 + \frac{lo}{hi}\right)\right)\right)\right), hi\right) \]

    7. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \left(\mathsf{neg}\left(\left(-1 + \frac{lo}{hi}\right)\right)\right)\right), hi\right) \]

    8. distribute-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{lo}{hi}\right)\right)\right)\right), hi\right) \]

    9. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \left(1 + \left(\mathsf{neg}\left(\frac{lo}{hi}\right)\right)\right)\right), hi\right) \]

    10. sub-neg N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \left(1 - \frac{lo}{hi}\right)\right), hi\right) \]

    11. --lowering--.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{\_.f64}\left(1, \left(\frac{lo}{hi}\right)\right)\right), hi\right) \]

    12. /-lowering-/.f64 99.4%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, hi\right)\right)\right), hi\right) \]

12. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\frac{\frac{x - lo}{1 - \frac{lo}{hi}}}{hi}} \]
13. Add Preprocessing

Alternative 2: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{lo}{hi}}{\frac{lo}{hi} + -1} \end{array} \]

FPCore

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

C

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

Fortran

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) / ((lo / hi) + (-1.0d0))
end function

Java

public static double code(double lo, double hi, double x) {
	return (lo / hi) / ((lo / hi) + -1.0);
}

Python

def code(lo, hi, x):
	return (lo / hi) / ((lo / hi) + -1.0)

Julia

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

MATLAB

function tmp = code(lo, hi, x)
	tmp = (lo / hi) / ((lo / hi) + -1.0);
end

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. distribute-lft-out N/A

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

   7. distribute-neg-frac N/A

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

   8. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\frac{-1 \cdot \left(x - lo\right) + -1 \cdot \frac{lo \cdot \left(x - lo\right)}{hi}}{hi}} \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{-1 \cdot \left(x - lo\right) + -1 \cdot \frac{lo \cdot \left(x - lo\right)}{hi}}{hi}\right)}\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(-1 \cdot \left(x - lo\right) + -1 \cdot \frac{lo \cdot \left(x - lo\right)}{hi}\right), \color{blue}{hi}\right)\right) \]

5. Simplified 10.3%

   \[\leadsto \color{blue}{0 - \frac{\left(x - lo\right) \cdot \left(-1 - \frac{lo}{hi}\right)}{hi}} \]
6. Step-by-step derivation

   1. flip-- N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \left(\frac{-1 \cdot -1 - \frac{lo}{hi} \cdot \frac{lo}{hi}}{-1 + \frac{lo}{hi}}\right)\right), hi\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\left(-1 \cdot -1 - \frac{lo}{hi} \cdot \frac{lo}{hi}\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\left(1 - \frac{lo}{hi} \cdot \frac{lo}{hi}\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo}{hi} \cdot \frac{lo}{hi}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo}{hi} \cdot \frac{1}{\frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   6. frac-times N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo \cdot 1}{hi \cdot \frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo \cdot \frac{1}{1}}{hi \cdot \frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   8. div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{lo}{1}}{hi \cdot \frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   9. /-rgt-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo}{hi \cdot \frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \left(hi \cdot \frac{hi}{lo}\right)\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo}\right)\right)\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \left(\frac{lo}{hi}\right)\right)\right)\right), hi\right)\right) \]

   14. /-lowering-/.f64 31.2%

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(lo, hi\right)\right)\right)\right), hi\right)\right) \]

7. Applied egg-rr 31.2%

   \[\leadsto 0 - \frac{\left(x - lo\right) \cdot \color{blue}{\frac{1 - \frac{lo}{hi \cdot \frac{hi}{lo}}}{-1 + \frac{lo}{hi}}}}{hi} \]

8. Taylor expanded in lo around 0

   \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(lo, hi\right)\right)\right)\right), hi\right)\right) \]
9. Step-by-step derivation

10. Simplified 99.3%

    \[\leadsto 0 - \frac{\left(x - lo\right) \cdot \frac{\color{blue}{1}}{-1 + \frac{lo}{hi}}}{hi} \]

11. Taylor expanded in x around 0

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

    1. associate-/r* N/A

       \[\leadsto \frac{\frac{lo}{hi}}{\color{blue}{\frac{lo}{hi} - 1}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{lo}{hi}\right), \color{blue}{\left(\frac{lo}{hi} - 1\right)}\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(lo, hi\right), \left(\color{blue}{\frac{lo}{hi}} - 1\right)\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(lo, hi\right), \left(\frac{lo}{hi} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(lo, hi\right), \left(\frac{lo}{hi} + -1\right)\right) \]

    6. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(lo, hi\right), \mathsf{+.f64}\left(\left(\frac{lo}{hi}\right), \color{blue}{-1}\right)\right) \]

    7. /-lowering-/.f64 99.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(lo, hi\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(lo, hi\right), -1\right)\right) \]

13. Simplified 99.1%

    \[\leadsto \color{blue}{\frac{\frac{lo}{hi}}{\frac{lo}{hi} + -1}} \]
14. Add Preprocessing

Alternative 3: 19.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return hi * ((hi / lo) / 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 = hi * ((hi / lo) / lo)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Simplified 18.9%

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

6. Taylor expanded in hi around inf

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
7. Step-by-step derivation

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(hi, \color{blue}{\left(\frac{hi}{{lo}^{2}}\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo \cdot \color{blue}{lo}}\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(hi, \left(\frac{\frac{hi}{lo}}{\color{blue}{lo}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\left(\frac{hi}{lo}\right), \color{blue}{lo}\right)\right) \]

   7. /-lowering-/.f64 19.1%

      \[\leadsto \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\mathsf{/.f64}\left(hi, lo\right), lo\right)\right) \]

8. Simplified 19.1%

   \[\leadsto \color{blue}{hi \cdot \frac{\frac{hi}{lo}}{lo}} \]
9. Add Preprocessing

Alternative 4: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x - lo\right), \color{blue}{hi}\right) \]

   2. --lowering--.f64 18.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, lo\right), hi\right) \]

5. Simplified 18.8%

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]
6. Add Preprocessing

Alternative 5: 18.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{lo}{hi} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (- 0.0 (/ lo hi)))

C

double code(double lo, double hi, double x) {
	return 0.0 - (lo / 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 = 0.0d0 - (lo / hi)
end function

Java

public static double code(double lo, double hi, double x) {
	return 0.0 - (lo / hi);
}

Python

def code(lo, hi, x):
	return 0.0 - (lo / hi)

Julia

function code(lo, hi, x)
	return Float64(0.0 - Float64(lo / hi))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 0.0 - (lo / hi);
end

Wolfram

code[lo_, hi_, x_] := N[(0.0 - N[(lo / hi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x - lo\right), \color{blue}{hi}\right) \]

   2. --lowering--.f64 18.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, lo\right), hi\right) \]

5. Simplified 18.8%

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{lo}{hi}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{-1 \cdot lo}{\color{blue}{hi}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(lo\right)}{hi} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(lo\right)\right), \color{blue}{hi}\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(0 - lo\right), hi\right) \]

   5. --lowering--.f64 18.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, lo\right), hi\right) \]

8. Simplified 18.8%

   \[\leadsto \color{blue}{\frac{0 - lo}{hi}} \]

9. Final simplification 18.8%

   \[\leadsto 0 - \frac{lo}{hi} \]
10. Add Preprocessing

Alternative 6: 18.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{x}{lo} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return 1.0 - (x / 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 = 1.0d0 - (x / lo)
end function

Java

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

Python

def code(lo, hi, x):
	return 1.0 - (x / lo)

Julia

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

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 - (x / lo);
end

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{x - lo}{lo}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{x - lo}{lo}\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\frac{x - lo}{lo}} \]

   3. div-sub N/A

      \[\leadsto 0 - \left(\frac{x}{lo} - \color{blue}{\frac{lo}{lo}}\right) \]

   4. sub-neg N/A

      \[\leadsto 0 - \left(\frac{x}{lo} + \color{blue}{\left(\mathsf{neg}\left(\frac{lo}{lo}\right)\right)}\right) \]

   5. *-inverses N/A

      \[\leadsto 0 - \left(\frac{x}{lo} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto 0 - \left(\frac{x}{lo} + -1\right) \]

   7. +-commutative N/A

      \[\leadsto 0 - \left(-1 + \color{blue}{\frac{x}{lo}}\right) \]

   8. associate--r+ N/A

      \[\leadsto \left(0 - -1\right) - \color{blue}{\frac{x}{lo}} \]

   9. metadata-eval N/A

      \[\leadsto 1 - \frac{\color{blue}{x}}{lo} \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{x}{lo}\right)}\right) \]

   11. /-lowering-/.f64 18.7%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{lo}\right)\right) \]

5. Simplified 18.7%

   \[\leadsto \color{blue}{1 - \frac{x}{lo}} \]
6. Add Preprocessing

Alternative 7: 18.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 1.0)

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0;
}

Python

def code(lo, hi, x):
	return 1.0

Julia

function code(lo, hi, x)
	return 1.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0;
end

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 18.7%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))