xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 18.3s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[(N[(hi * N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] + x), $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. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 18.9

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

5. Simplified 18.9%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 18.9

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

8. Simplified 18.9%

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

9. Taylor expanded in hi around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

    5. +-commutative N/A

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

    6. mul-1-neg N/A

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

    7. unsub-neg N/A

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

    8. lower--.f64 N/A

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

    9. lower-/.f64 18.9

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

11. Simplified 18.9%

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

Alternative 2: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(hi * N[(hi / lo), $MachinePrecision] + 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. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 18.9

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

5. Simplified 18.9%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 18.9

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

8. Simplified 18.9%

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

9. Taylor expanded in x around -inf

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

11. Simplified 18.9%

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

12. Taylor expanded in x around 0

    \[\leadsto \color{blue}{1 - -1 \cdot \frac{hi + \frac{{hi}^{2}}{lo}}{lo}} \]
13. Step-by-step derivation

    1. cancel-sign-sub-inv N/A

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

    2. metadata-eval N/A

       \[\leadsto 1 + \color{blue}{1} \cdot \frac{hi + \frac{{hi}^{2}}{lo}}{lo} \]

    3. *-lft-identity N/A

       \[\leadsto 1 + \color{blue}{\frac{hi + \frac{{hi}^{2}}{lo}}{lo}} \]

    4. +-commutative N/A

       \[\leadsto \color{blue}{\frac{hi + \frac{{hi}^{2}}{lo}}{lo} + 1} \]

    5. lower-+.f64 N/A

       \[\leadsto \color{blue}{\frac{hi + \frac{{hi}^{2}}{lo}}{lo} + 1} \]

    6. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{hi + \frac{{hi}^{2}}{lo}}{lo}} + 1 \]

    7. +-commutative N/A

       \[\leadsto \frac{\color{blue}{\frac{{hi}^{2}}{lo} + hi}}{lo} + 1 \]

    8. unpow2 N/A

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

    9. associate-*r/ N/A

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

    10. lower-fma.f64 N/A

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

    11. lower-/.f64 18.9

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

14. Simplified 18.9%

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

15. Final simplification 18.9%

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

Alternative 3: 18.8% accurate, 1.2× 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. lower-/.f64 N/A

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

   2. lower--.f64 18.8

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

5. Simplified 18.8%

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

Alternative 4: 18.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[((-lo) / 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. lower-/.f64 N/A

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

   2. lower--.f64 18.8

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 18.8

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

8. Simplified 18.8%

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

Alternative 5: 18.7% accurate, 18.0× 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 2024207 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))