xlohi (overflows)

Percentage Accurate: 3.1% → 19.4%

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: 19.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{hi}{lo}}{lo} \cdot \left(hi - x\right) \end{array} \]

FPCore

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

C

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

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 / lo) / lo) * (hi - x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(hi / lo), $MachinePrecision] / lo), $MachinePrecision] * N[(hi - x), $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 18.9

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

5. Applied rewrites 18.9%

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

6. Taylor expanded in lo around 0

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

8. Applied rewrites 19.5%

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

9. Taylor expanded in lo around 0

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

11. Applied rewrites 9.7%

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

12. Taylor expanded in lo around 0

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

14. Applied rewrites 19.5%

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

15. Final simplification 19.5%

    \[\leadsto \frac{\frac{hi}{lo}}{lo} \cdot \left(hi - x\right) \]
16. Add Preprocessing

Alternative 2: 19.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(hi / lo), $MachinePrecision] / lo), $MachinePrecision] * hi), $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 18.9

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

5. Applied rewrites 18.9%

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

6. Taylor expanded in hi around inf

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

8. Applied rewrites 19.5%

   \[\leadsto \frac{\frac{hi}{lo}}{lo} \cdot \color{blue}{hi} \]
9. 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. Applied rewrites 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. Applied rewrites 18.8%

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

6. Taylor expanded in lo around inf

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

8. Applied rewrites 18.8%

   \[\leadsto \frac{-lo}{hi} \]
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. Applied rewrites 18.7%

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

Reproduce

herbie shell --seed 2024288 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))