xlohi (overflows)

Percentage Accurate: 3.1% → 18.8%

Time: 8.5s

Alternatives: 3

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: 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 2: 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 lo around inf

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

   1. associate-/l/ N/A

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

   2. div-sub N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

   9. /-lowering-/.f64 18.8%

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

8. Simplified 18.8%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(lo, \color{blue}{\left(\frac{-1}{hi}\right)}\right) \]
10. Step-by-step derivation

    1. /-lowering-/.f64 18.8%

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

11. Simplified 18.8%

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

    1. frac-2neg N/A

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

    2. metadata-eval N/A

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

    3. un-div-inv N/A

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

    4. distribute-neg-frac2 N/A

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

    5. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\frac{lo}{hi}\right)\right) \]

    6. /-lowering-/.f64 18.8%

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

13. Applied egg-rr 18.8%

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

14. Final simplification 18.8%

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

Alternative 3: 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 2024159 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))