xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 16.1s

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: 19.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. *-inverses N/A

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

   6. metadata-eval N/A

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

   7. distribute-lft-in N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

5. Applied rewrites 18.9%

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

6. Taylor expanded in hi around inf

   \[\leadsto {hi}^{2} \cdot \color{blue}{\left(\frac{1}{{lo}^{2}} - \frac{x}{{lo}^{3}}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 19.5%

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

10. Applied rewrites 19.5%

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

11. Taylor expanded in lo around inf

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

13. Applied rewrites 19.5%

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

14. Final simplification 19.5%

    \[\leadsto \frac{hi}{lo} \cdot \left(\frac{1 - \frac{x}{lo}}{lo} \cdot hi\right) \]
15. Add Preprocessing

Alternative 2: 19.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. *-inverses N/A

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

   6. metadata-eval N/A

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

   7. distribute-lft-in N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

5. Applied rewrites 18.9%

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

6. Taylor expanded in hi around inf

   \[\leadsto {hi}^{2} \cdot \color{blue}{\left(\frac{1}{{lo}^{2}} - \frac{x}{{lo}^{3}}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 19.5%

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

10. Applied rewrites 19.5%

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

11. Taylor expanded in lo around inf

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

13. Applied rewrites 19.5%

    \[\leadsto \frac{hi}{lo} \cdot \frac{hi}{lo} \]
14. 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 2024255 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))