xlohi (overflows)

Percentage Accurate: 3.1% → 99.2%

Time: 15.8s

Alternatives: 6

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: 99.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(hi / N[(lo * lo), $MachinePrecision]), $MachinePrecision] * hi + -1.0), $MachinePrecision] * N[(N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] / N[(N[(hi / lo), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $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. Applied rewrites 18.9%

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

7. Applied rewrites 99.3%

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

9. Applied rewrites 99.3%

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

Alternative 2: 99.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[(-1.0 / N[(N[(hi / lo), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] + 1.0), $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. Applied rewrites 18.9%

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in lo around inf

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

10. Applied rewrites 99.3%

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

Alternative 3: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision] * hi + 1.0), $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 x around 0

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

8. Applied rewrites 18.9%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 18.9%

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

12. Final simplification 18.9%

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

Alternative 4: 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 5: 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 6: 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 2024240 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))