xlohi (overflows)

Percentage Accurate: 3.1% → 98.9%

Time: 17.9s

Alternatives: 10

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: 98.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 9.3

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

5. Applied rewrites 9.3%

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

7. Applied rewrites 9.3%

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

8. Taylor expanded in hi around inf

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

10. Applied rewrites 98.8%

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

12. Applied rewrites 99.0%

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

13. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(lo, hi, x):
	return math.pow(((((-lo / (x - lo)) / hi) + math.pow((x - lo), -1.0)) * hi), -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 9.3

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

5. Applied rewrites 9.3%

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

7. Applied rewrites 9.3%

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

8. Taylor expanded in hi around inf

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

10. Applied rewrites 98.8%

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

11. Taylor expanded in hi around inf

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

13. Applied rewrites 98.8%

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

14. Final simplification 98.8%

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

Alternative 3: 19.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(lo, hi, x):
	return math.pow(((((1.0 - (lo / hi)) / hi) + math.pow((x - lo), -1.0)) * hi), -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 9.3

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

5. Applied rewrites 9.3%

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

7. Applied rewrites 9.3%

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

8. Taylor expanded in hi around inf

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

10. Applied rewrites 98.8%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 19.8%

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

14. Final simplification 19.8%

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

Alternative 4: 19.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(lo, hi, x):
	return math.pow((-lo / hi), -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 9.3

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

5. Applied rewrites 9.3%

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

7. Applied rewrites 9.3%

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

8. Taylor expanded in hi around inf

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

10. Applied rewrites 98.8%

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

11. Taylor expanded in lo around inf

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

13. Applied rewrites 19.5%

    \[\leadsto \frac{1}{\frac{-lo}{hi}} \]

14. Final simplification 19.5%

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

Alternative 5: 19.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(N[(hi - x), $MachinePrecision] / 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 19.0

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

5. Applied rewrites 19.0%

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

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. Add Preprocessing

Alternative 6: 19.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(hi - x), $MachinePrecision] / 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 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 19.0

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

5. Applied rewrites 19.0%

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

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. Step-by-step derivation

10. Applied rewrites 19.5%

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

Alternative 7: 19.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(hi * N[(N[(hi / lo), $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. *-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 19.0

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

5. Applied rewrites 19.0%

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

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%

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

Alternative 8: 19.4% 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 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 19.0

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

5. Applied rewrites 19.0%

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

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 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 19.0%

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

12. Taylor expanded in lo around 0

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

14. Applied rewrites 19.5%

    \[\leadsto \frac{hi}{lo} \cdot \frac{hi}{\color{blue}{lo}} \]
15. Add Preprocessing

Alternative 9: 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 10: 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 2024342 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))