xlohi (overflows)

Percentage Accurate: 3.1% → 19.4%

Time: 6.1s

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: 19.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{-lo}{hi}\right)}^{3} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (pow (/ (- lo) hi) 3.0))

C

double code(double lo, double hi, double x) {
	return pow((-lo / hi), 3.0);
}

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) ** 3.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.pow((-lo / hi), 3.0);
}

Python

def code(lo, hi, x):
	return math.pow((-lo / hi), 3.0)

Julia

function code(lo, hi, x)
	return Float64(Float64(-lo) / hi) ^ 3.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (-lo / hi) ^ 3.0;
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[((-lo) / hi), $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0

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

   1. mul-1-neg N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-neg-in N/A

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

   4. associate-*r/ N/A

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

   5. *-rgt-identity N/A

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

   6. mul-1-neg N/A

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

   7. associate-+l+ N/A

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in lo around -inf

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

8. Applied rewrites 18.8%

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

9. Taylor expanded in hi around inf

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

    1. lower-/.f64 N/A

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

11. Applied rewrites 15.2%

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

12. Taylor expanded in lo around inf

    \[\leadsto -1 \cdot \color{blue}{\frac{{lo}^{3}}{{hi}^{3}}} \]
13. Step-by-step derivation

14. Applied rewrites 19.5%

    \[\leadsto {\left(\frac{-lo}{hi}\right)}^{\color{blue}{3}} \]
15. Add Preprocessing

Alternative 2: 19.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(N[(N[(N[Power[lo, -1.0], $MachinePrecision] - N[(x / N[(lo * lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * hi), $MachinePrecision] / lo), $MachinePrecision] - N[(N[(x / lo), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] * hi + N[((-x) / 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. lower-fma.f64 N/A

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 13.6%

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

9. Taylor expanded in lo around 0

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

11. Applied rewrites 6.1%

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

12. Taylor expanded in hi around inf

    \[\leadsto \mathsf{fma}\left(\frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo} - \frac{\frac{x}{lo}}{lo}, hi, \frac{-x}{lo}\right) \]
13. Step-by-step derivation

14. Applied rewrites 19.2%

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

15. Final simplification 19.2%

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

Alternative 3: 18.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0

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

   1. mul-1-neg N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-neg-in N/A

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

   4. associate-*r/ N/A

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

   5. *-rgt-identity N/A

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

   6. mul-1-neg N/A

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

   7. associate-+l+ N/A

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in lo around -inf

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

8. Applied rewrites 18.8%

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

9. Taylor expanded in lo around inf

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

11. Applied rewrites 18.8%

    \[\leadsto \frac{\frac{x}{hi} - 1}{hi} \cdot \color{blue}{lo} \]
12. 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 18.6%

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

Reproduce

herbie shell --seed 2024344 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))