xlohi (overflows)

Percentage Accurate: 3.1% → 19.3%

Time: 16.1s

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: 19.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 14.9%

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

6. Taylor expanded in lo around inf

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

8. Applied rewrites 19.6%

   \[\leadsto -{\left(\frac{lo}{hi}\right)}^{3} \]
9. Step-by-step derivation

10. Applied rewrites 14.5%

    \[\leadsto -\frac{{\left(\frac{hi}{lo}\right)}^{-2} \cdot lo}{hi} \]
11. Step-by-step derivation

12. Applied rewrites 19.6%

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

13. Final simplification 19.6%

    \[\leadsto \frac{\frac{lo}{hi}}{\frac{hi}{lo}} \cdot \frac{-lo}{hi} \]
14. Add Preprocessing

Alternative 2: 19.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(N[(N[(1.0 - N[(x / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision] * hi), $MachinePrecision] / 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. Taylor expanded in hi around 0

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

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

8. Applied rewrites 18.8%

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

9. Taylor expanded in hi around inf

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

11. Applied rewrites 19.2%

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

Alternative 3: 19.5% 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

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in hi around inf

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

8. Applied rewrites 19.2%

   \[\leadsto hi \cdot \color{blue}{\frac{\frac{hi}{lo}}{lo}} \]
9. 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 18.6%

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

Reproduce

herbie shell --seed 2024307 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))