xlohi (overflows)

Percentage Accurate: 3.1% → 18.7%

Time: 17.1s

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: 18.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(N[(x - lo), $MachinePrecision] * N[(N[(lo * N[(lo / hi), $MachinePrecision] + lo), $MachinePrecision] / hi), $MachinePrecision] + x), $MachinePrecision] / hi), $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. remove-double-neg N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. associate--r+ N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 15.2%

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

7. Applied rewrites 18.8%

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

Alternative 2: 18.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(lo * N[(N[(lo * N[(lo / hi), $MachinePrecision] + lo), $MachinePrecision] / (-hi)), $MachinePrecision]), $MachinePrecision] / hi), $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. remove-double-neg N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. associate--r+ N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 15.2%

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

7. Applied rewrites 18.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 18.8%

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

11. Final simplification 18.8%

    \[\leadsto \frac{lo \cdot \frac{\mathsf{fma}\left(lo, \frac{lo}{hi}, lo\right)}{-hi}}{hi} - \frac{lo}{hi} \]
12. 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 x around 0

   \[\leadsto \frac{-1 \cdot lo}{hi} \]
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 2024220 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))