xlohi (overflows)

Percentage Accurate: 3.1% → 99.4%

Time: 16.5s

Alternatives: 4

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: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

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 / ((hi / (x - lo)) + (lo / (lo - x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 / N[(N[(hi / N[(x - lo), $MachinePrecision]), $MachinePrecision] + N[(lo / N[(lo - x), $MachinePrecision]), $MachinePrecision]), $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 14.3%

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

7. Applied rewrites 14.3%

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

8. Taylor expanded in hi around -inf

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

10. Applied rewrites 98.8%

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

11. Taylor expanded in hi around 0

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

13. Applied rewrites 99.5%

    \[\leadsto \frac{1}{\frac{hi}{x - lo} - \frac{lo}{\color{blue}{x - lo}}} \]

14. Final simplification 99.5%

    \[\leadsto \frac{1}{\frac{hi}{x - lo} + \frac{lo}{lo - x}} \]
15. Add Preprocessing

Alternative 2: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 18.9%

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

Alternative 3: 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 4: 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 2024219 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))