xlohi (overflows)

Percentage Accurate: 3.1% → 99.4%

Time: 12.6s

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 / N[(N[(lo / N[(lo - x), $MachinePrecision]), $MachinePrecision] + N[(hi / N[(x - lo), $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. 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 13.5%

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

7. Applied rewrites 13.5%

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

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 98.9%

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

11. Taylor expanded in hi around 0

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

13. Applied rewrites 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 18.9% accurate, 0.5× 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(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

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{\left(x + \frac{hi \cdot \left(x - hi\right)}{lo}\right) - hi}{lo}} \]
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 19.1

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

5. Applied rewrites 19.1%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 - \frac{-1 \cdot \frac{{hi}^{2}}{lo} - hi}{lo} \]
7. Step-by-step derivation

8. Applied rewrites 19.1%

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

Alternative 3: 18.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 - 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

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. *-inverses N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-in N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

      \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{lo}\right)\right)} \]

   9. unsub-neg N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 18.7

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

5. Applied rewrites 18.7%

   \[\leadsto \color{blue}{1 - \frac{x}{lo}} \]
6. 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 2024238 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))