xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 16.1s

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: 18.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(N[((-hi) / lo), $MachinePrecision] / lo), $MachinePrecision] - N[(N[(-1.0 - N[(N[(hi * N[(hi / lo), $MachinePrecision] + hi), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * x), $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 19.0

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

5. Applied rewrites 19.0%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 19.0%

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

9. Taylor expanded in lo around 0

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

11. Applied rewrites 19.0%

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

12. Final simplification 19.0%

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

Alternative 2: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(hi / lo), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] + 1.0), $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 19.0%

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

Alternative 3: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(N[(hi / lo), $MachinePrecision] * N[(N[(hi / lo), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $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 19.0

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

5. Applied rewrites 19.0%

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

7. Applied rewrites 19.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 19.0%

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

11. Final simplification 19.0%

    \[\leadsto \mathsf{fma}\left(\frac{hi}{lo}, \frac{hi}{lo} + 1, 1\right) \]
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 18.7%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))