xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 13.5s

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: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(1.0 + N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] * N[(hi / lo), $MachinePrecision]), $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. lower-+.f64 N/A

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

   2. associate-*r/ N/A

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

   3. lower-/.f64 N/A

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

   4. mul-1-neg N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

   8. remove-double-neg N/A

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

   9. sub-neg N/A

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

   10. lower--.f64 N/A

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

   11. +-commutative N/A

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

   12. associate-/l* N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 15.2

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

5. Simplified 15.2%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. div-sub N/A

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

   5. lift-/.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. div-inv N/A

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

   10. associate-/r* N/A

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

   11. frac-2neg N/A

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

   12. metadata-eval N/A

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

   13. lift-neg.f64 N/A

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

   14. div-inv N/A

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

   15. div-inv N/A

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

   16. times-frac N/A

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

   17. metadata-eval N/A

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

   18. lower-fma.f64 N/A

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

7. Applied egg-rr 15.2%

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

8. Taylor expanded in hi around inf

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

10. Simplified 18.9%

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

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 18.9

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

8. Simplified 18.9%

   \[\leadsto \color{blue}{1 + \frac{\mathsf{fma}\left(hi, \frac{hi}{lo}, hi\right)}{lo}} \]
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. Simplified 18.8%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 18.8

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

   \[\leadsto -\frac{lo}{hi} \]
10. 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. Simplified 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)))