xlohi (overflows)

Percentage Accurate: 3.1% → 99.2%

Time: 10.1s

Alternatives: 5

Speedup: 7.0×

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: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around inf

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

5. Simplified 18.9%

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

   1. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{lo}{hi}}\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{hi}, x\right), lo\right)\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{lo} \cdot hi\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{hi}, x\right), lo\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{lo}\right), hi\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{hi}, x\right), lo\right)\right)\right) \]

   4. /-lowering-/.f64 18.9%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, lo\right), hi\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right)\right)\right) \]

7. Applied egg-rr 18.9%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi - x}{lo} \cdot \color{blue}{\left(\frac{1}{lo} \cdot hi + 1\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi - x}{lo} \cdot \left(1 + \color{blue}{\frac{1}{lo} \cdot hi}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi - x}{lo} \cdot \left(1 + hi \cdot \color{blue}{\frac{1}{lo}}\right)\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi - x}{lo} \cdot \left(1 + \frac{hi}{\color{blue}{lo}}\right)\right)\right) \]

   5. flip-+ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi - x}{lo} \cdot \frac{1 \cdot 1 - \frac{hi}{lo} \cdot \frac{hi}{lo}}{\color{blue}{1 - \frac{hi}{lo}}}\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{hi - x}{lo} \cdot \left(1 \cdot 1 - \frac{hi}{lo} \cdot \frac{hi}{lo}\right)}{\color{blue}{1 - \frac{hi}{lo}}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{hi - x}{lo} \cdot \left(1 \cdot 1 - \frac{hi}{lo} \cdot \frac{hi}{lo}\right)\right), \color{blue}{\left(1 - \frac{hi}{lo}\right)}\right)\right) \]

9. Applied egg-rr 40.3%

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

10. Taylor expanded in hi around 0

    \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{x}{lo} + \frac{hi}{lo}\right)}, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{hi}{lo} + -1 \cdot \frac{x}{lo}\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]

    2. mul-1-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{hi}{lo} + \left(\mathsf{neg}\left(\frac{x}{lo}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]

    3. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{hi}{lo} - \frac{x}{lo}\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]

    4. div-sub N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{hi - x}{lo}\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(hi - x\right), lo\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]

    6. --lowering--.f64 99.3%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]

12. Simplified 99.3%

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

Alternative 2: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around inf

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

5. Simplified 18.9%

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

6. Final simplification 18.9%

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

Alternative 3: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around inf

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

5. Simplified 18.9%

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

6. Taylor expanded in hi around inf

   \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(hi, lo\right), 1\right), \color{blue}{\left(\frac{hi}{lo}\right)}\right)\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 18.9%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(hi, lo\right), 1\right), \mathsf{/.f64}\left(hi, \color{blue}{lo}\right)\right)\right) \]

8. Simplified 18.9%

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

9. Final simplification 18.9%

   \[\leadsto 1 + \frac{hi}{lo} \cdot \left(1 + \frac{hi}{lo}\right) \]
10. Add Preprocessing

Alternative 4: 18.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around 0

   \[\leadsto \color{blue}{lo \cdot \left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) - \left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right) + \frac{x}{hi}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{x}{hi} + \color{blue}{lo \cdot \left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) - \left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x}{hi}\right), \color{blue}{\left(lo \cdot \left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) - \left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \left(\color{blue}{lo} \cdot \left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) - \left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \mathsf{*.f64}\left(lo, \color{blue}{\left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) - \left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \mathsf{*.f64}\left(lo, \left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)\right)}\right)\right)\right) \]

5. Simplified 18.8%

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

6. Taylor expanded in hi around inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \mathsf{*.f64}\left(lo, \color{blue}{\left(\frac{-1}{hi}\right)}\right)\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 18.8%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \mathsf{*.f64}\left(lo, \mathsf{/.f64}\left(-1, \color{blue}{hi}\right)\right)\right) \]

8. Simplified 18.8%

   \[\leadsto \frac{x}{hi} + lo \cdot \color{blue}{\frac{-1}{hi}} \]
9. Add Preprocessing

Alternative 5: 18.7% accurate, 7.0× 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 2024150 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))