xlohi (overflows)

Percentage Accurate: 3.1% → 99.5%

Time: 13.9s

Alternatives: 6

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: 99.5% accurate, 0.5× 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 0.7%

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

   1. +-commutative 0.7%

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

   2. associate--l+ 0.7%

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

   3. +-commutative 0.7%

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

   4. *-rgt-identity 0.7%

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

   5. *-commutative 0.7%

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

   6. associate-/l* 10.3%

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

   7. distribute-lft-out 10.5%

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

5. Simplified 10.5%

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

   1. clear-num 10.5%

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

   2. inv-pow 10.5%

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

7. Applied egg-rr 10.5%

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

   1. unpow-1 10.5%

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

   2. associate-/r* 10.5%

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

   3. +-commutative 10.5%

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

9. Simplified 10.5%

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

10. Taylor expanded in hi around -inf 18.8%

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

    1. associate-*r* 18.8%

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

    2. neg-mul-1 18.8%

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

    3. associate-/r* 98.8%

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

12. Simplified 98.8%

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

13. Taylor expanded in hi around 0 99.5%

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

    1. +-commutative 99.5%

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

    2. mul-1-neg 99.5%

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

    3. unsub-neg 99.5%

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

15. Simplified 99.5%

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

16. Final simplification 99.5%

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

Alternative 2: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 / N[(1.0 - N[(hi * N[(N[(1.0 - N[(x / hi), $MachinePrecision]), $MachinePrecision] / 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 0.7%

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

   1. +-commutative 0.7%

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

   2. associate--l+ 0.7%

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

   3. +-commutative 0.7%

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

   4. *-rgt-identity 0.7%

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

   5. *-commutative 0.7%

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

   6. associate-/l* 10.3%

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

   7. distribute-lft-out 10.5%

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

5. Simplified 10.5%

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

   1. clear-num 10.5%

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

   2. inv-pow 10.5%

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

7. Applied egg-rr 10.5%

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

   1. unpow-1 10.5%

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

   2. associate-/r* 10.5%

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

   3. +-commutative 10.5%

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

9. Simplified 10.5%

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

10. Taylor expanded in hi around -inf 18.8%

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

    1. associate-*r* 18.8%

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

    2. neg-mul-1 18.8%

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

    3. associate-/r* 98.8%

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

12. Simplified 98.8%

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

13. Taylor expanded in lo around inf 98.8%

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

    1. mul-1-neg 98.8%

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

    2. unsub-neg 98.8%

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

    3. associate-/l* 98.2%

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

15. Simplified 98.2%

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

Alternative 3: 97.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(-1.0 / N[(hi * N[(N[(1.0 / lo), $MachinePrecision] + N[(-1.0 / hi), $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 0.7%

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

   1. +-commutative 0.7%

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

   2. associate--l+ 0.7%

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

   3. +-commutative 0.7%

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

   4. *-rgt-identity 0.7%

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

   5. *-commutative 0.7%

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

   6. associate-/l* 10.3%

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

   7. distribute-lft-out 10.5%

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

5. Simplified 10.5%

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

   1. clear-num 10.5%

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

   2. inv-pow 10.5%

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

7. Applied egg-rr 10.5%

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

   1. unpow-1 10.5%

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

   2. associate-/r* 10.5%

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

   3. +-commutative 10.5%

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

9. Simplified 10.5%

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

10. Taylor expanded in hi around -inf 18.8%

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

    1. associate-*r* 18.8%

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

    2. neg-mul-1 18.8%

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

    3. associate-/r* 98.8%

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

12. Simplified 98.8%

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

13. Taylor expanded in x around 0 98.1%

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

14. Final simplification 98.1%

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

Alternative 4: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

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

Alternative 5: 18.8% accurate, 1.8× 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(lo / Float64(-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 lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac2 18.8%

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

8. Simplified 18.8%

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

Alternative 6: 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 18.6%

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

Reproduce

herbie shell --seed 2024108 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))