xlohi (overflows)

Percentage Accurate: 3.1% → 19.2%

Time: 8.1s

Alternatives: 4

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: 19.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\left(\left(x - lo\right) \cdot \frac{\frac{lo}{hi}}{hi}\right)}^{2}} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (sqrt (pow (* (- x lo) (/ (/ lo hi) hi)) 2.0)))

C

double code(double lo, double hi, double x) {
	return sqrt(pow(((x - lo) * ((lo / hi) / hi)), 2.0));
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = sqrt((((x - lo) * ((lo / hi) / hi)) ** 2.0d0))
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.sqrt(Math.pow(((x - lo) * ((lo / hi) / hi)), 2.0));
}

Python

def code(lo, hi, x):
	return math.sqrt(math.pow(((x - lo) * ((lo / hi) / hi)), 2.0))

Julia

function code(lo, hi, x)
	return sqrt((Float64(Float64(x - lo) * Float64(Float64(lo / hi) / hi)) ^ 2.0))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = sqrt((((x - lo) * ((lo / hi) / hi)) ^ 2.0));
end

Wolfram

code[lo_, hi_, x_] := N[Sqrt[N[Power[N[(N[(x - lo), $MachinePrecision] * N[(N[(lo / hi), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision], 2.0], $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* 9.8%

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

   7. distribute-lft-out 10.0%

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

5. Simplified 10.0%

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

   1. clear-num 10.0%

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

   2. associate-/r/ 10.0%

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

   3. *-commutative 10.0%

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

7. Applied egg-rr 10.0%

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

   1. add-sqr-sqrt 9.3%

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

   2. sqrt-unprod 17.9%

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

   3. associate-*l/ 17.9%

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

   4. *-un-lft-identity 17.9%

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

   5. associate-*l/ 17.9%

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

   6. *-un-lft-identity 17.9%

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

   7. frac-times 0.0%

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

9. Applied egg-rr 17.9%

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

    1. +-commutative 17.9%

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

11. Simplified 17.9%

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

12. Taylor expanded in lo around inf 19.4%

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

13. Final simplification 19.4%

    \[\leadsto \sqrt{{\left(\left(x - lo\right) \cdot \frac{\frac{lo}{hi}}{hi}\right)}^{2}} \]
14. Add Preprocessing

Alternative 2: 18.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(lo * N[(N[(x / hi), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / hi), $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* 9.8%

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

   7. distribute-lft-out 10.0%

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

5. Simplified 10.0%

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

6. Taylor expanded in lo around 0 18.8%

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

7. Taylor expanded in lo around inf 18.8%

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

8. Final simplification 18.8%

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

Alternative 3: 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 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* 9.8%

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

   7. distribute-lft-out 10.0%

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

5. Simplified 10.0%

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

6. Taylor expanded in lo around 0 18.8%

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

7. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

9. Simplified 18.8%

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

10. Final simplification 18.8%

    \[\leadsto \frac{lo}{-hi} \]
11. Add Preprocessing

Alternative 4: 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. Final simplification 18.6%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))