xlohi (overflows)

Percentage Accurate: 3.1% → 19.3%

Time: 6.0s

Alternatives: 3

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: 19.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left|\frac{hi}{lo}\right| \end{array} \]

FPCore

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

C

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

Java

public static double code(double lo, double hi, double x) {
	return Math.abs((hi / lo));
}

Python

def code(lo, hi, x):
	return math.fabs((hi / lo))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around inf 9.9%

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

   1. +-commutative 9.9%

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

   2. associate--l+ 9.9%

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

   3. associate-*r/ 9.9%

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

   4. associate-*r/ 9.9%

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

   5. div-sub 9.9%

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

   6. distribute-lft-out-- 9.9%

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

   7. associate-*r/ 9.9%

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

   8. mul-1-neg 9.9%

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

   9. unsub-neg 9.9%

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

4. Simplified 9.9%

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

5. Taylor expanded in x around 0 9.9%

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

   1. add-sqr-sqrt 9.2%

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

   2. sqrt-unprod 17.7%

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

   3. pow2 17.7%

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

7. Applied egg-rr 17.7%

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

   1. unpow2 17.7%

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

   2. rem-sqrt-square 17.7%

      \[\leadsto \color{blue}{\left|\frac{hi}{lo} + 1\right|} \]

   3. +-commutative 17.7%

      \[\leadsto \left|\color{blue}{1 + \frac{hi}{lo}}\right| \]

9. Simplified 17.7%

   \[\leadsto \color{blue}{\left|1 + \frac{hi}{lo}\right|} \]

10. Taylor expanded in hi around inf 19.4%

    \[\leadsto \left|\color{blue}{\frac{hi}{lo}}\right| \]

11. Final simplification 19.4%

    \[\leadsto \left|\frac{hi}{lo}\right| \]

Alternative 2: 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(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. Taylor expanded in hi around inf 18.8%

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

3. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

5. Simplified 18.8%

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

6. Final simplification 18.8%

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

Alternative 3: 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. Taylor expanded in lo around inf 18.7%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023229 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))