xlohi (overflows)

Percentage Accurate: 3.1% → 98.7%

Time: 12.0s

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: 98.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(1.0 / N[(lo * N[(N[(N[(hi / lo), $MachinePrecision] / N[(x - hi), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(hi - x), $MachinePrecision]), $MachinePrecision]), $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 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 15.5%

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

5. Simplified 15.5%

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

   1. clear-num 15.5%

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

   2. inv-pow 15.5%

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

7. Applied egg-rr 15.5%

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

   1. unpow-1 15.5%

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

   2. associate-+r- 15.5%

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

   3. +-commutative 15.5%

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

   4. fma-define 15.5%

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

9. Simplified 15.5%

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

10. Taylor expanded in lo around -inf 9.8%

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

    1. mul-1-neg 9.8%

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

    2. associate-/r* 98.7%

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

12. Simplified 98.7%

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

13. Final simplification 98.7%

    \[\leadsto 1 + \frac{1}{lo \cdot \left(\frac{\frac{hi}{lo}}{x - hi} + \frac{1}{hi - x}\right)} \]
14. 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 0.0%

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

5. Simplified 19.0%

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

6. Final simplification 19.0%

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

Alternative 3: 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. Final simplification 18.8%

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

Alternative 4: 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 18.8%

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

4. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

   \[\leadsto \frac{lo}{-hi} \]
8. 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 18.7%

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

4. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024082 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))