xlohi (overflows)

Percentage Accurate: 3.1% → 19.4%

Time: 24.7s

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: 19.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{hi}{e^{\log \left(-lo\right)}}\right)}^{2} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (pow (/ hi (exp (log (- lo)))) 2.0))

C

double code(double lo, double hi, double x) {
	return pow((hi / exp(log(-lo))), 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 = (hi / exp(log(-lo))) ** 2.0d0
end function

Java

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

Python

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

Julia

function code(lo, hi, x)
	return Float64(hi / exp(log(Float64(-lo)))) ^ 2.0
end

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[Power[N[(hi / N[Exp[N[Log[(-lo)], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $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* 14.7%

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

5. Simplified 14.7%

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

   1. div-inv 14.7%

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

   2. fmm-def 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in hi around inf 0.0%

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 0.0%

      \[\leadsto \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} \]

   2. unpow2 0.0%

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

   3. times-frac 19.5%

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

   4. sqr-neg 19.5%

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

   5. distribute-frac-neg 19.5%

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

   6. distribute-frac-neg 19.5%

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

   7. unpow1 19.5%

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

   8. pow-plus 19.5%

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

   9. distribute-frac-neg 19.5%

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

   10. distribute-neg-frac2 19.5%

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

   11. metadata-eval 19.5%

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

10. Simplified 19.5%

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

    1. add-exp-log 19.5%

       \[\leadsto {\left(\frac{hi}{\color{blue}{e^{\log \left(-lo\right)}}}\right)}^{2} \]

12. Applied egg-rr 19.5%

    \[\leadsto {\left(\frac{hi}{\color{blue}{e^{\log \left(-lo\right)}}}\right)}^{2} \]
13. Add Preprocessing

Alternative 2: 19.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(hi / lo), $MachinePrecision] * N[(hi / lo), $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* 14.7%

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

5. Simplified 14.7%

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

   1. div-inv 14.7%

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

   2. fmm-def 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in hi around inf 0.0%

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 0.0%

      \[\leadsto \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} \]

   2. unpow2 0.0%

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

   3. times-frac 19.5%

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

   4. sqr-neg 19.5%

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

   5. distribute-frac-neg 19.5%

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

   6. distribute-frac-neg 19.5%

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

   7. unpow1 19.5%

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

   8. pow-plus 19.5%

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

   9. distribute-frac-neg 19.5%

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

   10. distribute-neg-frac2 19.5%

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

   11. metadata-eval 19.5%

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

10. Simplified 19.5%

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

    1. unpow2 19.5%

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

12. Applied egg-rr 19.5%

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

13. Final simplification 19.5%

    \[\leadsto \frac{hi}{lo} \cdot \frac{hi}{lo} \]
14. 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. Add Preprocessing

Alternative 4: 18.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.7%

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

   1. associate-*r/ 18.7%

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

   2. neg-mul-1 18.7%

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

   3. sub-neg 18.7%

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

   4. +-commutative 18.7%

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

   5. distribute-neg-in 18.7%

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

   6. remove-double-neg 18.7%

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

   7. sub-neg 18.7%

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

5. Simplified 18.7%

   \[\leadsto \color{blue}{\frac{lo - x}{lo}} \]
6. 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 18.6%

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

Reproduce

herbie shell --seed 2024181 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))