xlohi (overflows)

Percentage Accurate: 3.1% → 40.3%

Time: 13.6s

Alternatives: 10

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: 40.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;hi \leq 1.368 \cdot 10^{+308}:\\ \;\;\;\;x \cdot \frac{-1}{lo} + hi \cdot \left(\frac{\frac{hi}{lo}}{lo} \cdot \left(1 - \frac{x}{lo}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - lo\right) + lo \cdot \frac{\frac{\left(x - lo\right) \cdot \left(-1 + \frac{lo}{\frac{hi}{\frac{lo}{hi}}}\right)}{-1 + \frac{lo}{hi}}}{hi}}{hi}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (if (<= hi 1.368e+308)
   (+ (* x (/ -1.0 lo)) (* hi (* (/ (/ hi lo) lo) (- 1.0 (/ x lo)))))
   (/
    (+
     (- x lo)
     (*
      lo
      (/
       (/ (* (- x lo) (+ -1.0 (/ lo (/ hi (/ lo hi))))) (+ -1.0 (/ lo hi)))
       hi)))
    hi)))

C

double code(double lo, double hi, double x) {
	double tmp;
	if (hi <= 1.368e+308) {
		tmp = (x * (-1.0 / lo)) + (hi * (((hi / lo) / lo) * (1.0 - (x / lo))));
	} else {
		tmp = ((x - lo) + (lo * ((((x - lo) * (-1.0 + (lo / (hi / (lo / hi))))) / (-1.0 + (lo / hi))) / hi))) / hi;
	}
	return tmp;
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: tmp
    if (hi <= 1.368d+308) then
        tmp = (x * ((-1.0d0) / lo)) + (hi * (((hi / lo) / lo) * (1.0d0 - (x / lo))))
    else
        tmp = ((x - lo) + (lo * ((((x - lo) * ((-1.0d0) + (lo / (hi / (lo / hi))))) / ((-1.0d0) + (lo / hi))) / hi))) / hi
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double tmp;
	if (hi <= 1.368e+308) {
		tmp = (x * (-1.0 / lo)) + (hi * (((hi / lo) / lo) * (1.0 - (x / lo))));
	} else {
		tmp = ((x - lo) + (lo * ((((x - lo) * (-1.0 + (lo / (hi / (lo / hi))))) / (-1.0 + (lo / hi))) / hi))) / hi;
	}
	return tmp;
}

Python

def code(lo, hi, x):
	tmp = 0
	if hi <= 1.368e+308:
		tmp = (x * (-1.0 / lo)) + (hi * (((hi / lo) / lo) * (1.0 - (x / lo))))
	else:
		tmp = ((x - lo) + (lo * ((((x - lo) * (-1.0 + (lo / (hi / (lo / hi))))) / (-1.0 + (lo / hi))) / hi))) / hi
	return tmp

Julia

function code(lo, hi, x)
	tmp = 0.0
	if (hi <= 1.368e+308)
		tmp = Float64(Float64(x * Float64(-1.0 / lo)) + Float64(hi * Float64(Float64(Float64(hi / lo) / lo) * Float64(1.0 - Float64(x / lo)))));
	else
		tmp = Float64(Float64(Float64(x - lo) + Float64(lo * Float64(Float64(Float64(Float64(x - lo) * Float64(-1.0 + Float64(lo / Float64(hi / Float64(lo / hi))))) / Float64(-1.0 + Float64(lo / hi))) / hi))) / hi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	tmp = 0.0;
	if (hi <= 1.368e+308)
		tmp = (x * (-1.0 / lo)) + (hi * (((hi / lo) / lo) * (1.0 - (x / lo))));
	else
		tmp = ((x - lo) + (lo * ((((x - lo) * (-1.0 + (lo / (hi / (lo / hi))))) / (-1.0 + (lo / hi))) / hi))) / hi;
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := If[LessEqual[hi, 1.368e+308], N[(N[(x * N[(-1.0 / lo), $MachinePrecision]), $MachinePrecision] + N[(hi * N[(N[(N[(hi / lo), $MachinePrecision] / lo), $MachinePrecision] * N[(1.0 - N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - lo), $MachinePrecision] + N[(lo * N[(N[(N[(N[(x - lo), $MachinePrecision] * N[(-1.0 + N[(lo / N[(hi / N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.36800000000000002e308

   1. Initial program 3.1%

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

   3. Taylor expanded in hi around 0

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

   5. Simplified 19.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot \frac{x}{lo}\right)}, \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1 \cdot x}{lo}\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot -1}{lo}\right), \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{-1}{lo}\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{-1}{lo}\right)\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 21.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

   8. Simplified 21.3%

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

   9. Taylor expanded in hi around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \color{blue}{\left(hi \cdot \left(\frac{1}{{lo}^{2}} - \frac{x}{{lo}^{3}}\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out-- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(hi \cdot \frac{1}{{lo}^{2}} - \color{blue}{hi \cdot \frac{x}{{lo}^{3}}}\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot 1}{{lo}^{2}} - \color{blue}{hi} \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot 1}{lo \cdot lo} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       4. times-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{1}{lo} - \color{blue}{hi} \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       5. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{\frac{lo}{lo}}{lo} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{lo}{lo \cdot lo} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{lo}{{lo}^{2}} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       8. times-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{lo \cdot {lo}^{2}} - \color{blue}{hi} \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{lo \cdot \left(lo \cdot lo\right)} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{{lo}^{3}} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       11. associate-/l* N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{{lo}^{3}} - \frac{hi \cdot x}{\color{blue}{{lo}^{3}}}\right)\right)\right) \]

       12. div-sub N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo - hi \cdot x}{\color{blue}{{lo}^{3}}}\right)\right)\right) \]

       13. distribute-lft-out-- N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot \left(lo - x\right)}{{\color{blue}{lo}}^{3}}\right)\right)\right) \]

       14. unpow3 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot \left(lo - x\right)}{\left(lo \cdot lo\right) \cdot \color{blue}{lo}}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot \left(lo - x\right)}{{lo}^{2} \cdot lo}\right)\right)\right) \]

       16. times-frac N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{{lo}^{2}} \cdot \color{blue}{\frac{lo - x}{lo}}\right)\right)\right) \]

       17. div-sub N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{{lo}^{2}} \cdot \left(\frac{lo}{lo} - \color{blue}{\frac{x}{lo}}\right)\right)\right)\right) \]

       18. *-inverses N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{{lo}^{2}} \cdot \left(1 - \frac{\color{blue}{x}}{lo}\right)\right)\right)\right) \]

       19. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \mathsf{*.f64}\left(\left(\frac{hi}{{lo}^{2}}\right), \color{blue}{\left(1 - \frac{x}{lo}\right)}\right)\right)\right) \]

   11. Simplified 21.3%

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

   if 1.36800000000000002e308 < hi

   1. Initial program 3.1%

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

   3. Taylor expanded in hi around inf

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

   5. Simplified 17.5%

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

      1. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{*.f64}\left(lo, \mathsf{/.f64}\left(\left(\left(x - lo\right) \cdot \frac{-1 \cdot -1 - \frac{lo}{hi} \cdot \frac{lo}{hi}}{-1 + \frac{lo}{hi}}\right), hi\right)\right)\right), hi\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{*.f64}\left(lo, \mathsf{/.f64}\left(\left(\frac{\left(x - lo\right) \cdot \left(-1 \cdot -1 - \frac{lo}{hi} \cdot \frac{lo}{hi}\right)}{-1 + \frac{lo}{hi}}\right), hi\right)\right)\right), hi\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{*.f64}\left(lo, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - lo\right) \cdot \left(-1 \cdot -1 - \frac{lo}{hi} \cdot \frac{lo}{hi}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right), hi\right)\right)\right), hi\right) \]

   7. Applied egg-rr 59.6%

      \[\leadsto \frac{\left(x - lo\right) - lo \cdot \frac{\color{blue}{\frac{\left(x - lo\right) \cdot \left(1 - \frac{lo}{\frac{hi}{\frac{lo}{hi}}}\right)}{-1 + \frac{lo}{hi}}}}{hi}}{hi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.368 \cdot 10^{+308}:\\ \;\;\;\;x \cdot \frac{-1}{lo} + hi \cdot \left(\frac{\frac{hi}{lo}}{lo} \cdot \left(1 - \frac{x}{lo}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - lo\right) + lo \cdot \frac{\frac{\left(x - lo\right) \cdot \left(-1 + \frac{lo}{\frac{hi}{\frac{lo}{hi}}}\right)}{-1 + \frac{lo}{hi}}}{hi}}{hi}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 38.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;hi \leq 1.38 \cdot 10^{+308}:\\ \;\;\;\;x \cdot \frac{-1}{lo} + hi \cdot \left(\frac{\frac{hi}{lo}}{lo} \cdot \left(1 - \frac{x}{lo}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x - lo\right) \cdot \left(-1 + \frac{lo}{\frac{hi}{\frac{lo}{hi}}}\right)}{-1 + \frac{lo}{hi}}}{hi}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (if (<= hi 1.38e+308)
   (+ (* x (/ -1.0 lo)) (* hi (* (/ (/ hi lo) lo) (- 1.0 (/ x lo)))))
   (/
    (/ (* (- x lo) (+ -1.0 (/ lo (/ hi (/ lo hi))))) (+ -1.0 (/ lo hi)))
    hi)))

C

double code(double lo, double hi, double x) {
	double tmp;
	if (hi <= 1.38e+308) {
		tmp = (x * (-1.0 / lo)) + (hi * (((hi / lo) / lo) * (1.0 - (x / lo))));
	} else {
		tmp = (((x - lo) * (-1.0 + (lo / (hi / (lo / hi))))) / (-1.0 + (lo / hi))) / hi;
	}
	return tmp;
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: tmp
    if (hi <= 1.38d+308) then
        tmp = (x * ((-1.0d0) / lo)) + (hi * (((hi / lo) / lo) * (1.0d0 - (x / lo))))
    else
        tmp = (((x - lo) * ((-1.0d0) + (lo / (hi / (lo / hi))))) / ((-1.0d0) + (lo / hi))) / hi
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double tmp;
	if (hi <= 1.38e+308) {
		tmp = (x * (-1.0 / lo)) + (hi * (((hi / lo) / lo) * (1.0 - (x / lo))));
	} else {
		tmp = (((x - lo) * (-1.0 + (lo / (hi / (lo / hi))))) / (-1.0 + (lo / hi))) / hi;
	}
	return tmp;
}

Python

def code(lo, hi, x):
	tmp = 0
	if hi <= 1.38e+308:
		tmp = (x * (-1.0 / lo)) + (hi * (((hi / lo) / lo) * (1.0 - (x / lo))))
	else:
		tmp = (((x - lo) * (-1.0 + (lo / (hi / (lo / hi))))) / (-1.0 + (lo / hi))) / hi
	return tmp

Julia

function code(lo, hi, x)
	tmp = 0.0
	if (hi <= 1.38e+308)
		tmp = Float64(Float64(x * Float64(-1.0 / lo)) + Float64(hi * Float64(Float64(Float64(hi / lo) / lo) * Float64(1.0 - Float64(x / lo)))));
	else
		tmp = Float64(Float64(Float64(Float64(x - lo) * Float64(-1.0 + Float64(lo / Float64(hi / Float64(lo / hi))))) / Float64(-1.0 + Float64(lo / hi))) / hi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	tmp = 0.0;
	if (hi <= 1.38e+308)
		tmp = (x * (-1.0 / lo)) + (hi * (((hi / lo) / lo) * (1.0 - (x / lo))));
	else
		tmp = (((x - lo) * (-1.0 + (lo / (hi / (lo / hi))))) / (-1.0 + (lo / hi))) / hi;
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := If[LessEqual[hi, 1.38e+308], N[(N[(x * N[(-1.0 / lo), $MachinePrecision]), $MachinePrecision] + N[(hi * N[(N[(N[(hi / lo), $MachinePrecision] / lo), $MachinePrecision] * N[(1.0 - N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x - lo), $MachinePrecision] * N[(-1.0 + N[(lo / N[(hi / N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.38e308

   1. Initial program 3.1%

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

   3. Taylor expanded in hi around 0

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

   5. Simplified 19.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot \frac{x}{lo}\right)}, \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1 \cdot x}{lo}\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot -1}{lo}\right), \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{-1}{lo}\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{-1}{lo}\right)\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 21.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

   8. Simplified 21.3%

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

   9. Taylor expanded in hi around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \color{blue}{\left(hi \cdot \left(\frac{1}{{lo}^{2}} - \frac{x}{{lo}^{3}}\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out-- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(hi \cdot \frac{1}{{lo}^{2}} - \color{blue}{hi \cdot \frac{x}{{lo}^{3}}}\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot 1}{{lo}^{2}} - \color{blue}{hi} \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot 1}{lo \cdot lo} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       4. times-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{1}{lo} - \color{blue}{hi} \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       5. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{\frac{lo}{lo}}{lo} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{lo}{lo \cdot lo} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{lo}{{lo}^{2}} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       8. times-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{lo \cdot {lo}^{2}} - \color{blue}{hi} \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{lo \cdot \left(lo \cdot lo\right)} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{{lo}^{3}} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       11. associate-/l* N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{{lo}^{3}} - \frac{hi \cdot x}{\color{blue}{{lo}^{3}}}\right)\right)\right) \]

       12. div-sub N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo - hi \cdot x}{\color{blue}{{lo}^{3}}}\right)\right)\right) \]

       13. distribute-lft-out-- N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot \left(lo - x\right)}{{\color{blue}{lo}}^{3}}\right)\right)\right) \]

       14. unpow3 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot \left(lo - x\right)}{\left(lo \cdot lo\right) \cdot \color{blue}{lo}}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot \left(lo - x\right)}{{lo}^{2} \cdot lo}\right)\right)\right) \]

       16. times-frac N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{{lo}^{2}} \cdot \color{blue}{\frac{lo - x}{lo}}\right)\right)\right) \]

       17. div-sub N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{{lo}^{2}} \cdot \left(\frac{lo}{lo} - \color{blue}{\frac{x}{lo}}\right)\right)\right)\right) \]

       18. *-inverses N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{{lo}^{2}} \cdot \left(1 - \frac{\color{blue}{x}}{lo}\right)\right)\right)\right) \]

       19. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \mathsf{*.f64}\left(\left(\frac{hi}{{lo}^{2}}\right), \color{blue}{\left(1 - \frac{x}{lo}\right)}\right)\right)\right) \]

   11. Simplified 21.3%

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

   if 1.38e308 < hi

   1. Initial program 3.1%

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

   3. Taylor expanded in hi around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-out N/A

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

      7. distribute-neg-frac N/A

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

      8. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{-1 \cdot \left(x - lo\right) + -1 \cdot \frac{lo \cdot \left(x - lo\right)}{hi}}{hi}} \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{-1 \cdot \left(x - lo\right) + -1 \cdot \frac{lo \cdot \left(x - lo\right)}{hi}}{hi}\right)}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(-1 \cdot \left(x - lo\right) + -1 \cdot \frac{lo \cdot \left(x - lo\right)}{hi}\right), \color{blue}{hi}\right)\right) \]

   5. Simplified 13.4%

      \[\leadsto \color{blue}{0 - \frac{\left(x - lo\right) \cdot \left(-1 - \frac{lo}{hi}\right)}{hi}} \]
   6. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\left(x - lo\right) \cdot \frac{-1 \cdot -1 - \frac{lo}{hi} \cdot \frac{lo}{hi}}{-1 + \frac{lo}{hi}}\right), hi\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\frac{\left(x - lo\right) \cdot \left(-1 \cdot -1 - \frac{lo}{hi} \cdot \frac{lo}{hi}\right)}{-1 + \frac{lo}{hi}}\right), hi\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(x - lo\right) \cdot \left(-1 \cdot -1 - \frac{lo}{hi} \cdot \frac{lo}{hi}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right), hi\right)\right) \]

   7. Applied egg-rr 57.9%

      \[\leadsto 0 - \frac{\color{blue}{\frac{\left(x - lo\right) \cdot \left(1 - \frac{lo}{\frac{hi}{\frac{lo}{hi}}}\right)}{-1 + \frac{lo}{hi}}}}{hi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.38 \cdot 10^{+308}:\\ \;\;\;\;x \cdot \frac{-1}{lo} + hi \cdot \left(\frac{\frac{hi}{lo}}{lo} \cdot \left(1 - \frac{x}{lo}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x - lo\right) \cdot \left(-1 + \frac{lo}{\frac{hi}{\frac{lo}{hi}}}\right)}{-1 + \frac{lo}{hi}}}{hi}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 38.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;hi \leq 1.38 \cdot 10^{+308}:\\ \;\;\;\;x \cdot \frac{-1}{lo} + hi \cdot \left(\frac{\frac{hi}{lo}}{lo} \cdot \left(1 - \frac{x}{lo}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - lo\right) \cdot \frac{-1 + \frac{lo}{\frac{hi}{\frac{lo}{hi}}}}{-1 + \frac{lo}{hi}}}{hi}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (if (<= hi 1.38e+308)
   (+ (* x (/ -1.0 lo)) (* hi (* (/ (/ hi lo) lo) (- 1.0 (/ x lo)))))
   (/
    (* (- x lo) (/ (+ -1.0 (/ lo (/ hi (/ lo hi)))) (+ -1.0 (/ lo hi))))
    hi)))

C

double code(double lo, double hi, double x) {
	double tmp;
	if (hi <= 1.38e+308) {
		tmp = (x * (-1.0 / lo)) + (hi * (((hi / lo) / lo) * (1.0 - (x / lo))));
	} else {
		tmp = ((x - lo) * ((-1.0 + (lo / (hi / (lo / hi)))) / (-1.0 + (lo / hi)))) / hi;
	}
	return tmp;
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: tmp
    if (hi <= 1.38d+308) then
        tmp = (x * ((-1.0d0) / lo)) + (hi * (((hi / lo) / lo) * (1.0d0 - (x / lo))))
    else
        tmp = ((x - lo) * (((-1.0d0) + (lo / (hi / (lo / hi)))) / ((-1.0d0) + (lo / hi)))) / hi
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double tmp;
	if (hi <= 1.38e+308) {
		tmp = (x * (-1.0 / lo)) + (hi * (((hi / lo) / lo) * (1.0 - (x / lo))));
	} else {
		tmp = ((x - lo) * ((-1.0 + (lo / (hi / (lo / hi)))) / (-1.0 + (lo / hi)))) / hi;
	}
	return tmp;
}

Python

def code(lo, hi, x):
	tmp = 0
	if hi <= 1.38e+308:
		tmp = (x * (-1.0 / lo)) + (hi * (((hi / lo) / lo) * (1.0 - (x / lo))))
	else:
		tmp = ((x - lo) * ((-1.0 + (lo / (hi / (lo / hi)))) / (-1.0 + (lo / hi)))) / hi
	return tmp

Julia

function code(lo, hi, x)
	tmp = 0.0
	if (hi <= 1.38e+308)
		tmp = Float64(Float64(x * Float64(-1.0 / lo)) + Float64(hi * Float64(Float64(Float64(hi / lo) / lo) * Float64(1.0 - Float64(x / lo)))));
	else
		tmp = Float64(Float64(Float64(x - lo) * Float64(Float64(-1.0 + Float64(lo / Float64(hi / Float64(lo / hi)))) / Float64(-1.0 + Float64(lo / hi)))) / hi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	tmp = 0.0;
	if (hi <= 1.38e+308)
		tmp = (x * (-1.0 / lo)) + (hi * (((hi / lo) / lo) * (1.0 - (x / lo))));
	else
		tmp = ((x - lo) * ((-1.0 + (lo / (hi / (lo / hi)))) / (-1.0 + (lo / hi)))) / hi;
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := If[LessEqual[hi, 1.38e+308], N[(N[(x * N[(-1.0 / lo), $MachinePrecision]), $MachinePrecision] + N[(hi * N[(N[(N[(hi / lo), $MachinePrecision] / lo), $MachinePrecision] * N[(1.0 - N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - lo), $MachinePrecision] * N[(N[(-1.0 + N[(lo / N[(hi / N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.38e308

   1. Initial program 3.1%

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

   3. Taylor expanded in hi around 0

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

   5. Simplified 19.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot \frac{x}{lo}\right)}, \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1 \cdot x}{lo}\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot -1}{lo}\right), \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{-1}{lo}\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{-1}{lo}\right)\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 21.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

   8. Simplified 21.3%

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

   9. Taylor expanded in hi around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \color{blue}{\left(hi \cdot \left(\frac{1}{{lo}^{2}} - \frac{x}{{lo}^{3}}\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out-- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(hi \cdot \frac{1}{{lo}^{2}} - \color{blue}{hi \cdot \frac{x}{{lo}^{3}}}\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot 1}{{lo}^{2}} - \color{blue}{hi} \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot 1}{lo \cdot lo} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       4. times-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{1}{lo} - \color{blue}{hi} \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       5. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{\frac{lo}{lo}}{lo} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{lo}{lo \cdot lo} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{lo}{{lo}^{2}} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       8. times-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{lo \cdot {lo}^{2}} - \color{blue}{hi} \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{lo \cdot \left(lo \cdot lo\right)} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{{lo}^{3}} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

       11. associate-/l* N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{{lo}^{3}} - \frac{hi \cdot x}{\color{blue}{{lo}^{3}}}\right)\right)\right) \]

       12. div-sub N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo - hi \cdot x}{\color{blue}{{lo}^{3}}}\right)\right)\right) \]

       13. distribute-lft-out-- N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot \left(lo - x\right)}{{\color{blue}{lo}}^{3}}\right)\right)\right) \]

       14. unpow3 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot \left(lo - x\right)}{\left(lo \cdot lo\right) \cdot \color{blue}{lo}}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot \left(lo - x\right)}{{lo}^{2} \cdot lo}\right)\right)\right) \]

       16. times-frac N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{{lo}^{2}} \cdot \color{blue}{\frac{lo - x}{lo}}\right)\right)\right) \]

       17. div-sub N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{{lo}^{2}} \cdot \left(\frac{lo}{lo} - \color{blue}{\frac{x}{lo}}\right)\right)\right)\right) \]

       18. *-inverses N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{{lo}^{2}} \cdot \left(1 - \frac{\color{blue}{x}}{lo}\right)\right)\right)\right) \]

       19. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \mathsf{*.f64}\left(\left(\frac{hi}{{lo}^{2}}\right), \color{blue}{\left(1 - \frac{x}{lo}\right)}\right)\right)\right) \]

   11. Simplified 21.3%

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

   if 1.38e308 < hi

   1. Initial program 3.1%

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

   3. Taylor expanded in hi around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-out N/A

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

      7. distribute-neg-frac N/A

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

      8. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{-1 \cdot \left(x - lo\right) + -1 \cdot \frac{lo \cdot \left(x - lo\right)}{hi}}{hi}} \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{-1 \cdot \left(x - lo\right) + -1 \cdot \frac{lo \cdot \left(x - lo\right)}{hi}}{hi}\right)}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(-1 \cdot \left(x - lo\right) + -1 \cdot \frac{lo \cdot \left(x - lo\right)}{hi}\right), \color{blue}{hi}\right)\right) \]

   5. Simplified 13.4%

      \[\leadsto \color{blue}{0 - \frac{\left(x - lo\right) \cdot \left(-1 - \frac{lo}{hi}\right)}{hi}} \]
   6. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \left(\frac{-1 \cdot -1 - \frac{lo}{hi} \cdot \frac{lo}{hi}}{-1 + \frac{lo}{hi}}\right)\right), hi\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\left(-1 \cdot -1 - \frac{lo}{hi} \cdot \frac{lo}{hi}\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\left(1 - \frac{lo}{hi} \cdot \frac{lo}{hi}\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo}{hi} \cdot \frac{lo}{hi}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo}{hi} \cdot \frac{1}{\frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      6. frac-times N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo \cdot 1}{hi \cdot \frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo \cdot \frac{1}{1}}{hi \cdot \frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{lo}{1}}{hi \cdot \frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      9. /-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo}{hi \cdot \frac{hi}{lo}}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{lo}{\frac{hi}{lo} \cdot hi}\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \left(\frac{hi}{lo} \cdot hi\right)\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \left(hi \cdot \frac{hi}{lo}\right)\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \left(hi \cdot \frac{1}{\frac{lo}{hi}}\right)\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      14. un-div-inv N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \left(\frac{hi}{\frac{lo}{hi}}\right)\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \mathsf{/.f64}\left(hi, \left(\frac{lo}{hi}\right)\right)\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \mathsf{/.f64}\left(hi, \mathsf{/.f64}\left(lo, hi\right)\right)\right)\right), \left(-1 + \frac{lo}{hi}\right)\right)\right), hi\right)\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \mathsf{/.f64}\left(hi, \mathsf{/.f64}\left(lo, hi\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \left(\frac{lo}{hi}\right)\right)\right)\right), hi\right)\right) \]

      18. /-lowering-/.f64 57.8%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, lo\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(lo, \mathsf{/.f64}\left(hi, \mathsf{/.f64}\left(lo, hi\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(lo, hi\right)\right)\right)\right), hi\right)\right) \]

   7. Applied egg-rr 57.8%

      \[\leadsto 0 - \frac{\left(x - lo\right) \cdot \color{blue}{\frac{1 - \frac{lo}{\frac{hi}{\frac{lo}{hi}}}}{-1 + \frac{lo}{hi}}}}{hi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.38 \cdot 10^{+308}:\\ \;\;\;\;x \cdot \frac{-1}{lo} + hi \cdot \left(\frac{\frac{hi}{lo}}{lo} \cdot \left(1 - \frac{x}{lo}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - lo\right) \cdot \frac{-1 + \frac{lo}{\frac{hi}{\frac{lo}{hi}}}}{-1 + \frac{lo}{hi}}}{hi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 19.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0

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

5. Simplified 18.9%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot \frac{x}{lo}\right)}, \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1 \cdot x}{lo}\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot -1}{lo}\right), \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{-1}{lo}\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{-1}{lo}\right)\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f64 19.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

8. Simplified 19.7%

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

9. Taylor expanded in hi around inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \color{blue}{\left(hi \cdot \left(\frac{1}{{lo}^{2}} - \frac{x}{{lo}^{3}}\right)\right)}\right)\right) \]
10. Step-by-step derivation

    1. distribute-lft-out-- N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(hi \cdot \frac{1}{{lo}^{2}} - \color{blue}{hi \cdot \frac{x}{{lo}^{3}}}\right)\right)\right) \]

    2. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot 1}{{lo}^{2}} - \color{blue}{hi} \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot 1}{lo \cdot lo} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

    4. times-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{1}{lo} - \color{blue}{hi} \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

    5. *-inverses N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{\frac{lo}{lo}}{lo} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

    6. associate-/r* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{lo}{lo \cdot lo} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo} \cdot \frac{lo}{{lo}^{2}} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

    8. times-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{lo \cdot {lo}^{2}} - \color{blue}{hi} \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{lo \cdot \left(lo \cdot lo\right)} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

    10. cube-mult N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{{lo}^{3}} - hi \cdot \frac{x}{{lo}^{3}}\right)\right)\right) \]

    11. associate-/l* N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo}{{lo}^{3}} - \frac{hi \cdot x}{\color{blue}{{lo}^{3}}}\right)\right)\right) \]

    12. div-sub N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot lo - hi \cdot x}{\color{blue}{{lo}^{3}}}\right)\right)\right) \]

    13. distribute-lft-out-- N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot \left(lo - x\right)}{{\color{blue}{lo}}^{3}}\right)\right)\right) \]

    14. unpow3 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot \left(lo - x\right)}{\left(lo \cdot lo\right) \cdot \color{blue}{lo}}\right)\right)\right) \]

    15. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi \cdot \left(lo - x\right)}{{lo}^{2} \cdot lo}\right)\right)\right) \]

    16. times-frac N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{{lo}^{2}} \cdot \color{blue}{\frac{lo - x}{lo}}\right)\right)\right) \]

    17. div-sub N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{{lo}^{2}} \cdot \left(\frac{lo}{lo} - \color{blue}{\frac{x}{lo}}\right)\right)\right)\right) \]

    18. *-inverses N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \left(\frac{hi}{{lo}^{2}} \cdot \left(1 - \frac{\color{blue}{x}}{lo}\right)\right)\right)\right) \]

    19. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \mathsf{*.f64}\left(\left(\frac{hi}{{lo}^{2}}\right), \color{blue}{\left(1 - \frac{x}{lo}\right)}\right)\right)\right) \]

11. Simplified 19.7%

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

Alternative 5: 19.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0

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

5. Simplified 18.9%

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

6. Taylor expanded in hi around inf

   \[\leadsto \color{blue}{{hi}^{2} \cdot \left(\frac{1}{{lo}^{2}} - \frac{x}{{lo}^{3}}\right)} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto {hi}^{2} \cdot \left(\frac{1}{lo \cdot lo} - \frac{x}{{lo}^{3}}\right) \]

   2. associate-/r* N/A

      \[\leadsto {hi}^{2} \cdot \left(\frac{\frac{1}{lo}}{lo} - \frac{\color{blue}{x}}{{lo}^{3}}\right) \]

   3. unpow3 N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. div-sub N/A

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

   7. associate-/l* N/A

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

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({hi}^{2} \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)\right), \color{blue}{lo}\right) \]

8. Simplified 15.9%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{hi - hi \cdot \frac{x}{lo}}{lo}\right), \color{blue}{\left(\frac{hi}{lo}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(hi - hi \cdot \frac{x}{lo}\right), lo\right), \left(\frac{\color{blue}{hi}}{lo}\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, \left(hi \cdot \frac{x}{lo}\right)\right), lo\right), \left(\frac{hi}{lo}\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, \left(hi \cdot \frac{1}{\frac{lo}{x}}\right)\right), lo\right), \left(\frac{hi}{lo}\right)\right) \]

   7. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, \left(\frac{hi}{\frac{lo}{x}}\right)\right), lo\right), \left(\frac{hi}{lo}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, \mathsf{/.f64}\left(hi, \left(\frac{lo}{x}\right)\right)\right), lo\right), \left(\frac{hi}{lo}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, \mathsf{/.f64}\left(hi, \mathsf{/.f64}\left(lo, x\right)\right)\right), lo\right), \left(\frac{hi}{lo}\right)\right) \]

   10. /-lowering-/.f64 19.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, \mathsf{/.f64}\left(hi, \mathsf{/.f64}\left(lo, x\right)\right)\right), lo\right), \mathsf{/.f64}\left(hi, \color{blue}{lo}\right)\right) \]

10. Applied egg-rr 19.7%

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

11. Final simplification 19.7%

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

Alternative 6: 19.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0

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

5. Simplified 18.9%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(-1 \cdot \frac{x}{lo}\right)}, \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1 \cdot x}{lo}\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x \cdot -1}{lo}\right), \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{-1}{lo}\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{-1}{lo}\right)\right), \mathsf{*.f64}\left(\color{blue}{hi}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f64 19.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(-1, lo\right)\right), \mathsf{*.f64}\left(hi, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(hi, lo\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(lo, lo\right)\right)\right)\right)\right) \]

8. Simplified 19.7%

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

9. Taylor expanded in x around 0

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

    1. unpow2 N/A

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

    2. associate-/l* N/A

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(hi, \color{blue}{\left(\frac{hi}{{lo}^{2}}\right)}\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo \cdot \color{blue}{lo}}\right)\right) \]

    5. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(hi, \left(\frac{\frac{hi}{lo}}{\color{blue}{lo}}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\left(\frac{hi}{lo}\right), \color{blue}{lo}\right)\right) \]

    7. /-lowering-/.f64 19.7%

       \[\leadsto \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\mathsf{/.f64}\left(hi, lo\right), lo\right)\right) \]

11. Simplified 19.7%

    \[\leadsto \color{blue}{hi \cdot \frac{\frac{hi}{lo}}{lo}} \]
12. Add Preprocessing

Alternative 7: 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

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x - lo\right), \color{blue}{hi}\right) \]

   2. --lowering--.f64 18.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, lo\right), hi\right) \]

5. Simplified 18.8%

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

Alternative 8: 18.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{lo}{0 - hi} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ lo (- 0.0 hi)))

C

double code(double lo, double hi, double x) {
	return lo / (0.0 - 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 / (0.0d0 - hi)
end function

Java

public static double code(double lo, double hi, double x) {
	return lo / (0.0 - hi);
}

Python

def code(lo, hi, x):
	return lo / (0.0 - hi)

Julia

function code(lo, hi, x)
	return Float64(lo / Float64(0.0 - hi))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = lo / (0.0 - hi);
end

Wolfram

code[lo_, hi_, x_] := N[(lo / N[(0.0 - hi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x - lo\right), \color{blue}{hi}\right) \]

   2. --lowering--.f64 18.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, lo\right), hi\right) \]

5. Simplified 18.8%

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

6. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(lo\right)}{hi} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(lo\right)\right), \color{blue}{hi}\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(0 - lo\right), hi\right) \]

   5. --lowering--.f64 18.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, lo\right), hi\right) \]

8. Simplified 18.8%

   \[\leadsto \color{blue}{\frac{0 - lo}{hi}} \]

9. Final simplification 18.8%

   \[\leadsto \frac{lo}{0 - hi} \]
10. Add Preprocessing

Alternative 9: 18.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[(x / lo), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{x - lo}{lo}\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\frac{x - lo}{lo}} \]

   3. div-sub N/A

      \[\leadsto 0 - \left(\frac{x}{lo} - \color{blue}{\frac{lo}{lo}}\right) \]

   4. sub-neg N/A

      \[\leadsto 0 - \left(\frac{x}{lo} + \color{blue}{\left(\mathsf{neg}\left(\frac{lo}{lo}\right)\right)}\right) \]

   5. *-inverses N/A

      \[\leadsto 0 - \left(\frac{x}{lo} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto 0 - \left(\frac{x}{lo} + -1\right) \]

   7. +-commutative N/A

      \[\leadsto 0 - \left(-1 + \color{blue}{\frac{x}{lo}}\right) \]

   8. associate--r+ N/A

      \[\leadsto \left(0 - -1\right) - \color{blue}{\frac{x}{lo}} \]

   9. metadata-eval N/A

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

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{x}{lo}\right)}\right) \]

   11. /-lowering-/.f64 18.7%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{lo}\right)\right) \]

5. Simplified 18.7%

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

Alternative 10: 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

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

5. Simplified 18.7%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))