xlohi (overflows)

Percentage Accurate: 3.1% → 98.0%

Time: 11.4s

Alternatives: 8

Speedup: 7.0×

Specification

\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{lo} + -1\\ 1 - \frac{x + hi \cdot \frac{t\_0 \cdot t\_0}{t\_0 + \frac{hi}{lo}}}{lo} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (+ (/ x lo) -1.0)))
   (- 1.0 (/ (+ x (* hi (/ (* t_0 t_0) (+ t_0 (/ hi lo))))) lo))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x / lo) + -1.0;
	return 1.0 - ((x + (hi * ((t_0 * t_0) / (t_0 + (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
    real(8) :: t_0
    t_0 = (x / lo) + (-1.0d0)
    code = 1.0d0 - ((x + (hi * ((t_0 * t_0) / (t_0 + (hi / lo))))) / lo)
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x / lo) + -1.0;
	return 1.0 - ((x + (hi * ((t_0 * t_0) / (t_0 + (hi / lo))))) / lo);
}

Python

def code(lo, hi, x):
	t_0 = (x / lo) + -1.0
	return 1.0 - ((x + (hi * ((t_0 * t_0) / (t_0 + (hi / lo))))) / lo)

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x / lo) + -1.0)
	return Float64(1.0 - Float64(Float64(x + Float64(hi * Float64(Float64(t_0 * t_0) / Float64(t_0 + Float64(hi / lo))))) / lo))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (x / lo) + -1.0;
	tmp = 1.0 - ((x + (hi * ((t_0 * t_0) / (t_0 + (hi / lo))))) / lo);
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x / lo), $MachinePrecision] + -1.0), $MachinePrecision]}, N[(1.0 - N[(N[(x + N[(hi * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(t$95$0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

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

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

   5. associate--l+ N/A

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

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

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

   7. sub-neg N/A

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

   8. associate-/l* N/A

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

   9. mul-1-neg N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-out N/A

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

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

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

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

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

   14. /-lowering-/.f64 N/A

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

   15. --lowering--.f64 18.9%

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

5. Simplified 18.9%

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

6. Taylor expanded in hi around inf

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

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

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

   2. associate--r+ N/A

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

   3. sub-neg N/A

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

   4. associate-/l/ N/A

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

   5. div-sub N/A

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

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

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

   7. /-lowering-/.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

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

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

   12. /-lowering-/.f64 N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f64 18.9%

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

8. Simplified 18.9%

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

   1. distribute-lft-in N/A

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

   2. flip-+ N/A

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

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

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

10. Applied egg-rr 40.3%

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

11. Taylor expanded in hi around 0

    \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\color{blue}{\left({\left(\frac{x}{lo} - 1\right)}^{2}\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, lo\right), -1\right), 1\right), \mathsf{/.f64}\left(hi, \mathsf{\_.f64}\left(0, lo\right)\right)\right)\right)\right)\right), lo\right)\right) \]
12. Step-by-step derivation

    1. unpow2 N/A

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

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

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

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

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

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

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

    7. sub-neg N/A

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

    8. metadata-eval N/A

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

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

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

    10. /-lowering-/.f64 97.1%

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

13. Simplified 97.1%

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

14. Final simplification 97.1%

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

Alternative 2: 38.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;hi \leq 1.388 \cdot 10^{+308}:\\ \;\;\;\;hi \cdot \frac{\frac{hi}{lo}}{lo}\\ \mathbf{else}:\\ \;\;\;\;\left(x - lo\right) \cdot \frac{\frac{-1 + \frac{lo}{hi \cdot \frac{hi}{lo}}}{-1 + \frac{lo}{hi}}}{hi}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (if (<= hi 1.388e+308)
   (* hi (/ (/ hi lo) lo))
   (*
    (- x lo)
    (/ (/ (+ -1.0 (/ lo (* hi (/ hi lo)))) (+ -1.0 (/ lo hi))) hi))))

C

double code(double lo, double hi, double x) {
	double tmp;
	if (hi <= 1.388e+308) {
		tmp = hi * ((hi / lo) / lo);
	} else {
		tmp = (x - lo) * (((-1.0 + (lo / (hi * (hi / lo)))) / (-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 <= 1.388d+308) then
        tmp = hi * ((hi / lo) / lo)
    else
        tmp = (x - lo) * ((((-1.0d0) + (lo / (hi * (hi / lo)))) / ((-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 <= 1.388e+308) {
		tmp = hi * ((hi / lo) / lo);
	} else {
		tmp = (x - lo) * (((-1.0 + (lo / (hi * (hi / lo)))) / (-1.0 + (lo / hi))) / hi);
	}
	return tmp;
}

Python

def code(lo, hi, x):
	tmp = 0
	if hi <= 1.388e+308:
		tmp = hi * ((hi / lo) / lo)
	else:
		tmp = (x - lo) * (((-1.0 + (lo / (hi * (hi / lo)))) / (-1.0 + (lo / hi))) / hi)
	return tmp

Julia

function code(lo, hi, x)
	tmp = 0.0
	if (hi <= 1.388e+308)
		tmp = Float64(hi * Float64(Float64(hi / lo) / lo));
	else
		tmp = Float64(Float64(x - lo) * Float64(Float64(Float64(-1.0 + Float64(lo / Float64(hi * Float64(hi / lo)))) / Float64(-1.0 + Float64(lo / hi))) / hi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	tmp = 0.0;
	if (hi <= 1.388e+308)
		tmp = hi * ((hi / lo) / lo);
	else
		tmp = (x - lo) * (((-1.0 + (lo / (hi * (hi / lo)))) / (-1.0 + (lo / hi))) / hi);
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := If[LessEqual[hi, 1.388e+308], N[(hi * N[(N[(hi / lo), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], N[(N[(x - lo), $MachinePrecision] * N[(N[(N[(-1.0 + N[(lo / N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.388e308

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around inf

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

   5. Simplified 19.6%

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

   6. Taylor expanded in hi around inf

      \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
   7. 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 20.8%

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

   8. Simplified 20.8%

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

   if 1.388e308 < 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

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

      5. mul-1-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-lft-out N/A

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

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

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

   5. Simplified 13.5%

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

      1. sub0-neg N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. distribute-frac-neg2 N/A

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

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

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

      7. /-lowering-/.f64 N/A

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

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

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

      9. /-lowering-/.f64 13.5%

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

   7. Applied egg-rr 13.5%

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

      1. flip-- N/A

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

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

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

      3. metadata-eval N/A

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

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

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

      5. associate-*r/ N/A

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

      6. associate-/r/ N/A

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

      7. associate-/l/ N/A

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

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

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

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

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

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

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

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

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

      12. /-lowering-/.f64 51.0%

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

   9. Applied egg-rr 51.0%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.388 \cdot 10^{+308}:\\ \;\;\;\;hi \cdot \frac{\frac{hi}{lo}}{lo}\\ \mathbf{else}:\\ \;\;\;\;\left(x - lo\right) \cdot \frac{\frac{-1 + \frac{lo}{hi \cdot \frac{hi}{lo}}}{-1 + \frac{lo}{hi}}}{hi}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{hi \cdot \frac{-1 - \frac{x \cdot -2}{lo}}{\left(\frac{x}{lo} + -1\right) + \frac{hi}{lo}} - x}{lo} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+
  1.0
  (/
   (- (* hi (/ (- -1.0 (/ (* x -2.0) lo)) (+ (+ (/ x lo) -1.0) (/ hi lo)))) x)
   lo)))

C

double code(double lo, double hi, double x) {
	return 1.0 + (((hi * ((-1.0 - ((x * -2.0) / lo)) / (((x / lo) + -1.0) + (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 = 1.0d0 + (((hi * (((-1.0d0) - ((x * (-2.0d0)) / lo)) / (((x / lo) + (-1.0d0)) + (hi / lo)))) - x) / lo)
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 + (((hi * ((-1.0 - ((x * -2.0) / lo)) / (((x / lo) + -1.0) + (hi / lo)))) - x) / lo);
}

Python

def code(lo, hi, x):
	return 1.0 + (((hi * ((-1.0 - ((x * -2.0) / lo)) / (((x / lo) + -1.0) + (hi / lo)))) - x) / lo)

Julia

function code(lo, hi, x)
	return Float64(1.0 + Float64(Float64(Float64(hi * Float64(Float64(-1.0 - Float64(Float64(x * -2.0) / lo)) / Float64(Float64(Float64(x / lo) + -1.0) + Float64(hi / lo)))) - x) / lo))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 + (((hi * ((-1.0 - ((x * -2.0) / lo)) / (((x / lo) + -1.0) + (hi / lo)))) - x) / lo);
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[(hi * N[(N[(-1.0 - N[(N[(x * -2.0), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x / lo), $MachinePrecision] + -1.0), $MachinePrecision] + N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

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

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

   5. associate--l+ N/A

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

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

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

   7. sub-neg N/A

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

   8. associate-/l* N/A

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

   9. mul-1-neg N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-out N/A

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

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

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

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

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

   14. /-lowering-/.f64 N/A

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

   15. --lowering--.f64 18.9%

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

5. Simplified 18.9%

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

6. Taylor expanded in hi around inf

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

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

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

   2. associate--r+ N/A

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

   3. sub-neg N/A

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

   4. associate-/l/ N/A

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

   5. div-sub N/A

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

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

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

   7. /-lowering-/.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

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

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

   12. /-lowering-/.f64 N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f64 18.9%

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

8. Simplified 18.9%

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

   1. distribute-lft-in N/A

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

   2. flip-+ N/A

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

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

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

10. Applied egg-rr 40.3%

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

11. Taylor expanded in lo around inf

    \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\color{blue}{\left(1 + -2 \cdot \frac{x}{lo}\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, lo\right), -1\right), 1\right), \mathsf{/.f64}\left(hi, \mathsf{\_.f64}\left(0, lo\right)\right)\right)\right)\right)\right), lo\right)\right) \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(-2 \cdot \frac{x}{lo}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, lo\right), -1\right), 1\right), \mathsf{/.f64}\left(hi, \mathsf{\_.f64}\left(0, lo\right)\right)\right)\right)\right)\right), lo\right)\right) \]

    2. associate-*r/ N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-2 \cdot x}{lo}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, lo\right), -1\right), 1\right), \mathsf{/.f64}\left(hi, \mathsf{\_.f64}\left(0, lo\right)\right)\right)\right)\right)\right), lo\right)\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-2 \cdot x\right), lo\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, lo\right), -1\right), 1\right), \mathsf{/.f64}\left(hi, \mathsf{\_.f64}\left(0, lo\right)\right)\right)\right)\right)\right), lo\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot -2\right), lo\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, lo\right), -1\right), 1\right), \mathsf{/.f64}\left(hi, \mathsf{\_.f64}\left(0, lo\right)\right)\right)\right)\right)\right), lo\right)\right) \]

    5. *-lowering-*.f64 97.1%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, -2\right), lo\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, lo\right), -1\right), 1\right), \mathsf{/.f64}\left(hi, \mathsf{\_.f64}\left(0, lo\right)\right)\right)\right)\right)\right), lo\right)\right) \]

13. Simplified 97.1%

    \[\leadsto 1 - \frac{x + hi \cdot \frac{\color{blue}{1 + \frac{x \cdot -2}{lo}}}{\left(\frac{x}{lo} + -1\right) \cdot 1 - \frac{hi}{0 - lo}}}{lo} \]

14. Final simplification 97.1%

    \[\leadsto 1 + \frac{hi \cdot \frac{-1 - \frac{x \cdot -2}{lo}}{\left(\frac{x}{lo} + -1\right) + \frac{hi}{lo}} - x}{lo} \]
15. Add Preprocessing

Alternative 4: 98.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

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

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

   5. associate--l+ N/A

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

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

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

   7. sub-neg N/A

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

   8. associate-/l* N/A

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

   9. mul-1-neg N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-out N/A

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

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

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

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

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

   14. /-lowering-/.f64 N/A

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

   15. --lowering--.f64 18.9%

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

5. Simplified 18.9%

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

6. Taylor expanded in hi around inf

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

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

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

   2. associate--r+ N/A

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

   3. sub-neg N/A

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

   4. associate-/l/ N/A

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

   5. div-sub N/A

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

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

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

   7. /-lowering-/.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

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

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

   12. /-lowering-/.f64 N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f64 18.9%

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

8. Simplified 18.9%

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

   1. distribute-lft-in N/A

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

   2. flip-+ N/A

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

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

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

10. Applied egg-rr 40.3%

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

11. Taylor expanded in lo around inf

    \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, lo\right), -1\right), 1\right), \mathsf{/.f64}\left(hi, \mathsf{\_.f64}\left(0, lo\right)\right)\right)\right)\right)\right), lo\right)\right) \]
12. Step-by-step derivation

13. Simplified 97.0%

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

14. Final simplification 97.0%

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

Alternative 5: 19.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ hi \cdot \frac{\frac{\frac{1}{lo}}{\frac{1}{hi}}}{lo} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return hi * (((1.0 / lo) / (1.0 / hi)) / lo);
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = hi * (((1.0d0 / lo) / (1.0d0 / hi)) / lo)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf

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

5. Simplified 18.9%

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

6. Taylor expanded in hi around inf

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
7. 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 19.5%

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

8. Simplified 19.5%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

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

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

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

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

   6. /-lowering-/.f64 19.5%

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

10. Applied egg-rr 19.5%

    \[\leadsto hi \cdot \frac{\color{blue}{\frac{\frac{1}{lo}}{\frac{1}{hi}}}}{lo} \]
11. 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 lo around inf

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

5. Simplified 18.9%

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

6. Taylor expanded in hi around inf

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
7. 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 19.5%

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

8. Simplified 19.5%

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

Alternative 7: 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 18.7%

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

5. Simplified 18.7%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 18.8%

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

8. Simplified 18.8%

   \[\leadsto \frac{\color{blue}{0 - lo}}{hi} \]
9. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 18.8%

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

10. Applied egg-rr 18.8%

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

11. Final simplification 18.8%

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

Alternative 8: 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 2024163 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))