xlohi (overflows)

Percentage Accurate: 3.1% → 39.9%

Time: 12.1s

Alternatives: 7

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: 39.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   8. *-rgt-identity N/A

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

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

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval 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

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

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-rgt-in N/A

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

   2. flip-+ N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

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

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

10. Applied egg-rr 41.5%

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

11. Final simplification 41.5%

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

Alternative 2: 37.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;lo \leq -1.36 \cdot 10^{+308}:\\ \;\;\;\;\frac{-1 + \frac{hi}{\frac{lo}{\frac{hi}{lo}}}}{-1 + \frac{hi}{lo}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - lo\right) + lo \cdot \frac{\left(x - lo\right) \cdot \left(\frac{lo}{hi} - -1\right)}{hi}}{hi}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (if (<= lo -1.36e+308)
   (/ (+ -1.0 (/ hi (/ lo (/ hi lo)))) (+ -1.0 (/ hi lo)))
   (/ (+ (- x lo) (* lo (/ (* (- x lo) (- (/ lo hi) -1.0)) hi))) hi)))

C

double code(double lo, double hi, double x) {
	double tmp;
	if (lo <= -1.36e+308) {
		tmp = (-1.0 + (hi / (lo / (hi / lo)))) / (-1.0 + (hi / lo));
	} else {
		tmp = ((x - lo) + (lo * (((x - lo) * ((lo / hi) - -1.0)) / 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 (lo <= (-1.36d+308)) then
        tmp = ((-1.0d0) + (hi / (lo / (hi / lo)))) / ((-1.0d0) + (hi / lo))
    else
        tmp = ((x - lo) + (lo * (((x - lo) * ((lo / hi) - (-1.0d0))) / hi))) / hi
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double tmp;
	if (lo <= -1.36e+308) {
		tmp = (-1.0 + (hi / (lo / (hi / lo)))) / (-1.0 + (hi / lo));
	} else {
		tmp = ((x - lo) + (lo * (((x - lo) * ((lo / hi) - -1.0)) / hi))) / hi;
	}
	return tmp;
}

Python

def code(lo, hi, x):
	tmp = 0
	if lo <= -1.36e+308:
		tmp = (-1.0 + (hi / (lo / (hi / lo)))) / (-1.0 + (hi / lo))
	else:
		tmp = ((x - lo) + (lo * (((x - lo) * ((lo / hi) - -1.0)) / hi))) / hi
	return tmp

Julia

function code(lo, hi, x)
	tmp = 0.0
	if (lo <= -1.36e+308)
		tmp = Float64(Float64(-1.0 + Float64(hi / Float64(lo / Float64(hi / lo)))) / Float64(-1.0 + Float64(hi / lo)));
	else
		tmp = Float64(Float64(Float64(x - lo) + Float64(lo * Float64(Float64(Float64(x - lo) * Float64(Float64(lo / hi) - -1.0)) / hi))) / hi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	tmp = 0.0;
	if (lo <= -1.36e+308)
		tmp = (-1.0 + (hi / (lo / (hi / lo)))) / (-1.0 + (hi / lo));
	else
		tmp = ((x - lo) + (lo * (((x - lo) * ((lo / hi) - -1.0)) / hi))) / hi;
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := If[LessEqual[lo, -1.36e+308], N[(N[(-1.0 + N[(hi / N[(lo / N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - lo), $MachinePrecision] + N[(lo * N[(N[(N[(x - lo), $MachinePrecision] * N[(N[(lo / hi), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.35999999999999991e308

   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%

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

   6. Taylor expanded in x around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. /-lowering-/.f64 19.6%

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

   8. Simplified 19.6%

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

   9. Taylor expanded in hi around 0

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

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

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

       2. /-lowering-/.f64 13.9%

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

   11. Simplified 13.9%

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

       1. +-commutative N/A

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

       2. flip-+ N/A

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

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

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

       4. metadata-eval N/A

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

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

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

       6. div-inv N/A

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

       7. associate-*l* N/A

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

       8. associate-/r/ N/A

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

       9. un-div-inv N/A

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

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

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

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

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

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

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

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

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

       14. /-lowering-/.f64 57.6%

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

   13. Applied egg-rr 57.6%

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

   if -1.35999999999999991e308 < lo

   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 19.5%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.36 \cdot 10^{+308}:\\ \;\;\;\;\frac{-1 + \frac{hi}{\frac{lo}{\frac{hi}{lo}}}}{-1 + \frac{hi}{lo}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - lo\right) + lo \cdot \frac{\left(x - lo\right) \cdot \left(\frac{lo}{hi} - -1\right)}{hi}}{hi}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;lo \leq -1.36 \cdot 10^{+308}:\\ \;\;\;\;\frac{-1 + \frac{hi}{\frac{lo}{\frac{hi}{lo}}}}{-1 + \frac{hi}{lo}}\\ \mathbf{else}:\\ \;\;\;\;\frac{lo}{hi} \cdot \left(-1 + \frac{x}{hi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (if (<= lo -1.36e+308)
   (/ (+ -1.0 (/ hi (/ lo (/ hi lo)))) (+ -1.0 (/ hi lo)))
   (* (/ lo hi) (+ -1.0 (/ x hi)))))

C

double code(double lo, double hi, double x) {
	double tmp;
	if (lo <= -1.36e+308) {
		tmp = (-1.0 + (hi / (lo / (hi / lo)))) / (-1.0 + (hi / lo));
	} else {
		tmp = (lo / hi) * (-1.0 + (x / 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 (lo <= (-1.36d+308)) then
        tmp = ((-1.0d0) + (hi / (lo / (hi / lo)))) / ((-1.0d0) + (hi / lo))
    else
        tmp = (lo / hi) * ((-1.0d0) + (x / hi))
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double tmp;
	if (lo <= -1.36e+308) {
		tmp = (-1.0 + (hi / (lo / (hi / lo)))) / (-1.0 + (hi / lo));
	} else {
		tmp = (lo / hi) * (-1.0 + (x / hi));
	}
	return tmp;
}

Python

def code(lo, hi, x):
	tmp = 0
	if lo <= -1.36e+308:
		tmp = (-1.0 + (hi / (lo / (hi / lo)))) / (-1.0 + (hi / lo))
	else:
		tmp = (lo / hi) * (-1.0 + (x / hi))
	return tmp

Julia

function code(lo, hi, x)
	tmp = 0.0
	if (lo <= -1.36e+308)
		tmp = Float64(Float64(-1.0 + Float64(hi / Float64(lo / Float64(hi / lo)))) / Float64(-1.0 + Float64(hi / lo)));
	else
		tmp = Float64(Float64(lo / hi) * Float64(-1.0 + Float64(x / hi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	tmp = 0.0;
	if (lo <= -1.36e+308)
		tmp = (-1.0 + (hi / (lo / (hi / lo)))) / (-1.0 + (hi / lo));
	else
		tmp = (lo / hi) * (-1.0 + (x / hi));
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := If[LessEqual[lo, -1.36e+308], N[(N[(-1.0 + N[(hi / N[(lo / N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(lo / hi), $MachinePrecision] * N[(-1.0 + N[(x / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.35999999999999991e308

   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%

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

   6. Taylor expanded in x around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. /-lowering-/.f64 19.6%

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

   8. Simplified 19.6%

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

   9. Taylor expanded in hi around 0

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

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

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

       2. /-lowering-/.f64 13.9%

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

   11. Simplified 13.9%

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

       1. +-commutative N/A

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

       2. flip-+ N/A

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

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

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

       4. metadata-eval N/A

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

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

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

       6. div-inv N/A

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

       7. associate-*l* N/A

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

       8. associate-/r/ N/A

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

       9. un-div-inv N/A

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

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

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

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

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

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

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

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

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

       14. /-lowering-/.f64 57.6%

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

   13. Applied egg-rr 57.6%

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

   if -1.35999999999999991e308 < lo

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

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

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

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

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

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

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

      14. mul-1-neg N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. unpow2 N/A

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

   5. Simplified 19.1%

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

   6. Taylor expanded in lo around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-/l* N/A

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

      9. associate-*l/ N/A

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

      10. distribute-lft-out N/A

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

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

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

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

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

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

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

      14. /-lowering-/.f64 19.1%

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

   8. Simplified 19.1%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.36 \cdot 10^{+308}:\\ \;\;\;\;\frac{-1 + \frac{hi}{\frac{lo}{\frac{hi}{lo}}}}{-1 + \frac{hi}{lo}}\\ \mathbf{else}:\\ \;\;\;\;\frac{lo}{hi} \cdot \left(-1 + \frac{x}{hi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 19.4% 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%

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

6. Taylor expanded in x around 0

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

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

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

   6. /-lowering-/.f64 18.9%

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

8. Simplified 18.9%

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

9. Taylor expanded in hi around inf

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

    1. unpow2 N/A

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

    2. associate-*r/ 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 19.6%

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

11. Simplified 19.6%

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

Alternative 5: 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 6: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. distribute-lft-in N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. +-commutative N/A

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

   8. associate--r+ N/A

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

   9. div-sub N/A

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

   10. sub-neg N/A

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

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

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

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

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

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

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

   14. mul-1-neg N/A

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

   15. distribute-rgt-neg-out N/A

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

   16. remove-double-neg N/A

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

   17. unpow2 N/A

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

5. Simplified 18.8%

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

6. Taylor expanded in lo around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. times-frac N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. associate-/l* N/A

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

   9. associate-*l/ N/A

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

   10. distribute-lft-out N/A

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

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

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

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

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

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

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

   14. /-lowering-/.f64 18.8%

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

8. Simplified 18.8%

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

   1. div-inv N/A

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

   2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   7. /-lowering-/.f64 18.8%

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

10. Applied egg-rr 18.8%

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

11. Taylor expanded in hi around inf

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

    1. /-lowering-/.f64 18.8%

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

13. Simplified 18.8%

    \[\leadsto lo \cdot \color{blue}{\frac{-1}{hi}} \]
14. Add Preprocessing

Alternative 7: 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 2024192 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))