xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 12.7s

Alternatives: 9

Speedup: 7.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 19.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ 1 - \left|\frac{x + \mathsf{fma}\left(hi, {\left(\sqrt[3]{hi \cdot \left(\frac{1}{lo} + \frac{x}{hi \cdot lo}\right)}\right)}^{3}, hi\right)}{lo}\right| \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (-
  1.0
  (fabs
   (/
    (+ x (fma hi (pow (cbrt (* hi (+ (/ 1.0 lo) (/ x (* hi lo))))) 3.0) hi))
    lo))))

C

double code(double lo, double hi, double x) {
	return 1.0 - fabs(((x + fma(hi, pow(cbrt((hi * ((1.0 / lo) + (x / (hi * lo))))), 3.0), hi)) / lo));
}

Julia

function code(lo, hi, x)
	return Float64(1.0 - abs(Float64(Float64(x + fma(hi, (cbrt(Float64(hi * Float64(Float64(1.0 / lo) + Float64(x / Float64(hi * lo))))) ^ 3.0), hi)) / lo)))
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[Abs[N[(N[(x + N[(hi * N[Power[N[Power[N[(hi * N[(N[(1.0 / lo), $MachinePrecision] + N[(x / N[(hi * lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] + hi), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.5%

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

5. Simplified 14.5%

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

   1. add-sqr-sqrt 10.2%

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

   2. sqrt-unprod 13.9%

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

   3. pow2 13.9%

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

7. Applied egg-rr 19.5%

   \[\leadsto 1 + \left(-\color{blue}{\sqrt{{\left(\frac{x + \mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)}{lo}\right)}^{2}}}\right) \]
8. Step-by-step derivation

   1. unpow2 19.5%

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

   2. rem-sqrt-square 19.5%

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

   3. +-commutative 19.5%

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

9. Simplified 19.5%

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

10. Taylor expanded in hi around inf 19.5%

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

    1. add-cube-cbrt 19.5%

       \[\leadsto 1 + \left(-\left|\frac{x + \mathsf{fma}\left(hi, \color{blue}{\left(\sqrt[3]{hi \cdot \left(\frac{1}{lo} + \frac{x}{hi \cdot lo}\right)} \cdot \sqrt[3]{hi \cdot \left(\frac{1}{lo} + \frac{x}{hi \cdot lo}\right)}\right) \cdot \sqrt[3]{hi \cdot \left(\frac{1}{lo} + \frac{x}{hi \cdot lo}\right)}}, hi\right)}{lo}\right|\right) \]

    2. pow3 19.5%

       \[\leadsto 1 + \left(-\left|\frac{x + \mathsf{fma}\left(hi, \color{blue}{{\left(\sqrt[3]{hi \cdot \left(\frac{1}{lo} + \frac{x}{hi \cdot lo}\right)}\right)}^{3}}, hi\right)}{lo}\right|\right) \]

12. Applied egg-rr 19.5%

    \[\leadsto 1 + \left(-\left|\frac{x + \mathsf{fma}\left(hi, \color{blue}{{\left(\sqrt[3]{hi \cdot \left(\frac{1}{lo} + \frac{x}{hi \cdot lo}\right)}\right)}^{3}}, hi\right)}{lo}\right|\right) \]

13. Final simplification 19.5%

    \[\leadsto 1 - \left|\frac{x + \mathsf{fma}\left(hi, {\left(\sqrt[3]{hi \cdot \left(\frac{1}{lo} + \frac{x}{hi \cdot lo}\right)}\right)}^{3}, hi\right)}{lo}\right| \]
14. Add Preprocessing

Alternative 2: 19.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return 1.0 - fabs(((x / lo) + ((hi / lo) + pow((hi / lo), 2.0))));
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = 1.0d0 - abs(((x / lo) + ((hi / lo) + ((hi / lo) ** 2.0d0))))
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 - Math.abs(((x / lo) + ((hi / lo) + Math.pow((hi / lo), 2.0))));
}

Python

def code(lo, hi, x):
	return 1.0 - math.fabs(((x / lo) + ((hi / lo) + math.pow((hi / lo), 2.0))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.5%

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

5. Simplified 14.5%

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

   1. add-sqr-sqrt 10.2%

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

   2. sqrt-unprod 13.9%

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

   3. pow2 13.9%

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

7. Applied egg-rr 19.5%

   \[\leadsto 1 + \left(-\color{blue}{\sqrt{{\left(\frac{x + \mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)}{lo}\right)}^{2}}}\right) \]
8. Step-by-step derivation

   1. unpow2 19.5%

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

   2. rem-sqrt-square 19.5%

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

   3. +-commutative 19.5%

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

9. Simplified 19.5%

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

10. Taylor expanded in hi around inf 19.5%

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

11. Taylor expanded in x around 0 0.0%

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

    1. associate-+r+ 0.0%

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

    2. +-commutative 0.0%

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

    3. associate-+l+ 0.0%

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

    4. unpow2 0.0%

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

    5. unpow2 0.0%

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

    6. times-frac 19.5%

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

    7. unpow2 19.5%

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

13. Simplified 19.5%

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

14. Final simplification 19.5%

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

Alternative 3: 19.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return 1.0 - fabs(((x + fma(hi, (hi / lo), hi)) / lo));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.5%

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

5. Simplified 14.5%

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

   1. add-sqr-sqrt 10.2%

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

   2. sqrt-unprod 13.9%

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

   3. pow2 13.9%

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

7. Applied egg-rr 19.5%

   \[\leadsto 1 + \left(-\color{blue}{\sqrt{{\left(\frac{x + \mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)}{lo}\right)}^{2}}}\right) \]
8. Step-by-step derivation

   1. unpow2 19.5%

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

   2. rem-sqrt-square 19.5%

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

   3. +-commutative 19.5%

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

9. Simplified 19.5%

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

10. Taylor expanded in hi around inf 19.5%

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

11. Final simplification 19.5%

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

Alternative 4: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.5%

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

5. Simplified 14.5%

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

   1. add-sqr-sqrt 10.2%

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

   2. sqrt-unprod 13.9%

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

   3. pow2 13.9%

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

7. Applied egg-rr 19.5%

   \[\leadsto 1 + \left(-\color{blue}{\sqrt{{\left(\frac{x + \mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)}{lo}\right)}^{2}}}\right) \]
8. Step-by-step derivation

   1. unpow2 19.5%

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

   2. rem-sqrt-square 19.5%

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

   3. +-commutative 19.5%

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

9. Simplified 19.5%

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

10. Taylor expanded in hi around 0 19.5%

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

    1. +-commutative 19.5%

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

12. Simplified 19.5%

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

13. Final simplification 19.5%

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

Alternative 5: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return pow((hi / lo), 2.0);
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (hi / lo) ** 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.5%

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

5. Simplified 14.5%

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

   1. div-sub 14.6%

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

7. Applied egg-rr 14.6%

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

8. Taylor expanded in hi around inf 0.0%

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

   1. unpow2 0.0%

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

   2. unpow2 0.0%

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

   3. times-frac 19.3%

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

   4. unpow2 19.3%

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

10. Simplified 19.3%

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

11. Final simplification 19.3%

    \[\leadsto {\left(\frac{hi}{lo}\right)}^{2} \]
12. Add Preprocessing

Alternative 6: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.5%

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

5. Simplified 14.5%

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

6. Taylor expanded in hi around 0 18.9%

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

   1. sub-neg 18.9%

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

   2. +-commutative 18.9%

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

   3. mul-1-neg 18.9%

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

   4. sub-neg 18.9%

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

   5. div-sub 18.8%

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

   6. metadata-eval 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

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

Alternative 7: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

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

4. Final simplification 18.8%

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

Alternative 8: 18.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

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

4. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 9: 18.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(lo, hi, x)
	return 1.0
end

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 18.7%

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

4. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))