xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 22.4s

Alternatives: 6

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[Abs[N[(hi * N[(N[(hi + x), $MachinePrecision] / lo), $MachinePrecision] + 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 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* 14.3%

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

5. Simplified 14.3%

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

   1. expm1-log1p-u 14.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)\right)\right)} \]

   2. expm1-undefine 14.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 + \left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)\right)} - 1} \]

   3. add-sqr-sqrt 5.4%

      \[\leadsto e^{\mathsf{log1p}\left(1 + \color{blue}{\sqrt{-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}} \cdot \sqrt{-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}}}\right)} - 1 \]

   4. sqrt-unprod 13.8%

      \[\leadsto e^{\mathsf{log1p}\left(1 + \color{blue}{\sqrt{\left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right) \cdot \left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}}\right)} - 1 \]

   5. sqr-neg 13.8%

      \[\leadsto e^{\mathsf{log1p}\left(1 + \sqrt{\color{blue}{\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo} \cdot \frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}}}\right)} - 1 \]

   6. sqrt-unprod 8.4%

      \[\leadsto e^{\mathsf{log1p}\left(1 + \color{blue}{\sqrt{\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}} \cdot \sqrt{\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}}}\right)} - 1 \]

   7. add-sqr-sqrt 13.0%

      \[\leadsto e^{\mathsf{log1p}\left(1 + \color{blue}{\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}}\right)} - 1 \]

7. Applied egg-rr 13.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 + \frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-define 13.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)\right)} \]

   2. associate-+r- 13.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{\color{blue}{\left(x + hi \cdot \frac{x - hi}{lo}\right) - hi}}{lo}\right)\right) \]

   3. div-sub 13.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \color{blue}{\left(\frac{x + hi \cdot \frac{x - hi}{lo}}{lo} - \frac{hi}{lo}\right)}\right)\right) \]

   4. associate-*r/ 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \left(\frac{x + \color{blue}{\frac{hi \cdot \left(x - hi\right)}{lo}}}{lo} - \frac{hi}{lo}\right)\right)\right) \]

   5. div-sub 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \color{blue}{\frac{\left(x + \frac{hi \cdot \left(x - hi\right)}{lo}\right) - hi}{lo}}\right)\right) \]

   6. associate-*r/ 13.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{\left(x + \color{blue}{hi \cdot \frac{x - hi}{lo}}\right) - hi}{lo}\right)\right) \]

   7. associate-+r- 13.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{\color{blue}{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}}{lo}\right)\right) \]

   8. fma-neg 19.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \color{blue}{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, -hi\right)}}{lo}\right)\right) \]

9. Simplified 19.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \mathsf{fma}\left(hi, \frac{x - hi}{lo}, -hi\right)}{lo}\right)\right)} \]
10. Step-by-step derivation

    1. fma-neg 13.0%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \color{blue}{\left(hi \cdot \frac{x - hi}{lo} - hi\right)}}{lo}\right)\right) \]

    2. add-sqr-sqrt 4.6%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \color{blue}{\sqrt{hi \cdot \frac{x - hi}{lo} - hi} \cdot \sqrt{hi \cdot \frac{x - hi}{lo} - hi}}}{lo}\right)\right) \]

    3. sqrt-unprod 0.0%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \color{blue}{\sqrt{\left(hi \cdot \frac{x - hi}{lo} - hi\right) \cdot \left(hi \cdot \frac{x - hi}{lo} - hi\right)}}}{lo}\right)\right) \]

    4. pow2 0.0%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{\color{blue}{{\left(hi \cdot \frac{x - hi}{lo} - hi\right)}^{2}}}}{lo}\right)\right) \]

    5. fma-neg 0.0%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{{\color{blue}{\left(\mathsf{fma}\left(hi, \frac{x - hi}{lo}, -hi\right)\right)}}^{2}}}{lo}\right)\right) \]

    6. sub-neg 0.0%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{{\left(\mathsf{fma}\left(hi, \frac{\color{blue}{x + \left(-hi\right)}}{lo}, -hi\right)\right)}^{2}}}{lo}\right)\right) \]

    7. add-sqr-sqrt 0.0%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{{\left(\mathsf{fma}\left(hi, \frac{x + \color{blue}{\sqrt{-hi} \cdot \sqrt{-hi}}}{lo}, -hi\right)\right)}^{2}}}{lo}\right)\right) \]

    8. sqrt-unprod 0.0%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{{\left(\mathsf{fma}\left(hi, \frac{x + \color{blue}{\sqrt{\left(-hi\right) \cdot \left(-hi\right)}}}{lo}, -hi\right)\right)}^{2}}}{lo}\right)\right) \]

    9. sqr-neg 0.0%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{{\left(\mathsf{fma}\left(hi, \frac{x + \sqrt{\color{blue}{hi \cdot hi}}}{lo}, -hi\right)\right)}^{2}}}{lo}\right)\right) \]

    10. sqrt-unprod 0.0%

        \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{{\left(\mathsf{fma}\left(hi, \frac{x + \color{blue}{\sqrt{hi} \cdot \sqrt{hi}}}{lo}, -hi\right)\right)}^{2}}}{lo}\right)\right) \]

    11. add-sqr-sqrt 0.0%

        \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{{\left(\mathsf{fma}\left(hi, \frac{x + \color{blue}{hi}}{lo}, -hi\right)\right)}^{2}}}{lo}\right)\right) \]

    12. add-sqr-sqrt 0.0%

        \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{{\left(\mathsf{fma}\left(hi, \frac{x + hi}{lo}, \color{blue}{\sqrt{-hi} \cdot \sqrt{-hi}}\right)\right)}^{2}}}{lo}\right)\right) \]

    13. sqrt-unprod 0.0%

        \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{{\left(\mathsf{fma}\left(hi, \frac{x + hi}{lo}, \color{blue}{\sqrt{\left(-hi\right) \cdot \left(-hi\right)}}\right)\right)}^{2}}}{lo}\right)\right) \]

    14. sqr-neg 0.0%

        \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{{\left(\mathsf{fma}\left(hi, \frac{x + hi}{lo}, \sqrt{\color{blue}{hi \cdot hi}}\right)\right)}^{2}}}{lo}\right)\right) \]

    15. sqrt-unprod 0.0%

        \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{{\left(\mathsf{fma}\left(hi, \frac{x + hi}{lo}, \color{blue}{\sqrt{hi} \cdot \sqrt{hi}}\right)\right)}^{2}}}{lo}\right)\right) \]

    16. add-sqr-sqrt 0.0%

        \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{{\left(\mathsf{fma}\left(hi, \frac{x + hi}{lo}, \color{blue}{hi}\right)\right)}^{2}}}{lo}\right)\right) \]

11. Applied egg-rr 0.0%

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \color{blue}{\sqrt{{\left(\mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)\right)}^{2}}}}{lo}\right)\right) \]
12. Step-by-step derivation

    1. unpow2 0.0%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \sqrt{\color{blue}{\mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right) \cdot \mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)}}}{lo}\right)\right) \]

    2. rem-sqrt-square 19.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \color{blue}{\left|\mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)\right|}}{lo}\right)\right) \]

    3. +-commutative 19.7%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \left|\mathsf{fma}\left(hi, \frac{\color{blue}{hi + x}}{lo}, hi\right)\right|}{lo}\right)\right) \]

13. Simplified 19.7%

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \color{blue}{\left|\mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi\right)\right|}}{lo}\right)\right) \]

14. Taylor expanded in x around 0 0.7%

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

    1. associate-*r/ 13.8%

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

    2. +-commutative 13.8%

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

    3. fma-define 19.7%

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

16. Simplified 19.7%

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

17. Final simplification 19.7%

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

Alternative 2: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[(N[(x + N[(hi * N[(N[(hi + x), $MachinePrecision] / lo), $MachinePrecision] + 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 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* 14.3%

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

5. Simplified 14.3%

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

   1. expm1-log1p-u 14.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)\right)\right)} \]

   2. expm1-undefine 14.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 + \left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)\right)} - 1} \]

   3. add-sqr-sqrt 5.4%

      \[\leadsto e^{\mathsf{log1p}\left(1 + \color{blue}{\sqrt{-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}} \cdot \sqrt{-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}}}\right)} - 1 \]

   4. sqrt-unprod 13.8%

      \[\leadsto e^{\mathsf{log1p}\left(1 + \color{blue}{\sqrt{\left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right) \cdot \left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}}\right)} - 1 \]

   5. sqr-neg 13.8%

      \[\leadsto e^{\mathsf{log1p}\left(1 + \sqrt{\color{blue}{\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo} \cdot \frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}}}\right)} - 1 \]

   6. sqrt-unprod 8.4%

      \[\leadsto e^{\mathsf{log1p}\left(1 + \color{blue}{\sqrt{\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}} \cdot \sqrt{\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}}}\right)} - 1 \]

   7. add-sqr-sqrt 13.0%

      \[\leadsto e^{\mathsf{log1p}\left(1 + \color{blue}{\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}}\right)} - 1 \]

7. Applied egg-rr 13.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 + \frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-define 13.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)\right)} \]

   2. associate-+r- 13.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{\color{blue}{\left(x + hi \cdot \frac{x - hi}{lo}\right) - hi}}{lo}\right)\right) \]

   3. div-sub 13.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \color{blue}{\left(\frac{x + hi \cdot \frac{x - hi}{lo}}{lo} - \frac{hi}{lo}\right)}\right)\right) \]

   4. associate-*r/ 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \left(\frac{x + \color{blue}{\frac{hi \cdot \left(x - hi\right)}{lo}}}{lo} - \frac{hi}{lo}\right)\right)\right) \]

   5. div-sub 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \color{blue}{\frac{\left(x + \frac{hi \cdot \left(x - hi\right)}{lo}\right) - hi}{lo}}\right)\right) \]

   6. associate-*r/ 13.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{\left(x + \color{blue}{hi \cdot \frac{x - hi}{lo}}\right) - hi}{lo}\right)\right) \]

   7. associate-+r- 13.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{\color{blue}{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}}{lo}\right)\right) \]

   8. fma-neg 19.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \color{blue}{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, -hi\right)}}{lo}\right)\right) \]

9. Simplified 19.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \mathsf{fma}\left(hi, \frac{x - hi}{lo}, -hi\right)}{lo}\right)\right)} \]
10. Step-by-step derivation

    1. expm1-log1p-u 19.2%

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

    2. div-inv 19.2%

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

    3. fma-neg 13.2%

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

    4. rem-cube-cbrt 13.2%

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

    5. rem-cube-cbrt 13.2%

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

    6. add-sqr-sqrt 8.4%

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

    7. sqrt-unprod 13.8%

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

    8. sqr-neg 13.8%

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

    9. sqrt-unprod 5.4%

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

11. Applied egg-rr 19.2%

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

12. Final simplification 19.2%

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

Alternative 3: 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* 14.3%

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

5. Simplified 14.3%

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

   1. unsub-neg 14.3%

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

   2. add-sqr-sqrt 8.9%

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

   3. sqrt-unprod 13.7%

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

   4. sqr-neg 13.7%

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

   5. sqrt-unprod 4.8%

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

   6. add-sqr-sqrt 13.2%

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

   7. expm1-log1p-u 13.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)\right)\right)} \]

   8. add-sqr-sqrt 4.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 - \color{blue}{\sqrt{-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}} \cdot \sqrt{-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}}}\right)\right) \]

   9. sqrt-unprod 13.5%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(1 - \color{blue}{\sqrt{\left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right) \cdot \left(-\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}}\right)\right) \]

7. Applied egg-rr 13.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)\right)} \]
8. Step-by-step derivation

   1. expm1-log1p-u 13.2%

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

9. Applied egg-rr 13.2%

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

    1. sub-neg 13.2%

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

    2. neg-mul-1 13.2%

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

    3. *-commutative 13.2%

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

    4. distribute-lft-in 19.2%

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

11. Simplified 19.2%

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

12. Final simplification 19.2%

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

Alternative 4: 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 5: 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(Float64(-lo) / 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 lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

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

Alternative 6: 18.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 18.6%

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

4. Final simplification 18.6%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))