xlohi (overflows)

Percentage Accurate: 3.1% → 26.8%

Time: 11.1s

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: 26.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi}{lo + x}\right)}^{-2}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (log1p
  (fabs
   (fma
    0.16666666666666666
    (pow (/ (+ lo x) hi) 3.0)
    (+ (fma 0.5 (pow (/ hi (+ lo x)) -2.0) (/ lo hi)) (/ x hi))))))

C

double code(double lo, double hi, double x) {
	return log1p(fabs(fma(0.16666666666666666, pow(((lo + x) / hi), 3.0), (fma(0.5, pow((hi / (lo + x)), -2.0), (lo / hi)) + (x / hi)))));
}

Julia

function code(lo, hi, x)
	return log1p(abs(fma(0.16666666666666666, (Float64(Float64(lo + x) / hi) ^ 3.0), Float64(fma(0.5, (Float64(hi / Float64(lo + x)) ^ -2.0), Float64(lo / hi)) + Float64(x / hi)))))
end

Wolfram

code[lo_, hi_, x_] := N[Log[1 + N[Abs[N[(0.16666666666666666 * N[Power[N[(N[(lo + x), $MachinePrecision] / hi), $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(0.5 * N[Power[N[(hi / N[(lo + x), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + N[(lo / hi), $MachinePrecision]), $MachinePrecision] + N[(x / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

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

   1. log1p-expm1-u 18.8%

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

5. Applied egg-rr 18.8%

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

   1. add-sqr-sqrt 18.8%

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

   2. sqrt-unprod 18.8%

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

   3. pow2 18.8%

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

   4. expm1-log1p-u 18.8%

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

   5. log1p-expm1-u 18.8%

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

   6. sub-neg 18.8%

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

   7. add-sqr-sqrt 18.8%

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

   8. sqrt-unprod 3.1%

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

   9. sqr-neg 3.1%

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

   10. sqrt-unprod 0.0%

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

   11. add-sqr-sqrt 26.8%

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

7. Applied egg-rr 26.8%

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

   1. unpow2 26.8%

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

   2. rem-sqrt-square 26.8%

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

   3. +-commutative 26.8%

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

9. Simplified 26.8%

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

10. Taylor expanded in hi around inf 0.0%

    \[\leadsto \mathsf{log1p}\left(\left|\color{blue}{0.16666666666666666 \cdot \frac{{\left(lo + x\right)}^{3}}{{hi}^{3}} + \left(0.5 \cdot \frac{{\left(lo + x\right)}^{2}}{{hi}^{2}} + \left(\frac{lo}{hi} + \frac{x}{hi}\right)\right)}\right|\right) \]
11. Step-by-step derivation

    1. fma-def 0.0%

       \[\leadsto \mathsf{log1p}\left(\left|\color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(lo + x\right)}^{3}}{{hi}^{3}}, 0.5 \cdot \frac{{\left(lo + x\right)}^{2}}{{hi}^{2}} + \left(\frac{lo}{hi} + \frac{x}{hi}\right)\right)}\right|\right) \]

    2. cube-div 0.0%

       \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left(\frac{lo + x}{hi}\right)}^{3}}, 0.5 \cdot \frac{{\left(lo + x\right)}^{2}}{{hi}^{2}} + \left(\frac{lo}{hi} + \frac{x}{hi}\right)\right)\right|\right) \]

    3. associate-+r+ 0.0%

       \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \color{blue}{\left(0.5 \cdot \frac{{\left(lo + x\right)}^{2}}{{hi}^{2}} + \frac{lo}{hi}\right) + \frac{x}{hi}}\right)\right|\right) \]

    4. fma-def 0.0%

       \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{\left(lo + x\right)}^{2}}{{hi}^{2}}, \frac{lo}{hi}\right)} + \frac{x}{hi}\right)\right|\right) \]

    5. unpow2 0.0%

       \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{\left(lo + x\right) \cdot \left(lo + x\right)}}{{hi}^{2}}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]

    6. unpow2 0.0%

       \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\left(lo + x\right) \cdot \left(lo + x\right)}{\color{blue}{hi \cdot hi}}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]

    7. times-frac 26.9%

       \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{lo + x}{hi} \cdot \frac{lo + x}{hi}}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]

    8. *-lft-identity 26.9%

       \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, \frac{\color{blue}{1 \cdot \left(lo + x\right)}}{hi} \cdot \frac{lo + x}{hi}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]

    9. associate-*l/ 26.9%

       \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\left(\frac{1}{hi} \cdot \left(lo + x\right)\right)} \cdot \frac{lo + x}{hi}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]

    10. associate-/r/ 26.9%

        \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{\frac{1}{\frac{hi}{lo + x}}} \cdot \frac{lo + x}{hi}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]

    11. unpow-1 26.9%

        \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi}{lo + x}\right)}^{-1}} \cdot \frac{lo + x}{hi}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]

    12. *-lft-identity 26.9%

        \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi}{lo + x}\right)}^{-1} \cdot \frac{\color{blue}{1 \cdot \left(lo + x\right)}}{hi}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]

    13. associate-*l/ 26.9%

        \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi}{lo + x}\right)}^{-1} \cdot \color{blue}{\left(\frac{1}{hi} \cdot \left(lo + x\right)\right)}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]

    14. associate-/r/ 26.9%

        \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi}{lo + x}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{hi}{lo + x}}}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]

    15. unpow-1 26.9%

        \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi}{lo + x}\right)}^{-1} \cdot \color{blue}{{\left(\frac{hi}{lo + x}\right)}^{-1}}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]

    16. pow-sqr 26.9%

        \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{hi}{lo + x}\right)}^{\left(2 \cdot -1\right)}}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]

    17. metadata-eval 26.9%

        \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi}{lo + x}\right)}^{\color{blue}{-2}}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]

12. Simplified 26.9%

    \[\leadsto \mathsf{log1p}\left(\left|\color{blue}{\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi}{lo + x}\right)}^{-2}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)}\right|\right) \]

13. Final simplification 26.9%

    \[\leadsto \mathsf{log1p}\left(\left|\mathsf{fma}\left(0.16666666666666666, {\left(\frac{lo + x}{hi}\right)}^{3}, \mathsf{fma}\left(0.5, {\left(\frac{hi}{lo + x}\right)}^{-2}, \frac{lo}{hi}\right) + \frac{x}{hi}\right)\right|\right) \]
14. Add Preprocessing

Alternative 2: 26.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\left|\mathsf{expm1}\left(\frac{lo + \left|x\right|}{hi}\right)\right|\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (log1p (fabs (expm1 (/ (+ lo (fabs x)) hi)))))

C

double code(double lo, double hi, double x) {
	return log1p(fabs(expm1(((lo + fabs(x)) / hi))));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p(Math.abs(Math.expm1(((lo + Math.abs(x)) / hi))));
}

Python

def code(lo, hi, x):
	return math.log1p(math.fabs(math.expm1(((lo + math.fabs(x)) / hi))))

Julia

function code(lo, hi, x)
	return log1p(abs(expm1(Float64(Float64(lo + abs(x)) / hi))))
end

Wolfram

code[lo_, hi_, x_] := N[Log[1 + N[Abs[N[(Exp[N[(N[(lo + N[Abs[x], $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $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. Step-by-step derivation

   1. log1p-expm1-u 18.8%

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

5. Applied egg-rr 18.8%

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

   1. add-sqr-sqrt 18.8%

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

   2. sqrt-unprod 18.8%

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

   3. pow2 18.8%

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

   4. expm1-log1p-u 18.8%

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

   5. log1p-expm1-u 18.8%

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

   6. sub-neg 18.8%

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

   7. add-sqr-sqrt 18.8%

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

   8. sqrt-unprod 3.1%

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

   9. sqr-neg 3.1%

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

   10. sqrt-unprod 0.0%

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

   11. add-sqr-sqrt 26.8%

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

7. Applied egg-rr 26.8%

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

   1. unpow2 26.8%

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

   2. rem-sqrt-square 26.8%

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

   3. +-commutative 26.8%

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

9. Simplified 26.8%

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

    1. add-sqr-sqrt 14.2%

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

    2. sqrt-unprod 20.4%

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

    3. pow2 20.4%

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

11. Applied egg-rr 20.4%

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

    1. unpow2 20.4%

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

    2. rem-sqrt-square 26.8%

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

13. Simplified 26.8%

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

14. Final simplification 26.8%

    \[\leadsto \mathsf{log1p}\left(\left|\mathsf{expm1}\left(\frac{lo + \left|x\right|}{hi}\right)\right|\right) \]
15. Add Preprocessing

Alternative 3: 26.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\left|\mathsf{expm1}\left(\frac{lo}{hi}\right)\right|\right) \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (log1p (fabs (expm1 (/ lo hi)))))

C

double code(double lo, double hi, double x) {
	return log1p(fabs(expm1((lo / hi))));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.log1p(Math.abs(Math.expm1((lo / hi))));
}

Python

def code(lo, hi, x):
	return math.log1p(math.fabs(math.expm1((lo / hi))))

Julia

function code(lo, hi, x)
	return log1p(abs(expm1(Float64(lo / hi))))
end

Wolfram

code[lo_, hi_, x_] := N[Log[1 + N[Abs[N[(Exp[N[(lo / hi), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $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. Step-by-step derivation

   1. log1p-expm1-u 18.8%

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

5. Applied egg-rr 18.8%

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

   1. add-sqr-sqrt 18.8%

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

   2. sqrt-unprod 18.8%

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

   3. pow2 18.8%

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

   4. expm1-log1p-u 18.8%

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

   5. log1p-expm1-u 18.8%

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

   6. sub-neg 18.8%

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

   7. add-sqr-sqrt 18.8%

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

   8. sqrt-unprod 3.1%

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

   9. sqr-neg 3.1%

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

   10. sqrt-unprod 0.0%

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

   11. add-sqr-sqrt 26.8%

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

7. Applied egg-rr 26.8%

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

   1. unpow2 26.8%

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

   2. rem-sqrt-square 26.8%

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

   3. +-commutative 26.8%

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

9. Simplified 26.8%

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

10. Taylor expanded in x around 0 26.8%

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

    1. expm1-def 26.8%

       \[\leadsto \mathsf{log1p}\left(\left|\color{blue}{\mathsf{expm1}\left(\frac{lo}{hi}\right)}\right|\right) \]

12. Simplified 26.8%

    \[\leadsto \mathsf{log1p}\left(\left|\color{blue}{\mathsf{expm1}\left(\frac{lo}{hi}\right)}\right|\right) \]

13. Final simplification 26.8%

    \[\leadsto \mathsf{log1p}\left(\left|\mathsf{expm1}\left(\frac{lo}{hi}\right)\right|\right) \]
14. 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)))