xlohi (overflows)

Percentage Accurate: 3.1% → 22.3%

Time: 9.9s

Alternatives: 5

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: 22.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(-0.5 \cdot {\left(\frac{hi}{lo}\right)}^{2} - hi \cdot \mathsf{fma}\left(-0.5, \frac{x}{lo \cdot lo} \cdot -4, \frac{1}{lo}\right)\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (expm1
  (-
   (* -0.5 (pow (/ hi lo) 2.0))
   (* hi (fma -0.5 (* (/ x (* lo lo)) -4.0) (/ 1.0 lo))))))

C

double code(double lo, double hi, double x) {
	return expm1(((-0.5 * pow((hi / lo), 2.0)) - (hi * fma(-0.5, ((x / (lo * lo)) * -4.0), (1.0 / lo)))));
}

Julia

function code(lo, hi, x)
	return expm1(Float64(Float64(-0.5 * (Float64(hi / lo) ^ 2.0)) - Float64(hi * fma(-0.5, Float64(Float64(x / Float64(lo * lo)) * -4.0), Float64(1.0 / lo)))))
end

Wolfram

code[lo_, hi_, x_] := N[(Exp[N[(N[(-0.5 * N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(hi * N[(-0.5 * N[(N[(x / N[(lo * lo), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] + N[(1.0 / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. expm1-log1p-u 18.8%

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

   2. associate-/r* 18.8%

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

   3. sub-div 18.8%

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

6. Applied egg-rr 18.8%

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

7. Taylor expanded in lo around -inf 0.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{-0.5 \cdot \frac{{\left(\frac{x}{hi} + 1\right)}^{2} \cdot {hi}^{2}}{{\left(1 - \frac{x}{hi}\right)}^{2} \cdot {lo}^{2}} + \left(-1 \cdot \log \left(\frac{-1}{lo}\right) + \left(-1 \cdot \frac{\left(\frac{x}{hi} + 1\right) \cdot hi}{\left(1 - \frac{x}{hi}\right) \cdot lo} + \log \left(--1 \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. fma-def 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(\frac{x}{hi} + 1\right)}^{2} \cdot {hi}^{2}}{{\left(1 - \frac{x}{hi}\right)}^{2} \cdot {lo}^{2}}, -1 \cdot \log \left(\frac{-1}{lo}\right) + \left(-1 \cdot \frac{\left(\frac{x}{hi} + 1\right) \cdot hi}{\left(1 - \frac{x}{hi}\right) \cdot lo} + \log \left(--1 \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)\right)}\right) \]

   2. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \frac{{\left(\frac{x}{hi} + 1\right)}^{2} \cdot \color{blue}{\left(hi \cdot hi\right)}}{{\left(1 - \frac{x}{hi}\right)}^{2} \cdot {lo}^{2}}, -1 \cdot \log \left(\frac{-1}{lo}\right) + \left(-1 \cdot \frac{\left(\frac{x}{hi} + 1\right) \cdot hi}{\left(1 - \frac{x}{hi}\right) \cdot lo} + \log \left(--1 \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)\right)\right) \]

   3. times-frac 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \color{blue}{\frac{{\left(\frac{x}{hi} + 1\right)}^{2}}{{\left(1 - \frac{x}{hi}\right)}^{2}} \cdot \frac{hi \cdot hi}{{lo}^{2}}}, -1 \cdot \log \left(\frac{-1}{lo}\right) + \left(-1 \cdot \frac{\left(\frac{x}{hi} + 1\right) \cdot hi}{\left(1 - \frac{x}{hi}\right) \cdot lo} + \log \left(--1 \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)\right)\right) \]

   4. +-commutative 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \frac{{\color{blue}{\left(1 + \frac{x}{hi}\right)}}^{2}}{{\left(1 - \frac{x}{hi}\right)}^{2}} \cdot \frac{hi \cdot hi}{{lo}^{2}}, -1 \cdot \log \left(\frac{-1}{lo}\right) + \left(-1 \cdot \frac{\left(\frac{x}{hi} + 1\right) \cdot hi}{\left(1 - \frac{x}{hi}\right) \cdot lo} + \log \left(--1 \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)\right)\right) \]

   5. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \frac{{\left(1 + \frac{x}{hi}\right)}^{2}}{{\left(1 - \frac{x}{hi}\right)}^{2}} \cdot \frac{hi \cdot hi}{\color{blue}{lo \cdot lo}}, -1 \cdot \log \left(\frac{-1}{lo}\right) + \left(-1 \cdot \frac{\left(\frac{x}{hi} + 1\right) \cdot hi}{\left(1 - \frac{x}{hi}\right) \cdot lo} + \log \left(--1 \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)\right)\right) \]

   6. times-frac 19.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \frac{{\left(1 + \frac{x}{hi}\right)}^{2}}{{\left(1 - \frac{x}{hi}\right)}^{2}} \cdot \color{blue}{\left(\frac{hi}{lo} \cdot \frac{hi}{lo}\right)}, -1 \cdot \log \left(\frac{-1}{lo}\right) + \left(-1 \cdot \frac{\left(\frac{x}{hi} + 1\right) \cdot hi}{\left(1 - \frac{x}{hi}\right) \cdot lo} + \log \left(--1 \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right)\right)\right) \]

   7. +-commutative 19.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, \frac{{\left(1 + \frac{x}{hi}\right)}^{2}}{{\left(1 - \frac{x}{hi}\right)}^{2}} \cdot \left(\frac{hi}{lo} \cdot \frac{hi}{lo}\right), \color{blue}{\left(-1 \cdot \frac{\left(\frac{x}{hi} + 1\right) \cdot hi}{\left(1 - \frac{x}{hi}\right) \cdot lo} + \log \left(--1 \cdot \frac{1 - \frac{x}{hi}}{hi}\right)\right) + -1 \cdot \log \left(\frac{-1}{lo}\right)}\right)\right) \]

9. Simplified 19.6%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(1 + \frac{x}{hi}\right)}^{2}}{{\left(1 - \frac{x}{hi}\right)}^{2}} \cdot \left(\frac{hi}{lo} \cdot \frac{hi}{lo}\right), \left(\log \left(\frac{1 - \frac{x}{hi}}{hi}\right) - \frac{1 + \frac{x}{hi}}{1 - \frac{x}{hi}} \cdot \frac{hi}{lo}\right) - \log \left(\frac{-1}{lo}\right)\right)}\right) \]

10. Taylor expanded in hi around -inf 0.0%

    \[\leadsto \mathsf{expm1}\left(\color{blue}{-1 \cdot \left(\left(-0.5 \cdot \left(-2 \cdot \frac{x}{{lo}^{2}} - 2 \cdot \frac{x}{{lo}^{2}}\right) + \frac{1}{lo}\right) \cdot hi\right) + -0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}}}\right) \]
11. Step-by-step derivation

    1. +-commutative 0.0%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{-0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + -1 \cdot \left(\left(-0.5 \cdot \left(-2 \cdot \frac{x}{{lo}^{2}} - 2 \cdot \frac{x}{{lo}^{2}}\right) + \frac{1}{lo}\right) \cdot hi\right)}\right) \]

    2. mul-1-neg 0.0%

       \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} + \color{blue}{\left(-\left(-0.5 \cdot \left(-2 \cdot \frac{x}{{lo}^{2}} - 2 \cdot \frac{x}{{lo}^{2}}\right) + \frac{1}{lo}\right) \cdot hi\right)}\right) \]

    3. unsub-neg 0.0%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{-0.5 \cdot \frac{{hi}^{2}}{{lo}^{2}} - \left(-0.5 \cdot \left(-2 \cdot \frac{x}{{lo}^{2}} - 2 \cdot \frac{x}{{lo}^{2}}\right) + \frac{1}{lo}\right) \cdot hi}\right) \]

    4. unpow2 0.0%

       \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} - \left(-0.5 \cdot \left(-2 \cdot \frac{x}{{lo}^{2}} - 2 \cdot \frac{x}{{lo}^{2}}\right) + \frac{1}{lo}\right) \cdot hi\right) \]

    5. unpow2 0.0%

       \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \frac{hi \cdot hi}{\color{blue}{lo \cdot lo}} - \left(-0.5 \cdot \left(-2 \cdot \frac{x}{{lo}^{2}} - 2 \cdot \frac{x}{{lo}^{2}}\right) + \frac{1}{lo}\right) \cdot hi\right) \]

    6. times-frac 22.2%

       \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \color{blue}{\left(\frac{hi}{lo} \cdot \frac{hi}{lo}\right)} - \left(-0.5 \cdot \left(-2 \cdot \frac{x}{{lo}^{2}} - 2 \cdot \frac{x}{{lo}^{2}}\right) + \frac{1}{lo}\right) \cdot hi\right) \]

    7. unpow2 22.2%

       \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \color{blue}{{\left(\frac{hi}{lo}\right)}^{2}} - \left(-0.5 \cdot \left(-2 \cdot \frac{x}{{lo}^{2}} - 2 \cdot \frac{x}{{lo}^{2}}\right) + \frac{1}{lo}\right) \cdot hi\right) \]

    8. *-commutative 22.2%

       \[\leadsto \mathsf{expm1}\left(-0.5 \cdot {\left(\frac{hi}{lo}\right)}^{2} - \color{blue}{hi \cdot \left(-0.5 \cdot \left(-2 \cdot \frac{x}{{lo}^{2}} - 2 \cdot \frac{x}{{lo}^{2}}\right) + \frac{1}{lo}\right)}\right) \]

    9. fma-def 22.2%

       \[\leadsto \mathsf{expm1}\left(-0.5 \cdot {\left(\frac{hi}{lo}\right)}^{2} - hi \cdot \color{blue}{\mathsf{fma}\left(-0.5, -2 \cdot \frac{x}{{lo}^{2}} - 2 \cdot \frac{x}{{lo}^{2}}, \frac{1}{lo}\right)}\right) \]

    10. distribute-rgt-out-- 22.2%

        \[\leadsto \mathsf{expm1}\left(-0.5 \cdot {\left(\frac{hi}{lo}\right)}^{2} - hi \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{x}{{lo}^{2}} \cdot \left(-2 - 2\right)}, \frac{1}{lo}\right)\right) \]

    11. unpow2 22.2%

        \[\leadsto \mathsf{expm1}\left(-0.5 \cdot {\left(\frac{hi}{lo}\right)}^{2} - hi \cdot \mathsf{fma}\left(-0.5, \frac{x}{\color{blue}{lo \cdot lo}} \cdot \left(-2 - 2\right), \frac{1}{lo}\right)\right) \]

    12. metadata-eval 22.2%

        \[\leadsto \mathsf{expm1}\left(-0.5 \cdot {\left(\frac{hi}{lo}\right)}^{2} - hi \cdot \mathsf{fma}\left(-0.5, \frac{x}{lo \cdot lo} \cdot \color{blue}{-4}, \frac{1}{lo}\right)\right) \]

12. Simplified 22.2%

    \[\leadsto \mathsf{expm1}\left(\color{blue}{-0.5 \cdot {\left(\frac{hi}{lo}\right)}^{2} - hi \cdot \mathsf{fma}\left(-0.5, \frac{x}{lo \cdot lo} \cdot -4, \frac{1}{lo}\right)}\right) \]

13. Final simplification 22.2%

    \[\leadsto \mathsf{expm1}\left(-0.5 \cdot {\left(\frac{hi}{lo}\right)}^{2} - hi \cdot \mathsf{fma}\left(-0.5, \frac{x}{lo \cdot lo} \cdot -4, \frac{1}{lo}\right)\right) \]

Alternative 2: 19.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in hi around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. associate--l+ 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. times-frac 9.2%

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

   6. div-sub 9.2%

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

4. Simplified 9.2%

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

5. Taylor expanded in x around 0 0.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{{lo}^{2}}{{hi}^{2}} - \frac{lo}{hi}} \]
6. Step-by-step derivation

   1. sub-neg 0.0%

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

   2. +-commutative 0.0%

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

   3. mul-1-neg 0.0%

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

   4. unsub-neg 0.0%

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

   5. distribute-neg-frac 0.0%

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

   6. unpow2 0.0%

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

   7. unpow2 0.0%

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

   8. times-frac 9.2%

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

7. Simplified 9.2%

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

   1. *-un-lft-identity 9.2%

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

   2. *-un-lft-identity 9.2%

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

   3. prod-diff 9.2%

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

   4. add-sqr-sqrt 9.2%

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

   5. sqrt-unprod 3.1%

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

   6. sqr-neg 3.1%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 1.6%

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

   9. pow2 1.6%

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

9. Applied egg-rr 10.3%

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

    1. +-commutative 10.3%

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

    2. fma-udef 10.3%

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

    3. *-rgt-identity 10.3%

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

    4. *-rgt-identity 10.3%

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

    5. count-2 10.3%

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

    6. fma-udef 10.3%

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

    7. *-lft-identity 10.3%

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

    8. distribute-lft-neg-in 10.3%

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

    9. cancel-sign-sub-inv 10.3%

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

    10. *-rgt-identity 10.3%

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

11. Simplified 10.3%

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

12. Taylor expanded in lo around inf 0.0%

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

    1. unpow2 0.0%

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

    2. unpow2 0.0%

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

    3. times-frac 19.2%

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

    4. unpow2 19.2%

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

14. Simplified 19.2%

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

15. Final simplification 19.2%

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

Alternative 3: 18.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. expm1-log1p-u 18.8%

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

   2. associate-/r* 18.8%

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

   3. sub-div 18.8%

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

6. Applied egg-rr 18.8%

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

   1. expm1-udef 18.8%

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

   2. log1p-udef 18.8%

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

   3. add-exp-log 18.8%

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

   4. associate-*r/ 18.8%

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

   5. sub-div 18.8%

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

8. Applied egg-rr 18.8%

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

9. Final simplification 18.8%

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

Alternative 4: 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. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

5. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

Alternative 5: 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. Taylor expanded in lo around inf 18.7%

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

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023215 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))