xlohi (overflows)

Percentage Accurate: 3.1% → 95.2%

Time: 8.6s

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: 95.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(lo, 1 - {\left(\frac{hi}{lo \cdot lo} \cdot \left(x - hi\right)\right)}^{2}, hi - x\right)}{lo \cdot \mathsf{fma}\left(\frac{hi}{lo}, \frac{x - hi}{lo}, 1\right)}\right)\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (expm1
  (log1p
   (/
    (fma lo (- 1.0 (pow (* (/ hi (* lo lo)) (- x hi)) 2.0)) (- hi x))
    (* lo (fma (/ hi lo) (/ (- x hi) lo) 1.0))))))

C

double code(double lo, double hi, double x) {
	return expm1(log1p((fma(lo, (1.0 - pow(((hi / (lo * lo)) * (x - hi)), 2.0)), (hi - x)) / (lo * fma((hi / lo), ((x - hi) / lo), 1.0)))));
}

Julia

function code(lo, hi, x)
	return expm1(log1p(Float64(fma(lo, Float64(1.0 - (Float64(Float64(hi / Float64(lo * lo)) * Float64(x - hi)) ^ 2.0)), Float64(hi - x)) / Float64(lo * fma(Float64(hi / lo), Float64(Float64(x - hi) / lo), 1.0)))))
end

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. associate-*r* 0.0%

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

   6. *-commutative 0.0%

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

   7. associate-*r* 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. associate-+r+ 0.0%

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

   12. mul-1-neg 0.0%

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

4. Simplified 18.9%

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

   1. flip-- 18.9%

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

   2. frac-sub 16.1%

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

   3. metadata-eval 16.1%

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

   4. pow2 16.1%

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

6. Applied egg-rr 16.1%

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

8. Simplified 95.4%

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

   1. expm1-log1p-u 95.4%

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

10. Applied egg-rr 95.4%

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

11. Taylor expanded in hi around 0 95.4%

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

12. Final simplification 95.4%

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

Alternative 2: 95.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. associate-*r* 0.0%

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

   6. *-commutative 0.0%

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

   7. associate-*r* 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. associate-+r+ 0.0%

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

   12. mul-1-neg 0.0%

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

4. Simplified 18.9%

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

   1. flip-- 18.9%

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

   2. frac-sub 16.1%

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

   3. metadata-eval 16.1%

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

   4. pow2 16.1%

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

6. Applied egg-rr 16.1%

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

8. Simplified 95.4%

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

9. Taylor expanded in lo around inf 95.4%

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

    1. mul-1-neg 95.4%

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

    2. unsub-neg 95.4%

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

11. Simplified 95.4%

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

12. Final simplification 95.4%

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

Alternative 3: 19.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. associate-*r* 0.0%

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

   6. *-commutative 0.0%

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

   7. associate-*r* 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. associate-+r+ 0.0%

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

   12. mul-1-neg 0.0%

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

4. Simplified 18.9%

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

   1. flip-- 18.9%

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

   2. frac-sub 16.1%

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

   3. metadata-eval 16.1%

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

   4. pow2 16.1%

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

6. Applied egg-rr 16.1%

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

8. Simplified 95.4%

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

9. Taylor expanded in lo around 0 0.0%

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

    1. mul-1-neg 0.0%

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

    2. associate-*l/ 3.1%

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

    3. unpow2 3.1%

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

    4. *-commutative 3.1%

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

    5. distribute-rgt-neg-in 3.1%

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

    6. associate-/r* 19.6%

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

    7. distribute-neg-frac 19.6%

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

11. Simplified 19.6%

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

12. Final simplification 19.6%

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

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 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 9.1%

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

   6. div-sub 9.1%

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

4. Simplified 9.1%

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

5. Taylor expanded in lo around 0 18.8%

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

   1. fma-def 18.8%

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

   2. unpow2 18.8%

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

7. Simplified 18.8%

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

8. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

10. Simplified 18.8%

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

11. 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 2023291 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))