xlohi (overflows)

Percentage Accurate: 3.1% → 22.7%

Time: 8.8s

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: 22.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \mathbf{if}\;lo \leq -1.0902 \cdot 10^{+308}:\\ \;\;\;\;1 - \mathsf{expm1}\left(\left|\log \left(\frac{-hi}{lo}\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} - {t_0}^{2}}{\frac{lo \cdot t_0 + \left(lo - x\right)}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (if (<= lo -1.0902e+308)
     (- 1.0 (expm1 (fabs (log (/ (- hi) lo)))))
     (/
      (- (pow (/ (- x lo) (* hi (/ hi lo))) 2.0) (pow t_0 2.0))
      (/ (+ (* lo t_0) (- lo x)) hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.0902e+308) {
		tmp = 1.0 - expm1(fabs(log((-hi / lo))));
	} else {
		tmp = (pow(((x - lo) / (hi * (hi / lo))), 2.0) - pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / hi);
	}
	return tmp;
}

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.0902e+308) {
		tmp = 1.0 - Math.expm1(Math.abs(Math.log((-hi / lo))));
	} else {
		tmp = (Math.pow(((x - lo) / (hi * (hi / lo))), 2.0) - Math.pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / hi);
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	tmp = 0
	if lo <= -1.0902e+308:
		tmp = 1.0 - math.expm1(math.fabs(math.log((-hi / lo))))
	else:
		tmp = (math.pow(((x - lo) / (hi * (hi / lo))), 2.0) - math.pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / hi)
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	tmp = 0.0
	if (lo <= -1.0902e+308)
		tmp = Float64(1.0 - expm1(abs(log(Float64(Float64(-hi) / lo)))));
	else
		tmp = Float64(Float64((Float64(Float64(x - lo) / Float64(hi * Float64(hi / lo))) ^ 2.0) - (t_0 ^ 2.0)) / Float64(Float64(Float64(lo * t_0) + Float64(lo - x)) / hi));
	end
	return tmp
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, If[LessEqual[lo, -1.0902e+308], N[(1.0 - N[(Exp[N[Abs[N[Log[N[((-hi) / lo), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] / N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(lo * t$95$0), $MachinePrecision] + N[(lo - x), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.09019999999999994e308

   1. Initial program 3.1%

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

   2. Taylor expanded in lo around inf 10.6%

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

      1. +-commutative 10.6%

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

      2. associate--l+ 10.6%

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

      3. associate-*r/ 10.6%

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

      4. associate-*r/ 10.6%

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

      5. div-sub 10.6%

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

      6. distribute-lft-out-- 10.6%

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

      7. associate-*r/ 10.6%

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

      8. mul-1-neg 10.6%

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

      9. unsub-neg 10.6%

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

   4. Simplified 10.6%

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

      1. expm1-log1p-u 10.6%

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

   6. Applied egg-rr 10.6%

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

   7. Taylor expanded in hi around inf 18.9%

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

      1. +-commutative 18.9%

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

      2. distribute-neg-frac 18.9%

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

      3. metadata-eval 18.9%

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

      4. mul-1-neg 18.9%

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

      5. log-rec 18.9%

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

      6. remove-double-neg 18.9%

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

   9. Simplified 18.9%

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

       1. add-sqr-sqrt 8.2%

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

       2. sqrt-unprod 20.1%

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

       3. pow2 20.1%

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

       4. sum-log 20.1%

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

   11. Applied egg-rr 20.1%

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

       1. unpow2 20.1%

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

       2. rem-sqrt-square 20.1%

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

       3. associate-*l/ 20.1%

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

       4. neg-mul-1 20.1%

          \[\leadsto 1 - \mathsf{expm1}\left(\left|\log \left(\frac{\color{blue}{-hi}}{lo}\right)\right|\right) \]

   13. Simplified 20.1%

       \[\leadsto 1 - \mathsf{expm1}\left(\color{blue}{\left|\log \left(\frac{-hi}{lo}\right)\right|}\right) \]

   if -1.09019999999999994e308 < lo

   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 18.3%

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

      6. div-sub 18.3%

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

   4. Simplified 18.3%

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

      1. flip-+ 18.3%

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

      2. pow2 18.3%

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

      3. *-commutative 18.3%

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

      4. clear-num 18.3%

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

      5. frac-times 67.3%

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

      6. *-un-lft-identity 67.3%

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

      7. pow2 67.3%

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

      8. associate-*r/ 67.3%

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

      9. sub-div 46.3%

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

   6. Applied egg-rr 46.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{\frac{hi}{lo} \cdot hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{\frac{x - lo}{hi} \cdot lo - \left(x - lo\right)}{hi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.0902 \cdot 10^{+308}:\\ \;\;\;\;1 - \mathsf{expm1}\left(\left|\log \left(\frac{-hi}{lo}\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \]

Alternative 2: 22.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \mathbf{if}\;lo \leq -1.0902 \cdot 10^{+308}:\\ \;\;\;\;1 - \left|\frac{hi}{lo} + 1\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} - {t_0}^{2}}{\frac{lo \cdot t_0 + \left(lo - x\right)}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (if (<= lo -1.0902e+308)
     (- 1.0 (fabs (+ (/ hi lo) 1.0)))
     (/
      (- (pow (/ (- x lo) (* hi (/ hi lo))) 2.0) (pow t_0 2.0))
      (/ (+ (* lo t_0) (- lo x)) hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.0902e+308) {
		tmp = 1.0 - fabs(((hi / lo) + 1.0));
	} else {
		tmp = (pow(((x - lo) / (hi * (hi / lo))), 2.0) - pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / hi);
	}
	return tmp;
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - lo) / hi
    if (lo <= (-1.0902d+308)) then
        tmp = 1.0d0 - abs(((hi / lo) + 1.0d0))
    else
        tmp = ((((x - lo) / (hi * (hi / lo))) ** 2.0d0) - (t_0 ** 2.0d0)) / (((lo * t_0) + (lo - x)) / hi)
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.0902e+308) {
		tmp = 1.0 - Math.abs(((hi / lo) + 1.0));
	} else {
		tmp = (Math.pow(((x - lo) / (hi * (hi / lo))), 2.0) - Math.pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / hi);
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	tmp = 0
	if lo <= -1.0902e+308:
		tmp = 1.0 - math.fabs(((hi / lo) + 1.0))
	else:
		tmp = (math.pow(((x - lo) / (hi * (hi / lo))), 2.0) - math.pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / hi)
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	tmp = 0.0
	if (lo <= -1.0902e+308)
		tmp = Float64(1.0 - abs(Float64(Float64(hi / lo) + 1.0)));
	else
		tmp = Float64(Float64((Float64(Float64(x - lo) / Float64(hi * Float64(hi / lo))) ^ 2.0) - (t_0 ^ 2.0)) / Float64(Float64(Float64(lo * t_0) + Float64(lo - x)) / hi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	tmp = 0.0;
	if (lo <= -1.0902e+308)
		tmp = 1.0 - abs(((hi / lo) + 1.0));
	else
		tmp = ((((x - lo) / (hi * (hi / lo))) ^ 2.0) - (t_0 ^ 2.0)) / (((lo * t_0) + (lo - x)) / hi);
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, If[LessEqual[lo, -1.0902e+308], N[(1.0 - N[Abs[N[(N[(hi / lo), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] / N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(lo * t$95$0), $MachinePrecision] + N[(lo - x), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.09019999999999994e308

   1. Initial program 3.1%

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

   2. Taylor expanded in lo around inf 10.6%

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

      1. +-commutative 10.6%

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

      2. associate--l+ 10.6%

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

      3. associate-*r/ 10.6%

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

      4. associate-*r/ 10.6%

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

      5. div-sub 10.6%

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

      6. distribute-lft-out-- 10.6%

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

      7. associate-*r/ 10.6%

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

      8. mul-1-neg 10.6%

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

      9. unsub-neg 10.6%

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

   4. Simplified 10.6%

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

      1. expm1-log1p-u 10.6%

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

   6. Applied egg-rr 10.6%

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

   7. Taylor expanded in hi around inf 18.9%

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

      1. +-commutative 18.9%

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

      2. distribute-neg-frac 18.9%

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

      3. metadata-eval 18.9%

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

      4. mul-1-neg 18.9%

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

      5. log-rec 18.9%

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

      6. remove-double-neg 18.9%

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

   9. Simplified 18.9%

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

       1. add-sqr-sqrt 8.2%

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

       2. sqrt-unprod 20.0%

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

       3. pow2 20.0%

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

       4. expm1-udef 20.0%

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

       5. sum-log 20.0%

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

       6. add-exp-log 20.0%

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

   11. Applied egg-rr 20.0%

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

       1. unpow2 20.0%

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

       2. sqr-neg 20.0%

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

       3. rem-sqrt-square 20.0%

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

       4. sub-neg 20.0%

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

       5. metadata-eval 20.0%

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

       6. distribute-neg-in 20.0%

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

       7. associate-*l/ 20.0%

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

       8. neg-mul-1 20.0%

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

       9. distribute-neg-frac 20.0%

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

       10. remove-double-neg 20.0%

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

       11. metadata-eval 20.0%

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

   13. Simplified 20.0%

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

   if -1.09019999999999994e308 < lo

   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 18.3%

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

      6. div-sub 18.3%

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

   4. Simplified 18.3%

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

      1. flip-+ 18.3%

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

      2. pow2 18.3%

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

      3. *-commutative 18.3%

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

      4. clear-num 18.3%

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

      5. frac-times 67.3%

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

      6. *-un-lft-identity 67.3%

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

      7. pow2 67.3%

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

      8. associate-*r/ 67.3%

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

      9. sub-div 46.3%

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

   6. Applied egg-rr 46.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x - lo}{\frac{hi}{lo} \cdot hi}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{\frac{x - lo}{hi} \cdot lo - \left(x - lo\right)}{hi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.0902 \cdot 10^{+308}:\\ \;\;\;\;1 - \left|\frac{hi}{lo} + 1\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \]

Alternative 3: 20.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 9.5%

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

   1. +-commutative 9.5%

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

   2. associate--l+ 9.5%

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

   3. associate-*r/ 9.5%

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

   4. associate-*r/ 9.5%

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

   5. div-sub 9.5%

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

   6. distribute-lft-out-- 9.5%

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

   7. associate-*r/ 9.5%

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

   8. mul-1-neg 9.5%

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

   9. unsub-neg 9.5%

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

4. Simplified 9.5%

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

   1. expm1-log1p-u 9.5%

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

6. Applied egg-rr 9.5%

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

7. Taylor expanded in hi around inf 19.1%

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

   1. +-commutative 19.1%

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

   2. distribute-neg-frac 19.1%

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

   3. metadata-eval 19.1%

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

   4. mul-1-neg 19.1%

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

   5. log-rec 19.1%

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

   6. remove-double-neg 19.1%

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

9. Simplified 19.1%

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

    1. add-sqr-sqrt 9.7%

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

    2. sqrt-unprod 20.0%

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

    3. pow2 20.0%

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

    4. expm1-udef 20.0%

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

    5. sum-log 20.0%

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

    6. add-exp-log 20.0%

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

11. Applied egg-rr 20.0%

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

    1. unpow2 20.0%

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

    2. sqr-neg 20.0%

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

    3. rem-sqrt-square 20.0%

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

    4. sub-neg 20.0%

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

    5. metadata-eval 20.0%

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

    6. distribute-neg-in 20.0%

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

    7. associate-*l/ 20.0%

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

    8. neg-mul-1 20.0%

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

    9. distribute-neg-frac 20.0%

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

    10. remove-double-neg 20.0%

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

    11. metadata-eval 20.0%

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

13. Simplified 20.0%

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

14. Final simplification 20.0%

    \[\leadsto 1 - \left|1 + \frac{hi}{lo}\right| \]

Alternative 4: 19.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{hi}{lo} + 2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(lo, hi, x):
	return (hi / lo) + 2.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 9.5%

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

   1. +-commutative 9.5%

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

   2. associate--l+ 9.5%

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

   3. associate-*r/ 9.5%

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

   4. associate-*r/ 9.5%

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

   5. div-sub 9.5%

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

   6. distribute-lft-out-- 9.5%

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

   7. associate-*r/ 9.5%

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

   8. mul-1-neg 9.5%

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

   9. unsub-neg 9.5%

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

4. Simplified 9.5%

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

   1. expm1-log1p-u 9.5%

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

6. Applied egg-rr 9.5%

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

7. Taylor expanded in hi around inf 19.1%

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

   1. +-commutative 19.1%

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

   2. distribute-neg-frac 19.1%

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

   3. metadata-eval 19.1%

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

   4. mul-1-neg 19.1%

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

   5. log-rec 19.1%

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

   6. remove-double-neg 19.1%

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

9. Simplified 19.1%

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

10. Taylor expanded in lo around 0 0.0%

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

    1. exp-sum 0.0%

       \[\leadsto 2 - \color{blue}{e^{-1 \cdot \log lo} \cdot e^{\log -1 + \log hi}} \]

    2. mul-1-neg 0.0%

       \[\leadsto 2 - e^{\color{blue}{-\log lo}} \cdot e^{\log -1 + \log hi} \]

    3. log-rec 0.0%

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

    4. exp-sum 0.0%

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

    5. log-rec 0.0%

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

    6. mul-1-neg 0.0%

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

    7. +-commutative 0.0%

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

    8. +-commutative 0.0%

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

    9. associate-+l+ 0.0%

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

    10. mul-1-neg 0.0%

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

    11. sub-neg 0.0%

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

    12. log-div 19.1%

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

    13. +-commutative 19.1%

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

    14. log-prod 19.1%

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

    15. rem-exp-log 19.1%

        \[\leadsto 2 - \color{blue}{\frac{-1}{lo} \cdot hi} \]

12. Simplified 19.1%

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

13. Final simplification 19.1%

    \[\leadsto \frac{hi}{lo} + 2 \]

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. 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 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. Taylor expanded in lo around inf 18.7%

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

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023252 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))