xlohi (overflows)

Percentage Accurate: 3.1% → 22.3%

Time: 16.7s

Alternatives: 10

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + \mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi\right)}{lo}\\ \mathbf{if}\;hi \leq 1.445 \cdot 10^{+308}:\\ \;\;\;\;\frac{1 + \left(1 + \left({t\_0}^{3} + -1\right)\right)}{1 + \left({\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{2} + \frac{hi \cdot \frac{hi - x}{lo} - x}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\frac{1 - {t\_0}^{2}}{1 + t\_0}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (+ x (fma hi (/ (+ hi x) lo) hi)) lo)))
   (if (<= hi 1.445e+308)
     (/
      (+ 1.0 (+ 1.0 (+ (pow t_0 3.0) -1.0)))
      (+
       1.0
       (+
        (pow (/ (+ x (- (* hi (/ (- x hi) lo)) hi)) lo) 2.0)
        (/ (- (* hi (/ (- hi x) lo)) x) lo))))
     (sqrt (pow (/ (- 1.0 (pow t_0 2.0)) (+ 1.0 t_0)) 2.0)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x + fma(hi, ((hi + x) / lo), hi)) / lo;
	double tmp;
	if (hi <= 1.445e+308) {
		tmp = (1.0 + (1.0 + (pow(t_0, 3.0) + -1.0))) / (1.0 + (pow(((x + ((hi * ((x - hi) / lo)) - hi)) / lo), 2.0) + (((hi * ((hi - x) / lo)) - x) / lo)));
	} else {
		tmp = sqrt(pow(((1.0 - pow(t_0, 2.0)) / (1.0 + t_0)), 2.0));
	}
	return tmp;
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x + fma(hi, Float64(Float64(hi + x) / lo), hi)) / lo)
	tmp = 0.0
	if (hi <= 1.445e+308)
		tmp = Float64(Float64(1.0 + Float64(1.0 + Float64((t_0 ^ 3.0) + -1.0))) / Float64(1.0 + Float64((Float64(Float64(x + Float64(Float64(hi * Float64(Float64(x - hi) / lo)) - hi)) / lo) ^ 2.0) + Float64(Float64(Float64(hi * Float64(Float64(hi - x) / lo)) - x) / lo))));
	else
		tmp = sqrt((Float64(Float64(1.0 - (t_0 ^ 2.0)) / Float64(1.0 + t_0)) ^ 2.0));
	end
	return tmp
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x + N[(hi * N[(N[(hi + x), $MachinePrecision] / lo), $MachinePrecision] + hi), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]}, If[LessEqual[hi, 1.445e+308], N[(N[(1.0 + N[(1.0 + N[(N[Power[t$95$0, 3.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(N[(x + N[(N[(hi * N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - hi), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(hi * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Power[N[(N[(1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.445e308

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 19.1%

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

   5. Simplified 19.1%

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

      1. flip3-+ 19.0%

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

   7. Applied egg-rr 18.3%

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

   8. Taylor expanded in lo around 0 3.0%

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

      1. associate-*r/ 23.8%

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

   10. Simplified 23.8%

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

       1. expm1-log1p-u 23.8%

          \[\leadsto \frac{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{3}\right)\right)}}{1 + \left({\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{2} - \frac{x + hi \cdot \frac{x - hi}{lo}}{lo}\right)} \]

       2. log1p-define 23.8%

          \[\leadsto \frac{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + {\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{3}\right)}\right)}{1 + \left({\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{2} - \frac{x + hi \cdot \frac{x - hi}{lo}}{lo}\right)} \]

       3. expm1-undefine 23.8%

          \[\leadsto \frac{1 + \color{blue}{\left(e^{\log \left(1 + {\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{3}\right)} - 1\right)}}{1 + \left({\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{2} - \frac{x + hi \cdot \frac{x - hi}{lo}}{lo}\right)} \]

   12. Applied egg-rr 24.1%

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

       1. associate--l+ 24.1%

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

       2. +-commutative 24.1%

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

   14. Simplified 24.1%

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

   if 1.445e308 < hi

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 9.0%

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

   5. Simplified 9.0%

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

      1. add-sqr-sqrt 9.0%

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

      2. sqrt-unprod 9.0%

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

      3. pow2 9.0%

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

      4. add-sqr-sqrt 5.1%

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

      5. sqrt-unprod 9.0%

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

      6. sqr-neg 9.0%

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

      7. sqrt-unprod 3.9%

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

      8. add-sqr-sqrt 9.1%

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

   7. Applied egg-rr 9.1%

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

      1. fmm-def 19.9%

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

   9. Simplified 19.9%

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

   11. Applied egg-rr 19.9%

       \[\leadsto \sqrt{{\color{blue}{\left(\frac{1 - {\left(\frac{x + \mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)}{lo}\right)}^{2}}{1 + \frac{x + \mathsf{fma}\left(hi, \frac{x + hi}{lo}, hi\right)}{lo}}\right)}}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.445 \cdot 10^{+308}:\\ \;\;\;\;\frac{1 + \left(1 + \left({\left(\frac{x + \mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi\right)}{lo}\right)}^{3} + -1\right)\right)}{1 + \left({\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{2} + \frac{hi \cdot \frac{hi - x}{lo} - x}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\frac{1 - {\left(\frac{x + \mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi\right)}{lo}\right)}^{2}}{1 + \frac{x + \mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi\right)}{lo}}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 22.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - hi}{lo}\\ \mathbf{if}\;hi \leq 1.445 \cdot 10^{+308}:\\ \;\;\;\;\frac{1 + \left(1 + \left({\left(\frac{x + \mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi\right)}{lo}\right)}^{3} + -1\right)\right)}{1 + \left({\left(\frac{x + \left(hi \cdot t\_0 - hi\right)}{lo}\right)}^{2} + \frac{hi \cdot \frac{hi - x}{lo} - x}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(1 + \frac{x + hi \cdot \left(t\_0 + -1\right)}{lo}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x hi) lo)))
   (if (<= hi 1.445e+308)
     (/
      (+
       1.0
       (+ 1.0 (+ (pow (/ (+ x (fma hi (/ (+ hi x) lo) hi)) lo) 3.0) -1.0)))
      (+
       1.0
       (+
        (pow (/ (+ x (- (* hi t_0) hi)) lo) 2.0)
        (/ (- (* hi (/ (- hi x) lo)) x) lo))))
     (sqrt (pow (+ 1.0 (/ (+ x (* hi (+ t_0 -1.0))) lo)) 2.0)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double tmp;
	if (hi <= 1.445e+308) {
		tmp = (1.0 + (1.0 + (pow(((x + fma(hi, ((hi + x) / lo), hi)) / lo), 3.0) + -1.0))) / (1.0 + (pow(((x + ((hi * t_0) - hi)) / lo), 2.0) + (((hi * ((hi - x) / lo)) - x) / lo)));
	} else {
		tmp = sqrt(pow((1.0 + ((x + (hi * (t_0 + -1.0))) / lo)), 2.0));
	}
	return tmp;
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - hi) / lo)
	tmp = 0.0
	if (hi <= 1.445e+308)
		tmp = Float64(Float64(1.0 + Float64(1.0 + Float64((Float64(Float64(x + fma(hi, Float64(Float64(hi + x) / lo), hi)) / lo) ^ 3.0) + -1.0))) / Float64(1.0 + Float64((Float64(Float64(x + Float64(Float64(hi * t_0) - hi)) / lo) ^ 2.0) + Float64(Float64(Float64(hi * Float64(Float64(hi - x) / lo)) - x) / lo))));
	else
		tmp = sqrt((Float64(1.0 + Float64(Float64(x + Float64(hi * Float64(t_0 + -1.0))) / lo)) ^ 2.0));
	end
	return tmp
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]}, If[LessEqual[hi, 1.445e+308], N[(N[(1.0 + N[(1.0 + N[(N[Power[N[(N[(x + N[(hi * N[(N[(hi + x), $MachinePrecision] / lo), $MachinePrecision] + hi), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision], 3.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(N[(x + N[(N[(hi * t$95$0), $MachinePrecision] - hi), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(hi * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Power[N[(1.0 + N[(N[(x + N[(hi * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.445e308

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 19.1%

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

   5. Simplified 19.1%

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

      1. flip3-+ 19.0%

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

   7. Applied egg-rr 18.3%

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

   8. Taylor expanded in lo around 0 3.0%

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

      1. associate-*r/ 23.8%

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

   10. Simplified 23.8%

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

       1. expm1-log1p-u 23.8%

          \[\leadsto \frac{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{3}\right)\right)}}{1 + \left({\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{2} - \frac{x + hi \cdot \frac{x - hi}{lo}}{lo}\right)} \]

       2. log1p-define 23.8%

          \[\leadsto \frac{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + {\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{3}\right)}\right)}{1 + \left({\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{2} - \frac{x + hi \cdot \frac{x - hi}{lo}}{lo}\right)} \]

       3. expm1-undefine 23.8%

          \[\leadsto \frac{1 + \color{blue}{\left(e^{\log \left(1 + {\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{3}\right)} - 1\right)}}{1 + \left({\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{2} - \frac{x + hi \cdot \frac{x - hi}{lo}}{lo}\right)} \]

   12. Applied egg-rr 24.1%

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

       1. associate--l+ 24.1%

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

       2. +-commutative 24.1%

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

   14. Simplified 24.1%

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

   if 1.445e308 < hi

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 9.0%

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

   5. Simplified 9.0%

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

      1. add-sqr-sqrt 9.0%

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

      2. sqrt-unprod 9.0%

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

      3. pow2 9.0%

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

      4. add-sqr-sqrt 5.1%

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

      5. sqrt-unprod 9.0%

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

      6. sqr-neg 9.0%

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

      7. sqrt-unprod 3.9%

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

      8. add-sqr-sqrt 9.1%

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

   7. Applied egg-rr 9.1%

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

      1. fmm-def 19.9%

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

   9. Simplified 19.9%

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

   10. Taylor expanded in hi around 0 19.9%

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

       1. sub-neg 19.9%

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

       2. mul-1-neg 19.9%

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

       3. +-commutative 19.9%

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

       4. sub-neg 19.9%

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

       5. div-sub 19.9%

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

       6. metadata-eval 19.9%

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

   12. Simplified 19.9%

       \[\leadsto \sqrt{{\left(1 + \frac{x + \color{blue}{hi \cdot \left(\frac{x - hi}{lo} + -1\right)}}{lo}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.445 \cdot 10^{+308}:\\ \;\;\;\;\frac{1 + \left(1 + \left({\left(\frac{x + \mathsf{fma}\left(hi, \frac{hi + x}{lo}, hi\right)}{lo}\right)}^{3} + -1\right)\right)}{1 + \left({\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{2} + \frac{hi \cdot \frac{hi - x}{lo} - x}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(1 + \frac{x + hi \cdot \left(\frac{x - hi}{lo} + -1\right)}{lo}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 22.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - hi}{lo}\\ t_1 := hi \cdot \frac{hi - x}{lo}\\ \mathbf{if}\;hi \leq 1.445 \cdot 10^{+308}:\\ \;\;\;\;\frac{1 + {\left(\frac{\left(hi + t\_1\right) - x}{lo}\right)}^{3}}{1 + \left({\left(\frac{x + \left(hi \cdot t\_0 - hi\right)}{lo}\right)}^{2} + \frac{t\_1 - x}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(1 + \frac{x + hi \cdot \left(t\_0 + -1\right)}{lo}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x hi) lo)) (t_1 (* hi (/ (- hi x) lo))))
   (if (<= hi 1.445e+308)
     (/
      (+ 1.0 (pow (/ (- (+ hi t_1) x) lo) 3.0))
      (+ 1.0 (+ (pow (/ (+ x (- (* hi t_0) hi)) lo) 2.0) (/ (- t_1 x) lo))))
     (sqrt (pow (+ 1.0 (/ (+ x (* hi (+ t_0 -1.0))) lo)) 2.0)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double t_1 = hi * ((hi - x) / lo);
	double tmp;
	if (hi <= 1.445e+308) {
		tmp = (1.0 + pow((((hi + t_1) - x) / lo), 3.0)) / (1.0 + (pow(((x + ((hi * t_0) - hi)) / lo), 2.0) + ((t_1 - x) / lo)));
	} else {
		tmp = sqrt(pow((1.0 + ((x + (hi * (t_0 + -1.0))) / lo)), 2.0));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (x - hi) / lo
    t_1 = hi * ((hi - x) / lo)
    if (hi <= 1.445d+308) then
        tmp = (1.0d0 + ((((hi + t_1) - x) / lo) ** 3.0d0)) / (1.0d0 + ((((x + ((hi * t_0) - hi)) / lo) ** 2.0d0) + ((t_1 - x) / lo)))
    else
        tmp = sqrt(((1.0d0 + ((x + (hi * (t_0 + (-1.0d0)))) / lo)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double t_1 = hi * ((hi - x) / lo);
	double tmp;
	if (hi <= 1.445e+308) {
		tmp = (1.0 + Math.pow((((hi + t_1) - x) / lo), 3.0)) / (1.0 + (Math.pow(((x + ((hi * t_0) - hi)) / lo), 2.0) + ((t_1 - x) / lo)));
	} else {
		tmp = Math.sqrt(Math.pow((1.0 + ((x + (hi * (t_0 + -1.0))) / lo)), 2.0));
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - hi) / lo
	t_1 = hi * ((hi - x) / lo)
	tmp = 0
	if hi <= 1.445e+308:
		tmp = (1.0 + math.pow((((hi + t_1) - x) / lo), 3.0)) / (1.0 + (math.pow(((x + ((hi * t_0) - hi)) / lo), 2.0) + ((t_1 - x) / lo)))
	else:
		tmp = math.sqrt(math.pow((1.0 + ((x + (hi * (t_0 + -1.0))) / lo)), 2.0))
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - hi) / lo)
	t_1 = Float64(hi * Float64(Float64(hi - x) / lo))
	tmp = 0.0
	if (hi <= 1.445e+308)
		tmp = Float64(Float64(1.0 + (Float64(Float64(Float64(hi + t_1) - x) / lo) ^ 3.0)) / Float64(1.0 + Float64((Float64(Float64(x + Float64(Float64(hi * t_0) - hi)) / lo) ^ 2.0) + Float64(Float64(t_1 - x) / lo))));
	else
		tmp = sqrt((Float64(1.0 + Float64(Float64(x + Float64(hi * Float64(t_0 + -1.0))) / lo)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x - hi) / lo;
	t_1 = hi * ((hi - x) / lo);
	tmp = 0.0;
	if (hi <= 1.445e+308)
		tmp = (1.0 + ((((hi + t_1) - x) / lo) ^ 3.0)) / (1.0 + ((((x + ((hi * t_0) - hi)) / lo) ^ 2.0) + ((t_1 - x) / lo)));
	else
		tmp = sqrt(((1.0 + ((x + (hi * (t_0 + -1.0))) / lo)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]}, Block[{t$95$1 = N[(hi * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[hi, 1.445e+308], N[(N[(1.0 + N[Power[N[(N[(N[(hi + t$95$1), $MachinePrecision] - x), $MachinePrecision] / lo), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(N[(x + N[(N[(hi * t$95$0), $MachinePrecision] - hi), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(t$95$1 - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Power[N[(1.0 + N[(N[(x + N[(hi * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.445e308

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 19.1%

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

   5. Simplified 19.1%

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

      1. flip3-+ 19.0%

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

   7. Applied egg-rr 18.3%

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

   8. Taylor expanded in lo around 0 3.0%

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

      1. associate-*r/ 23.8%

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

   10. Simplified 23.8%

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

       1. add-sqr-sqrt 14.1%

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

       2. sqrt-unprod 19.3%

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

       3. sqr-neg 19.3%

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

       4. sqrt-unprod 5.2%

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

       5. add-sqr-sqrt 18.3%

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

   12. Applied egg-rr 24.0%

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

   if 1.445e308 < hi

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 9.0%

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

   5. Simplified 9.0%

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

      1. add-sqr-sqrt 9.0%

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

      2. sqrt-unprod 9.0%

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

      3. pow2 9.0%

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

      4. add-sqr-sqrt 5.1%

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

      5. sqrt-unprod 9.0%

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

      6. sqr-neg 9.0%

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

      7. sqrt-unprod 3.9%

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

      8. add-sqr-sqrt 9.1%

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

   7. Applied egg-rr 9.1%

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

      1. fmm-def 19.9%

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

   9. Simplified 19.9%

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

   10. Taylor expanded in hi around 0 19.9%

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

       1. sub-neg 19.9%

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

       2. mul-1-neg 19.9%

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

       3. +-commutative 19.9%

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

       4. sub-neg 19.9%

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

       5. div-sub 19.9%

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

       6. metadata-eval 19.9%

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

   12. Simplified 19.9%

       \[\leadsto \sqrt{{\left(1 + \frac{x + \color{blue}{hi \cdot \left(\frac{x - hi}{lo} + -1\right)}}{lo}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.445 \cdot 10^{+308}:\\ \;\;\;\;\frac{1 + {\left(\frac{\left(hi + hi \cdot \frac{hi - x}{lo}\right) - x}{lo}\right)}^{3}}{1 + \left({\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{2} + \frac{hi \cdot \frac{hi - x}{lo} - x}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(1 + \frac{x + hi \cdot \left(\frac{x - hi}{lo} + -1\right)}{lo}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 22.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - hi}{lo}\\ \mathbf{if}\;hi \leq 1.445 \cdot 10^{+308}:\\ \;\;\;\;\frac{1 + {\left(\frac{x}{lo}\right)}^{3}}{1 + \left({\left(\frac{x + \left(hi \cdot t\_0 - hi\right)}{lo}\right)}^{2} + \frac{hi \cdot \frac{hi - x}{lo} - x}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(1 + \frac{x + hi \cdot \left(t\_0 + -1\right)}{lo}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x hi) lo)))
   (if (<= hi 1.445e+308)
     (/
      (+ 1.0 (pow (/ x lo) 3.0))
      (+
       1.0
       (+
        (pow (/ (+ x (- (* hi t_0) hi)) lo) 2.0)
        (/ (- (* hi (/ (- hi x) lo)) x) lo))))
     (sqrt (pow (+ 1.0 (/ (+ x (* hi (+ t_0 -1.0))) lo)) 2.0)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double tmp;
	if (hi <= 1.445e+308) {
		tmp = (1.0 + pow((x / lo), 3.0)) / (1.0 + (pow(((x + ((hi * t_0) - hi)) / lo), 2.0) + (((hi * ((hi - x) / lo)) - x) / lo)));
	} else {
		tmp = sqrt(pow((1.0 + ((x + (hi * (t_0 + -1.0))) / lo)), 2.0));
	}
	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 - hi) / lo
    if (hi <= 1.445d+308) then
        tmp = (1.0d0 + ((x / lo) ** 3.0d0)) / (1.0d0 + ((((x + ((hi * t_0) - hi)) / lo) ** 2.0d0) + (((hi * ((hi - x) / lo)) - x) / lo)))
    else
        tmp = sqrt(((1.0d0 + ((x + (hi * (t_0 + (-1.0d0)))) / lo)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double tmp;
	if (hi <= 1.445e+308) {
		tmp = (1.0 + Math.pow((x / lo), 3.0)) / (1.0 + (Math.pow(((x + ((hi * t_0) - hi)) / lo), 2.0) + (((hi * ((hi - x) / lo)) - x) / lo)));
	} else {
		tmp = Math.sqrt(Math.pow((1.0 + ((x + (hi * (t_0 + -1.0))) / lo)), 2.0));
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - hi) / lo
	tmp = 0
	if hi <= 1.445e+308:
		tmp = (1.0 + math.pow((x / lo), 3.0)) / (1.0 + (math.pow(((x + ((hi * t_0) - hi)) / lo), 2.0) + (((hi * ((hi - x) / lo)) - x) / lo)))
	else:
		tmp = math.sqrt(math.pow((1.0 + ((x + (hi * (t_0 + -1.0))) / lo)), 2.0))
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - hi) / lo)
	tmp = 0.0
	if (hi <= 1.445e+308)
		tmp = Float64(Float64(1.0 + (Float64(x / lo) ^ 3.0)) / Float64(1.0 + Float64((Float64(Float64(x + Float64(Float64(hi * t_0) - hi)) / lo) ^ 2.0) + Float64(Float64(Float64(hi * Float64(Float64(hi - x) / lo)) - x) / lo))));
	else
		tmp = sqrt((Float64(1.0 + Float64(Float64(x + Float64(hi * Float64(t_0 + -1.0))) / lo)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x - hi) / lo;
	tmp = 0.0;
	if (hi <= 1.445e+308)
		tmp = (1.0 + ((x / lo) ^ 3.0)) / (1.0 + ((((x + ((hi * t_0) - hi)) / lo) ^ 2.0) + (((hi * ((hi - x) / lo)) - x) / lo)));
	else
		tmp = sqrt(((1.0 + ((x + (hi * (t_0 + -1.0))) / lo)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]}, If[LessEqual[hi, 1.445e+308], N[(N[(1.0 + N[Power[N[(x / lo), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(N[(x + N[(N[(hi * t$95$0), $MachinePrecision] - hi), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(hi * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Power[N[(1.0 + N[(N[(x + N[(hi * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.445e308

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 19.1%

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

   5. Simplified 19.1%

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

      1. flip3-+ 19.0%

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

   7. Applied egg-rr 18.3%

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

   8. Taylor expanded in lo around 0 3.0%

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

      1. associate-*r/ 23.8%

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

   10. Simplified 23.8%

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

   11. Taylor expanded in hi around 0 23.9%

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

   if 1.445e308 < hi

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 9.0%

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

   5. Simplified 9.0%

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

      1. add-sqr-sqrt 9.0%

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

      2. sqrt-unprod 9.0%

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

      3. pow2 9.0%

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

      4. add-sqr-sqrt 5.1%

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

      5. sqrt-unprod 9.0%

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

      6. sqr-neg 9.0%

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

      7. sqrt-unprod 3.9%

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

      8. add-sqr-sqrt 9.1%

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

   7. Applied egg-rr 9.1%

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

      1. fmm-def 19.9%

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

   9. Simplified 19.9%

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

   10. Taylor expanded in hi around 0 19.9%

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

       1. sub-neg 19.9%

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

       2. mul-1-neg 19.9%

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

       3. +-commutative 19.9%

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

       4. sub-neg 19.9%

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

       5. div-sub 19.9%

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

       6. metadata-eval 19.9%

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

   12. Simplified 19.9%

       \[\leadsto \sqrt{{\left(1 + \frac{x + \color{blue}{hi \cdot \left(\frac{x - hi}{lo} + -1\right)}}{lo}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.445 \cdot 10^{+308}:\\ \;\;\;\;\frac{1 + {\left(\frac{x}{lo}\right)}^{3}}{1 + \left({\left(\frac{x + \left(hi \cdot \frac{x - hi}{lo} - hi\right)}{lo}\right)}^{2} + \frac{hi \cdot \frac{hi - x}{lo} - x}{lo}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(1 + \frac{x + hi \cdot \left(\frac{x - hi}{lo} + -1\right)}{lo}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = 1.0d0 + ((x + abs((hi * (((x - hi) / lo) + (-1.0d0))))) * (1.0d0 / lo))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.8%

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

5. Simplified 14.8%

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

   1. add-sqr-sqrt 9.7%

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

   2. sqrt-unprod 14.2%

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

   3. sqr-neg 14.2%

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

   4. sqrt-unprod 4.5%

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

   5. add-sqr-sqrt 13.6%

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

7. Applied egg-rr 13.6%

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

   1. div-inv 13.6%

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

9. Applied egg-rr 13.6%

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

    1. add-sqr-sqrt 4.6%

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

    2. sqrt-unprod 0.7%

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

    3. pow2 0.7%

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

11. Applied egg-rr 0.7%

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

    1. unpow2 0.7%

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

    2. rem-sqrt-square 14.2%

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

    3. sub-neg 14.2%

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

    4. neg-mul-1 14.2%

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

    5. *-commutative 14.2%

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

    6. distribute-lft-in 19.7%

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

13. Simplified 19.7%

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

14. Final simplification 19.7%

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

Alternative 6: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 14.8%

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

5. Simplified 14.8%

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

   1. add-sqr-sqrt 9.7%

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

   2. sqrt-unprod 14.2%

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

   3. sqr-neg 14.2%

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

   4. sqrt-unprod 4.5%

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

   5. add-sqr-sqrt 13.6%

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

7. Applied egg-rr 13.6%

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

8. Taylor expanded in hi around 0 19.1%

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

   1. sub-neg 19.1%

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

   2. +-commutative 19.1%

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

   3. mul-1-neg 19.1%

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

   4. sub-neg 19.1%

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

   5. div-sub 19.1%

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

   6. metadata-eval 19.1%

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

10. Simplified 19.1%

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

11. Final simplification 19.1%

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

Alternative 7: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.9%

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

6. Final simplification 18.9%

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

Alternative 8: 18.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{x - lo}{hi} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

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

4. Final simplification 18.8%

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

Alternative 9: 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(lo / Float64(-hi))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

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

4. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 10: 18.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 18.7%

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

4. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))