xlohi (overflows)

Percentage Accurate: 3.1% → 98.1%

Time: 14.2s

Alternatives: 7

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: 98.1% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double lo, double hi, double x) {
	return (x / hi) - (Math.pow((lo / hi), 2.0) / ((lo / hi) * (1.0 + (Math.exp(Math.log1p(((x - lo) / hi))) + -1.0))));
}

Python

def code(lo, hi, x):
	return (x / hi) - (math.pow((lo / hi), 2.0) / ((lo / hi) * (1.0 + (math.exp(math.log1p(((x - lo) / hi))) + -1.0))))

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. associate-/l* 18.8%

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

   3. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. flip-- 18.8%

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

   2. pow2 18.8%

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

   3. associate-/r/ 18.8%

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

   4. pow2 18.8%

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

   5. associate-/r/ 18.8%

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

   6. fma-def 18.8%

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

6. Applied egg-rr 18.8%

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

   1. associate-*l/ 0.0%

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

   2. associate-/l* 18.8%

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

   3. associate-*l/ 10.0%

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

   4. associate-/r/ 10.0%

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

   5. *-rgt-identity 10.0%

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

   6. associate-*r/ 10.0%

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

   7. associate-/r/ 10.6%

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

   8. associate-*l/ 10.6%

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

   9. *-lft-identity 10.6%

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

   10. associate-/r* 18.8%

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

   11. fma-udef 18.8%

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

   12. associate-*l/ 0.0%

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

   13. times-frac 99.2%

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

   14. *-rgt-identity 99.2%

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

   15. distribute-lft-out 99.0%

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

8. Simplified 99.0%

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

   1. expm1-log1p-u 98.9%

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

   2. expm1-udef 99.0%

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

10. Applied egg-rr 99.0%

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

11. Taylor expanded in hi around inf 0.0%

    \[\leadsto \frac{x}{hi} + \frac{\color{blue}{-1 \cdot \frac{{lo}^{2}}{{hi}^{2}}}}{\frac{lo}{hi} \cdot \left(\left(e^{\mathsf{log1p}\left(\frac{x - lo}{hi}\right)} - 1\right) + 1\right)} \]
12. Step-by-step derivation

    1. mul-1-neg 0.0%

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

    2. unpow2 0.0%

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

    3. unpow2 0.0%

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

    4. times-frac 99.0%

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

    5. unpow2 99.0%

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

13. Simplified 99.0%

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

14. Final simplification 99.0%

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

Alternative 2: 98.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{x}{hi} - \frac{{\left(\frac{lo}{hi}\right)}^{2}}{\frac{lo}{hi} \cdot \left(\frac{x - lo}{hi} + 1\right)} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (- (/ x hi) (/ (pow (/ lo hi) 2.0) (* (/ lo hi) (+ (/ (- x lo) hi) 1.0)))))

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (x / hi) - (((lo / hi) ** 2.0d0) / ((lo / hi) * (((x - lo) / hi) + 1.0d0)))
end function

Java

public static double code(double lo, double hi, double x) {
	return (x / hi) - (Math.pow((lo / hi), 2.0) / ((lo / hi) * (((x - lo) / hi) + 1.0)));
}

Python

def code(lo, hi, x):
	return (x / hi) - (math.pow((lo / hi), 2.0) / ((lo / hi) * (((x - lo) / hi) + 1.0)))

Julia

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

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x / hi) - (((lo / hi) ^ 2.0) / ((lo / hi) * (((x - lo) / hi) + 1.0)));
end

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. associate-/l* 18.8%

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

   3. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. flip-- 18.8%

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

   2. pow2 18.8%

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

   3. associate-/r/ 18.8%

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

   4. pow2 18.8%

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

   5. associate-/r/ 18.8%

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

   6. fma-def 18.8%

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

6. Applied egg-rr 18.8%

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

   1. associate-*l/ 0.0%

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

   2. associate-/l* 18.8%

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

   3. associate-*l/ 10.0%

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

   4. associate-/r/ 10.0%

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

   5. *-rgt-identity 10.0%

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

   6. associate-*r/ 10.0%

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

   7. associate-/r/ 10.6%

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

   8. associate-*l/ 10.6%

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

   9. *-lft-identity 10.6%

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

   10. associate-/r* 18.8%

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

   11. fma-udef 18.8%

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

   12. associate-*l/ 0.0%

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

   13. times-frac 99.2%

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

   14. *-rgt-identity 99.2%

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

   15. distribute-lft-out 99.0%

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

8. Simplified 99.0%

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

9. Taylor expanded in hi around inf 0.0%

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

    1. mul-1-neg 0.0%

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

    2. unpow2 0.0%

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

    3. unpow2 0.0%

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

    4. times-frac 99.0%

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

    5. unpow2 99.0%

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

11. Simplified 99.0%

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

12. Final simplification 99.0%

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

Alternative 3: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

   1. div-inv 18.9%

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

6. Applied egg-rr 18.9%

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

7. Taylor expanded in hi around inf 18.9%

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

8. Final simplification 18.9%

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

Alternative 4: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[(hi - x), $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. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

5. Taylor expanded in lo around 0 0.0%

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

   1. +-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. times-frac 18.9%

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

   4. associate-+r- 18.9%

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

   5. div-sub 18.9%

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

   6. *-lft-identity 18.9%

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

   7. distribute-rgt-out 18.9%

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

   8. +-commutative 18.9%

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

7. Simplified 18.9%

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

8. Final simplification 18.9%

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

Alternative 5: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(hi / lo), $MachinePrecision] * 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 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. div-sub 0.0%

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

   6. mul-1-neg 0.0%

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

   7. sub-neg 0.0%

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

   8. unpow2 0.0%

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

   9. times-frac 18.9%

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

   10. distribute-lft-out-- 18.9%

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

   11. associate-*r/ 18.9%

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

   12. fma-neg 18.9%

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

4. Simplified 18.9%

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

   1. clear-num 18.9%

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

   2. inv-pow 18.9%

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

6. Applied egg-rr 18.9%

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

   1. unpow-1 18.9%

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

8. Simplified 18.9%

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

9. Taylor expanded in x around 0 0.0%

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

    1. *-rgt-identity 0.0%

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

    2. unpow2 0.0%

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

    3. unpow2 0.0%

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

    4. times-frac 18.9%

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

    5. distribute-lft-in 18.9%

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

11. Simplified 18.9%

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

12. Final simplification 18.9%

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

Alternative 6: 18.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around inf 18.8%

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

3. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

5. Simplified 18.8%

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

6. Final simplification 18.8%

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

Alternative 7: 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 2023279 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))