xlohi (overflows)

Percentage Accurate: 3.1% → 21.3%

Time: 7.7s

Alternatives: 5

Speedup: 7.0×

Specification

\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 21.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \mathbf{if}\;lo \leq -1.1025 \cdot 10^{+308}:\\ \;\;\;\;{\left({\left(x - lo\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{\sqrt[3]{x - lo}}{hi}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} - {t_0}^{2}}{\frac{lo \cdot t_0 + \left(lo - x\right)}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (if (<= lo -1.1025e+308)
     (* (pow (pow (- x lo) 0.3333333333333333) 2.0) (/ (cbrt (- x lo)) hi))
     (/
      (- (pow (/ (- x lo) (* hi (/ hi lo))) 2.0) (pow t_0 2.0))
      (/ (+ (* lo t_0) (- lo x)) hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.1025e+308) {
		tmp = pow(pow((x - lo), 0.3333333333333333), 2.0) * (cbrt((x - lo)) / hi);
	} else {
		tmp = (pow(((x - lo) / (hi * (hi / lo))), 2.0) - pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / hi);
	}
	return tmp;
}

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.1025e+308) {
		tmp = Math.pow(Math.pow((x - lo), 0.3333333333333333), 2.0) * (Math.cbrt((x - lo)) / hi);
	} else {
		tmp = (Math.pow(((x - lo) / (hi * (hi / lo))), 2.0) - Math.pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / hi);
	}
	return tmp;
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	tmp = 0.0
	if (lo <= -1.1025e+308)
		tmp = Float64(((Float64(x - lo) ^ 0.3333333333333333) ^ 2.0) * Float64(cbrt(Float64(x - lo)) / hi));
	else
		tmp = Float64(Float64((Float64(Float64(x - lo) / Float64(hi * Float64(hi / lo))) ^ 2.0) - (t_0 ^ 2.0)) / Float64(Float64(Float64(lo * t_0) + Float64(lo - x)) / hi));
	end
	return tmp
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, If[LessEqual[lo, -1.1025e+308], N[(N[Power[N[Power[N[(x - lo), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[(x - lo), $MachinePrecision], 1/3], $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] / N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(lo * t$95$0), $MachinePrecision] + N[(lo - x), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.10249999999999996e308

   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. Step-by-step derivation

      1. add-cube-cbrt 18.8%

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

      2. *-un-lft-identity 18.8%

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

      3. times-frac 18.8%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x - lo} \cdot \sqrt[3]{x - lo}}{1} \cdot \frac{\sqrt[3]{x - lo}}{hi}} \]

      4. pow2 18.8%

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

   4. Applied egg-rr 18.8%

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

      1. pow1/3 18.8%

         \[\leadsto \frac{{\color{blue}{\left({\left(x - lo\right)}^{0.3333333333333333}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{x - lo}}{hi} \]

   6. Applied egg-rr 18.8%

      \[\leadsto \frac{{\color{blue}{\left({\left(x - lo\right)}^{0.3333333333333333}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{x - lo}}{hi} \]

   if -1.10249999999999996e308 < lo

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 18.4%

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

      6. div-sub 18.4%

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

   4. Simplified 18.4%

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

      1. flip-+ 18.4%

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

      2. pow2 18.4%

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

      3. *-commutative 18.4%

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

      4. clear-num 18.4%

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

      5. frac-times 67.4%

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

      6. *-un-lft-identity 67.4%

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

      7. pow2 67.4%

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

      8. associate-*r/ 67.3%

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

      9. sub-div 48.9%

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

   6. Applied egg-rr 48.9%

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

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.1025 \cdot 10^{+308}:\\ \;\;\;\;{\left({\left(x - lo\right)}^{0.3333333333333333}\right)}^{2} \cdot \frac{\sqrt[3]{x - lo}}{hi}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \]

Alternative 2: 21.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \mathbf{if}\;lo \leq -1.1025 \cdot 10^{+308}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} - {t_0}^{2}}{\frac{lo \cdot t_0 + \left(lo - x\right)}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (if (<= lo -1.1025e+308)
     t_0
     (/
      (- (pow (/ (- x lo) (* hi (/ hi lo))) 2.0) (pow t_0 2.0))
      (/ (+ (* lo t_0) (- lo x)) hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.1025e+308) {
		tmp = t_0;
	} else {
		tmp = (pow(((x - lo) / (hi * (hi / lo))), 2.0) - pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / hi);
	}
	return tmp;
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - lo) / hi
    if (lo <= (-1.1025d+308)) then
        tmp = t_0
    else
        tmp = ((((x - lo) / (hi * (hi / lo))) ** 2.0d0) - (t_0 ** 2.0d0)) / (((lo * t_0) + (lo - x)) / hi)
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.1025e+308) {
		tmp = t_0;
	} else {
		tmp = (Math.pow(((x - lo) / (hi * (hi / lo))), 2.0) - Math.pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / hi);
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	tmp = 0
	if lo <= -1.1025e+308:
		tmp = t_0
	else:
		tmp = (math.pow(((x - lo) / (hi * (hi / lo))), 2.0) - math.pow(t_0, 2.0)) / (((lo * t_0) + (lo - x)) / hi)
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	tmp = 0.0
	if (lo <= -1.1025e+308)
		tmp = t_0;
	else
		tmp = Float64(Float64((Float64(Float64(x - lo) / Float64(hi * Float64(hi / lo))) ^ 2.0) - (t_0 ^ 2.0)) / Float64(Float64(Float64(lo * t_0) + Float64(lo - x)) / hi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	tmp = 0.0;
	if (lo <= -1.1025e+308)
		tmp = t_0;
	else
		tmp = ((((x - lo) / (hi * (hi / lo))) ^ 2.0) - (t_0 ^ 2.0)) / (((lo * t_0) + (lo - x)) / hi);
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, If[LessEqual[lo, -1.1025e+308], t$95$0, N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] / N[(hi * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(lo * t$95$0), $MachinePrecision] + N[(lo - x), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.10249999999999996e308

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 18.8%

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

   if -1.10249999999999996e308 < lo

   1. Initial program 3.1%

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

   2. Taylor expanded in hi around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. associate--l+ 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 18.4%

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

      6. div-sub 18.4%

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

   4. Simplified 18.4%

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

      1. flip-+ 18.4%

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

      2. pow2 18.4%

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

      3. *-commutative 18.4%

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

      4. clear-num 18.4%

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

      5. frac-times 67.4%

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

      6. *-un-lft-identity 67.4%

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

      7. pow2 67.4%

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

      8. associate-*r/ 67.3%

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

      9. sub-div 48.9%

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

   6. Applied egg-rr 48.9%

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

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.1025 \cdot 10^{+308}:\\ \;\;\;\;\frac{x - lo}{hi}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x - lo}{hi \cdot \frac{hi}{lo}}\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \]

Alternative 3: 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. Taylor expanded in hi around inf 18.8%

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

3. Final simplification 18.8%

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

Alternative 4: 18.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

5. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

Alternative 5: 18.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 18.6%

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

3. Final simplification 18.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023178 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))