xlohi (overflows)

Percentage Accurate: 3.1% → 24.2%

Time: 8.7s

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: 24.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - hi}{lo}\\ t_1 := \frac{x - lo}{hi}\\ \mathbf{if}\;lo \leq -1.08 \cdot 10^{+308}:\\ \;\;\;\;\frac{1}{1 + \left(t_0 + {t_0}^{2}\right)} \cdot \left(1 - {\left(\frac{hi}{lo}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(x - lo\right) \cdot \left(lo \cdot {hi}^{-2}\right)\right)}^{2} - {t_1}^{2}}{\frac{lo \cdot t_1 + \left(lo - x\right)}{hi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x hi) lo)) (t_1 (/ (- x lo) hi)))
   (if (<= lo -1.08e+308)
     (* (/ 1.0 (+ 1.0 (+ t_0 (pow t_0 2.0)))) (- 1.0 (pow (/ hi lo) 3.0)))
     (/
      (- (pow (* (- x lo) (* lo (pow hi -2.0))) 2.0) (pow t_1 2.0))
      (/ (+ (* lo t_1) (- lo x)) hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double t_1 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.08e+308) {
		tmp = (1.0 / (1.0 + (t_0 + pow(t_0, 2.0)))) * (1.0 - pow((hi / lo), 3.0));
	} else {
		tmp = (pow(((x - lo) * (lo * pow(hi, -2.0))), 2.0) - pow(t_1, 2.0)) / (((lo * t_1) + (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) :: t_1
    real(8) :: tmp
    t_0 = (x - hi) / lo
    t_1 = (x - lo) / hi
    if (lo <= (-1.08d+308)) then
        tmp = (1.0d0 / (1.0d0 + (t_0 + (t_0 ** 2.0d0)))) * (1.0d0 - ((hi / lo) ** 3.0d0))
    else
        tmp = ((((x - lo) * (lo * (hi ** (-2.0d0)))) ** 2.0d0) - (t_1 ** 2.0d0)) / (((lo * t_1) + (lo - x)) / hi)
    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 = (x - lo) / hi;
	double tmp;
	if (lo <= -1.08e+308) {
		tmp = (1.0 / (1.0 + (t_0 + Math.pow(t_0, 2.0)))) * (1.0 - Math.pow((hi / lo), 3.0));
	} else {
		tmp = (Math.pow(((x - lo) * (lo * Math.pow(hi, -2.0))), 2.0) - Math.pow(t_1, 2.0)) / (((lo * t_1) + (lo - x)) / hi);
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = (x - hi) / lo
	t_1 = (x - lo) / hi
	tmp = 0
	if lo <= -1.08e+308:
		tmp = (1.0 / (1.0 + (t_0 + math.pow(t_0, 2.0)))) * (1.0 - math.pow((hi / lo), 3.0))
	else:
		tmp = (math.pow(((x - lo) * (lo * math.pow(hi, -2.0))), 2.0) - math.pow(t_1, 2.0)) / (((lo * t_1) + (lo - x)) / hi)
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - hi) / lo)
	t_1 = Float64(Float64(x - lo) / hi)
	tmp = 0.0
	if (lo <= -1.08e+308)
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(t_0 + (t_0 ^ 2.0)))) * Float64(1.0 - (Float64(hi / lo) ^ 3.0)));
	else
		tmp = Float64(Float64((Float64(Float64(x - lo) * Float64(lo * (hi ^ -2.0))) ^ 2.0) - (t_1 ^ 2.0)) / Float64(Float64(Float64(lo * t_1) + Float64(lo - x)) / hi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = (x - hi) / lo;
	t_1 = (x - lo) / hi;
	tmp = 0.0;
	if (lo <= -1.08e+308)
		tmp = (1.0 / (1.0 + (t_0 + (t_0 ^ 2.0)))) * (1.0 - ((hi / lo) ^ 3.0));
	else
		tmp = ((((x - lo) * (lo * (hi ^ -2.0))) ^ 2.0) - (t_1 ^ 2.0)) / (((lo * t_1) + (lo - x)) / hi);
	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[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, If[LessEqual[lo, -1.08e+308], N[(N[(1.0 / N[(1.0 + N[(t$95$0 + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Power[N[(hi / lo), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(x - lo), $MachinePrecision] * N[(lo * N[Power[hi, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(lo * t$95$1), $MachinePrecision] + N[(lo - x), $MachinePrecision]), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lo < -1.0800000000000001e308

   1. Initial program 3.1%

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

   2. Taylor expanded in lo around inf 11.3%

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

      1. +-commutative 11.3%

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

      2. associate--l+ 11.3%

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

      3. associate-*r/ 11.3%

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

      4. associate-*r/ 11.3%

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

      5. div-sub 11.3%

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

      6. distribute-lft-out-- 11.3%

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

      7. associate-*r/ 11.3%

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

      8. mul-1-neg 11.3%

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

      9. unsub-neg 11.3%

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

   4. Simplified 11.3%

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

      1. add-cbrt-cube 11.3%

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

      2. pow3 11.3%

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

   6. Applied egg-rr 11.3%

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

   8. Applied egg-rr 22.1%

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

   9. Taylor expanded in x around 0 0.0%

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

       1. mul-1-neg 0.0%

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

       2. cube-div 22.1%

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

   11. Simplified 22.1%

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

   if -1.0800000000000001e308 < 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 17.8%

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

      6. div-sub 17.8%

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

   4. Simplified 17.8%

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

      1. flip-+ 17.8%

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

      2. div-sub 17.8%

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

   6. Applied egg-rr 0.0%

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

      1. div-sub 0.0%

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

      2. associate-*l* 42.1%

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

      3. associate-*r/ 3.1%

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

      4. associate-*l/ 42.1%

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

      5. *-commutative 42.1%

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

   8. Simplified 42.1%

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

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;lo \leq -1.08 \cdot 10^{+308}:\\ \;\;\;\;\frac{1}{1 + \left(\frac{x - hi}{lo} + {\left(\frac{x - hi}{lo}\right)}^{2}\right)} \cdot \left(1 - {\left(\frac{hi}{lo}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(x - lo\right) \cdot \left(lo \cdot {hi}^{-2}\right)\right)}^{2} - {\left(\frac{x - lo}{hi}\right)}^{2}}{\frac{lo \cdot \frac{x - lo}{hi} + \left(lo - x\right)}{hi}}\\ \end{array} \]

Alternative 2: 21.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - hi}{lo}\\ \frac{1}{1 + \left(t_0 + {t_0}^{2}\right)} \cdot \left(1 - {\left(\frac{hi}{lo}\right)}^{3}\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x hi) lo)))
   (* (/ 1.0 (+ 1.0 (+ t_0 (pow t_0 2.0)))) (- 1.0 (pow (/ hi lo) 3.0)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	return (1.0 / (1.0 + (t_0 + pow(t_0, 2.0)))) * (1.0 - pow((hi / lo), 3.0));
}

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
    t_0 = (x - hi) / lo
    code = (1.0d0 / (1.0d0 + (t_0 + (t_0 ** 2.0d0)))) * (1.0d0 - ((hi / lo) ** 3.0d0))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	return (1.0 / (1.0 + (t_0 + Math.pow(t_0, 2.0)))) * (1.0 - Math.pow((hi / lo), 3.0));
}

Python

def code(lo, hi, x):
	t_0 = (x - hi) / lo
	return (1.0 / (1.0 + (t_0 + math.pow(t_0, 2.0)))) * (1.0 - math.pow((hi / lo), 3.0))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - hi) / lo)
	return Float64(Float64(1.0 / Float64(1.0 + Float64(t_0 + (t_0 ^ 2.0)))) * Float64(1.0 - (Float64(hi / lo) ^ 3.0)))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (x - hi) / lo;
	tmp = (1.0 / (1.0 + (t_0 + (t_0 ^ 2.0)))) * (1.0 - ((hi / lo) ^ 3.0));
end

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 10.3%

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

   1. +-commutative 10.3%

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

   2. associate--l+ 10.3%

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

   3. associate-*r/ 10.3%

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

   4. associate-*r/ 10.3%

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

   5. div-sub 10.3%

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

   6. distribute-lft-out-- 10.3%

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

   7. associate-*r/ 10.3%

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

   8. mul-1-neg 10.3%

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

   9. unsub-neg 10.3%

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

4. Simplified 10.3%

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

   1. add-cbrt-cube 10.3%

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

   2. pow3 10.3%

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

6. Applied egg-rr 10.3%

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

8. Applied egg-rr 21.8%

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

9. Taylor expanded in x around 0 0.0%

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

    1. mul-1-neg 0.0%

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

    2. cube-div 21.8%

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

11. Simplified 21.8%

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

12. Final simplification 21.8%

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

Alternative 3: 21.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - {\left(\frac{hi}{lo}\right)}^{3}}{\mathsf{fma}\left(\frac{hi}{lo}, \frac{hi}{lo}, 1\right) - \frac{hi}{lo}} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (/ (- 1.0 (pow (/ hi lo) 3.0)) (- (fma (/ hi lo) (/ hi lo) 1.0) (/ hi lo))))

C

double code(double lo, double hi, double x) {
	return (1.0 - pow((hi / lo), 3.0)) / (fma((hi / lo), (hi / lo), 1.0) - (hi / lo));
}

Julia

function code(lo, hi, x)
	return Float64(Float64(1.0 - (Float64(hi / lo) ^ 3.0)) / Float64(fma(Float64(hi / lo), Float64(hi / lo), 1.0) - Float64(hi / lo)))
end

Wolfram

code[lo_, hi_, x_] := N[(N[(1.0 - N[Power[N[(hi / lo), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(hi / lo), $MachinePrecision] * N[(hi / lo), $MachinePrecision] + 1.0), $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 10.3%

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

   1. +-commutative 10.3%

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

   2. associate--l+ 10.3%

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

   3. associate-*r/ 10.3%

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

   4. associate-*r/ 10.3%

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

   5. div-sub 10.3%

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

   6. distribute-lft-out-- 10.3%

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

   7. associate-*r/ 10.3%

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

   8. mul-1-neg 10.3%

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

   9. unsub-neg 10.3%

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

4. Simplified 10.3%

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

   1. add-cbrt-cube 10.3%

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

   2. pow3 10.3%

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

6. Applied egg-rr 10.3%

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

8. Applied egg-rr 21.8%

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

9. Taylor expanded in x around 0 0.0%

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

    1. cube-div 0.0%

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

    2. mul-1-neg 0.0%

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

    3. unsub-neg 0.0%

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

    4. +-commutative 0.0%

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

    5. unpow2 0.0%

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

    6. unpow2 0.0%

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

    7. times-frac 21.8%

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

    8. fma-def 21.8%

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

11. Simplified 21.8%

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

12. Final simplification 21.8%

    \[\leadsto \frac{1 - {\left(\frac{hi}{lo}\right)}^{3}}{\mathsf{fma}\left(\frac{hi}{lo}, \frac{hi}{lo}, 1\right) - \frac{hi}{lo}} \]

Alternative 4: 21.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{hi - x}{lo}\\ \left(1 + {\left(\frac{x - hi}{lo}\right)}^{3}\right) \cdot \frac{1}{1 + \left(t_0 \cdot t_0 - t_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- hi x) lo)))
   (* (+ 1.0 (pow (/ (- x hi) lo) 3.0)) (/ 1.0 (+ 1.0 (- (* t_0 t_0) t_0))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	return (1.0 + pow(((x - hi) / lo), 3.0)) * (1.0 / (1.0 + ((t_0 * t_0) - t_0)));
}

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
    t_0 = (hi - x) / lo
    code = (1.0d0 + (((x - hi) / lo) ** 3.0d0)) * (1.0d0 / (1.0d0 + ((t_0 * t_0) - t_0)))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (hi - x) / lo;
	return (1.0 + Math.pow(((x - hi) / lo), 3.0)) * (1.0 / (1.0 + ((t_0 * t_0) - t_0)));
}

Python

def code(lo, hi, x):
	t_0 = (hi - x) / lo
	return (1.0 + math.pow(((x - hi) / lo), 3.0)) * (1.0 / (1.0 + ((t_0 * t_0) - t_0)))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(hi - x) / lo)
	return Float64(Float64(1.0 + (Float64(Float64(x - hi) / lo) ^ 3.0)) * Float64(1.0 / Float64(1.0 + Float64(Float64(t_0 * t_0) - t_0))))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (hi - x) / lo;
	tmp = (1.0 + (((x - hi) / lo) ^ 3.0)) * (1.0 / (1.0 + ((t_0 * t_0) - t_0)));
end

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 10.3%

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

   1. +-commutative 10.3%

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

   2. associate--l+ 10.3%

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

   3. associate-*r/ 10.3%

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

   4. associate-*r/ 10.3%

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

   5. div-sub 10.3%

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

   6. distribute-lft-out-- 10.3%

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

   7. associate-*r/ 10.3%

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

   8. mul-1-neg 10.3%

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

   9. unsub-neg 10.3%

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

4. Simplified 10.3%

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

   1. add-cbrt-cube 10.3%

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

   2. pow3 10.3%

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

6. Applied egg-rr 10.3%

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

8. Applied egg-rr 21.8%

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

   1. unpow2 21.8%

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

10. Applied egg-rr 21.8%

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

11. Final simplification 21.8%

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

Alternative 5: 21.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left(1 + {\left(\frac{x - hi}{lo}\right)}^{3}\right) \cdot \frac{1}{1 + \left(\frac{hi}{lo} \cdot \frac{hi}{lo} - \frac{hi - x}{lo}\right)} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (*
  (+ 1.0 (pow (/ (- x hi) lo) 3.0))
  (/ 1.0 (+ 1.0 (- (* (/ hi lo) (/ hi lo)) (/ (- hi x) lo))))))

C

double code(double lo, double hi, double x) {
	return (1.0 + pow(((x - hi) / lo), 3.0)) * (1.0 / (1.0 + (((hi / lo) * (hi / lo)) - ((hi - x) / 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) ** 3.0d0)) * (1.0d0 / (1.0d0 + (((hi / lo) * (hi / lo)) - ((hi - x) / lo))))
end function

Java

public static double code(double lo, double hi, double x) {
	return (1.0 + Math.pow(((x - hi) / lo), 3.0)) * (1.0 / (1.0 + (((hi / lo) * (hi / lo)) - ((hi - x) / lo))));
}

Python

def code(lo, hi, x):
	return (1.0 + math.pow(((x - hi) / lo), 3.0)) * (1.0 / (1.0 + (((hi / lo) * (hi / lo)) - ((hi - x) / lo))))

Julia

function code(lo, hi, x)
	return Float64(Float64(1.0 + (Float64(Float64(x - hi) / lo) ^ 3.0)) * Float64(1.0 / Float64(1.0 + Float64(Float64(Float64(hi / lo) * Float64(hi / lo)) - Float64(Float64(hi - x) / lo)))))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (1.0 + (((x - hi) / lo) ^ 3.0)) * (1.0 / (1.0 + (((hi / lo) * (hi / lo)) - ((hi - x) / lo))));
end

Wolfram

code[lo_, hi_, x_] := N[(N[(1.0 + N[Power[N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(N[(N[(hi / lo), $MachinePrecision] * N[(hi / lo), $MachinePrecision]), $MachinePrecision] - N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 10.3%

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

   1. +-commutative 10.3%

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

   2. associate--l+ 10.3%

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

   3. associate-*r/ 10.3%

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

   4. associate-*r/ 10.3%

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

   5. div-sub 10.3%

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

   6. distribute-lft-out-- 10.3%

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

   7. associate-*r/ 10.3%

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

   8. mul-1-neg 10.3%

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

   9. unsub-neg 10.3%

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

4. Simplified 10.3%

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

   1. add-cbrt-cube 10.3%

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

   2. pow3 10.3%

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

6. Applied egg-rr 10.3%

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

8. Applied egg-rr 21.8%

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

9. Taylor expanded in x around 0 0.0%

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

    1. unpow2 0.0%

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

    2. unpow2 0.0%

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

    3. times-frac 21.8%

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

11. Simplified 21.8%

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

12. Final simplification 21.8%

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

Alternative 6: 19.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return -(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 = -(hi / lo)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 10.3%

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

   1. +-commutative 10.3%

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

   2. associate--l+ 10.3%

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

   3. associate-*r/ 10.3%

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

   4. associate-*r/ 10.3%

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

   5. div-sub 10.3%

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

   6. distribute-lft-out-- 10.3%

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

   7. associate-*r/ 10.3%

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

   8. mul-1-neg 10.3%

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

   9. unsub-neg 10.3%

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

4. Simplified 10.3%

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

   1. add-cbrt-cube 10.3%

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

   2. pow3 10.3%

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

6. Applied egg-rr 10.3%

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

8. Applied egg-rr 21.8%

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

9. Taylor expanded in hi around inf 19.4%

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

    1. associate-*r/ 19.4%

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

    2. neg-mul-1 19.4%

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

11. Simplified 19.4%

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

12. Final simplification 19.4%

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

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 2023263 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))