xlohi (overflows)

Percentage Accurate: 3.1% → 92.0%

Time: 52.5s

Alternatives: 10

Speedup: 7.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 - \frac{hi - \left(x + x \cdot \frac{hi}{lo}\right)}{lo}}\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+154} \lor \neg \left(x \leq 8.1 \cdot 10^{+148}\right):\\ \;\;\;\;t\_0 + \frac{{\left(\frac{\left(x + \frac{hi}{lo} \cdot \left(x - hi\right)\right) - hi}{lo}\right)}^{2}}{-1 + \frac{hi + \left(\frac{hi \cdot \left(hi - x\right)}{lo} - x\right)}{lo}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{{x}^{2}}{{lo}^{2} \cdot \left(-1 - \frac{x}{lo}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- 1.0 (/ (- hi (+ x (* x (/ hi lo)))) lo)))))
   (if (or (<= x -1.32e+154) (not (<= x 8.1e+148)))
     (+
      t_0
      (/
       (pow (/ (- (+ x (* (/ hi lo) (- x hi))) hi) lo) 2.0)
       (+ -1.0 (/ (+ hi (- (/ (* hi (- hi x)) lo) x)) lo))))
     (+ t_0 (/ (pow x 2.0) (* (pow lo 2.0) (- -1.0 (/ x lo))))))))

C

double code(double lo, double hi, double x) {
	double t_0 = 1.0 / (1.0 - ((hi - (x + (x * (hi / lo)))) / lo));
	double tmp;
	if ((x <= -1.32e+154) || !(x <= 8.1e+148)) {
		tmp = t_0 + (pow((((x + ((hi / lo) * (x - hi))) - hi) / lo), 2.0) / (-1.0 + ((hi + (((hi * (hi - x)) / lo) - x)) / lo)));
	} else {
		tmp = t_0 + (pow(x, 2.0) / (pow(lo, 2.0) * (-1.0 - (x / lo))));
	}
	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 = 1.0d0 / (1.0d0 - ((hi - (x + (x * (hi / lo)))) / lo))
    if ((x <= (-1.32d+154)) .or. (.not. (x <= 8.1d+148))) then
        tmp = t_0 + (((((x + ((hi / lo) * (x - hi))) - hi) / lo) ** 2.0d0) / ((-1.0d0) + ((hi + (((hi * (hi - x)) / lo) - x)) / lo)))
    else
        tmp = t_0 + ((x ** 2.0d0) / ((lo ** 2.0d0) * ((-1.0d0) - (x / lo))))
    end if
    code = tmp
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = 1.0 / (1.0 - ((hi - (x + (x * (hi / lo)))) / lo));
	double tmp;
	if ((x <= -1.32e+154) || !(x <= 8.1e+148)) {
		tmp = t_0 + (Math.pow((((x + ((hi / lo) * (x - hi))) - hi) / lo), 2.0) / (-1.0 + ((hi + (((hi * (hi - x)) / lo) - x)) / lo)));
	} else {
		tmp = t_0 + (Math.pow(x, 2.0) / (Math.pow(lo, 2.0) * (-1.0 - (x / lo))));
	}
	return tmp;
}

Python

def code(lo, hi, x):
	t_0 = 1.0 / (1.0 - ((hi - (x + (x * (hi / lo)))) / lo))
	tmp = 0
	if (x <= -1.32e+154) or not (x <= 8.1e+148):
		tmp = t_0 + (math.pow((((x + ((hi / lo) * (x - hi))) - hi) / lo), 2.0) / (-1.0 + ((hi + (((hi * (hi - x)) / lo) - x)) / lo)))
	else:
		tmp = t_0 + (math.pow(x, 2.0) / (math.pow(lo, 2.0) * (-1.0 - (x / lo))))
	return tmp

Julia

function code(lo, hi, x)
	t_0 = Float64(1.0 / Float64(1.0 - Float64(Float64(hi - Float64(x + Float64(x * Float64(hi / lo)))) / lo)))
	tmp = 0.0
	if ((x <= -1.32e+154) || !(x <= 8.1e+148))
		tmp = Float64(t_0 + Float64((Float64(Float64(Float64(x + Float64(Float64(hi / lo) * Float64(x - hi))) - hi) / lo) ^ 2.0) / Float64(-1.0 + Float64(Float64(hi + Float64(Float64(Float64(hi * Float64(hi - x)) / lo) - x)) / lo))));
	else
		tmp = Float64(t_0 + Float64((x ^ 2.0) / Float64((lo ^ 2.0) * Float64(-1.0 - Float64(x / lo)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lo, hi, x)
	t_0 = 1.0 / (1.0 - ((hi - (x + (x * (hi / lo)))) / lo));
	tmp = 0.0;
	if ((x <= -1.32e+154) || ~((x <= 8.1e+148)))
		tmp = t_0 + (((((x + ((hi / lo) * (x - hi))) - hi) / lo) ^ 2.0) / (-1.0 + ((hi + (((hi * (hi - x)) / lo) - x)) / lo)));
	else
		tmp = t_0 + ((x ^ 2.0) / ((lo ^ 2.0) * (-1.0 - (x / lo))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 - N[(N[(hi - N[(x + N[(x * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.32e+154], N[Not[LessEqual[x, 8.1e+148]], $MachinePrecision]], N[(t$95$0 + N[(N[Power[N[(N[(N[(x + N[(N[(hi / lo), $MachinePrecision] * N[(x - hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - hi), $MachinePrecision] / lo), $MachinePrecision], 2.0], $MachinePrecision] / N[(-1.0 + N[(N[(hi + N[(N[(N[(hi * N[(hi - x), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[Power[x, 2.0], $MachinePrecision] / N[(N[Power[lo, 2.0], $MachinePrecision] * N[(-1.0 - N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.31999999999999998e154 or 8.1e148 < x

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 15.4%

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

   5. Simplified 15.4%

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

      1. flip-+ 14.6%

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

      2. metadata-eval 14.6%

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

      3. div-sub 14.6%

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

   7. Applied egg-rr 14.6%

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

      1. associate-+r- 14.3%

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

      2. associate-/r/ 14.3%

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

      3. associate-+r- 14.3%

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

      4. associate-/r/ 14.3%

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

   9. Simplified 14.3%

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

   10. Taylor expanded in x around inf 20.3%

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

   11. Taylor expanded in lo around inf 69.2%

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

   if -1.31999999999999998e154 < x < 8.1e148

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 15.5%

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

   5. Simplified 15.5%

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

      1. flip-+ 14.8%

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

      2. metadata-eval 14.8%

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

      3. div-sub 14.8%

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

   7. Applied egg-rr 14.8%

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

      1. associate-+r- 14.8%

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

      2. associate-/r/ 14.8%

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

      3. associate-+r- 14.8%

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

      4. associate-/r/ 14.8%

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

   9. Simplified 14.8%

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

   10. Taylor expanded in x around inf 21.0%

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

   11. Taylor expanded in hi around 0 99.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+154} \lor \neg \left(x \leq 8.1 \cdot 10^{+148}\right):\\ \;\;\;\;\frac{1}{1 - \frac{hi - \left(x + x \cdot \frac{hi}{lo}\right)}{lo}} + \frac{{\left(\frac{\left(x + \frac{hi}{lo} \cdot \left(x - hi\right)\right) - hi}{lo}\right)}^{2}}{-1 + \frac{hi + \left(\frac{hi \cdot \left(hi - x\right)}{lo} - x\right)}{lo}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{hi - \left(x + x \cdot \frac{hi}{lo}\right)}{lo}} + \frac{{x}^{2}}{{lo}^{2} \cdot \left(-1 - \frac{x}{lo}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.4% accurate, 0.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return (1.0 / (1.0 - ((hi - (x + (x * (hi / lo)))) / lo))) + (pow((((x + ((hi / lo) * (x - hi))) - hi) / lo), 2.0) / (-1.0 + ((hi + (((hi * (hi - x)) / lo) - 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 / (1.0d0 - ((hi - (x + (x * (hi / lo)))) / lo))) + (((((x + ((hi / lo) * (x - hi))) - hi) / lo) ** 2.0d0) / ((-1.0d0) + ((hi + (((hi * (hi - x)) / lo) - x)) / lo)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 15.5%

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

5. Simplified 15.5%

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

   1. flip-+ 14.7%

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

   2. metadata-eval 14.7%

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

   3. div-sub 14.7%

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

7. Applied egg-rr 14.7%

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

   1. associate-+r- 14.6%

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

   2. associate-/r/ 14.6%

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

   3. associate-+r- 14.6%

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

   4. associate-/r/ 14.6%

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

9. Simplified 14.6%

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

10. Taylor expanded in x around inf 20.8%

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

11. Taylor expanded in lo around inf 74.5%

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

12. Final simplification 74.5%

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

Alternative 3: 24.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - hi}{lo}\\ \mathbf{if}\;hi \leq 1.51 \cdot 10^{+308}:\\ \;\;\;\;\frac{1}{1 - \frac{hi - \left(x + x \cdot \frac{hi}{lo}\right)}{lo}} + \frac{{\left(\frac{\left(x + \frac{hi}{lo} \cdot \left(x - hi\right)\right) - hi}{lo}\right)}^{2}}{-1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\mathsf{fma}\left(hi, t\_0 + -1, x\right)}{lo}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x hi) lo)))
   (if (<= hi 1.51e+308)
     (+
      (/ 1.0 (- 1.0 (/ (- hi (+ x (* x (/ hi lo)))) lo)))
      (/ (pow (/ (- (+ x (* (/ hi lo) (- x hi))) hi) lo) 2.0) (- -1.0 t_0)))
     (+ 1.0 (/ (fma hi (+ t_0 -1.0) x) lo)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - hi) / lo;
	double tmp;
	if (hi <= 1.51e+308) {
		tmp = (1.0 / (1.0 - ((hi - (x + (x * (hi / lo)))) / lo))) + (pow((((x + ((hi / lo) * (x - hi))) - hi) / lo), 2.0) / (-1.0 - t_0));
	} else {
		tmp = 1.0 + (fma(hi, (t_0 + -1.0), x) / lo);
	}
	return tmp;
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - hi) / lo)
	tmp = 0.0
	if (hi <= 1.51e+308)
		tmp = Float64(Float64(1.0 / Float64(1.0 - Float64(Float64(hi - Float64(x + Float64(x * Float64(hi / lo)))) / lo))) + Float64((Float64(Float64(Float64(x + Float64(Float64(hi / lo) * Float64(x - hi))) - hi) / lo) ^ 2.0) / Float64(-1.0 - t_0)));
	else
		tmp = Float64(1.0 + Float64(fma(hi, Float64(t_0 + -1.0), x) / lo));
	end
	return tmp
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision]}, If[LessEqual[hi, 1.51e+308], N[(N[(1.0 / N[(1.0 - N[(N[(hi - N[(x + N[(x * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(N[(x + N[(N[(hi / lo), $MachinePrecision] * N[(x - hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - hi), $MachinePrecision] / lo), $MachinePrecision], 2.0], $MachinePrecision] / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(hi * N[(t$95$0 + -1.0), $MachinePrecision] + x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.51e308

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 18.6%

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

   5. Simplified 18.6%

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

      1. flip-+ 18.4%

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

      2. metadata-eval 18.4%

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

      3. div-sub 18.4%

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

   7. Applied egg-rr 18.4%

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

      1. associate-+r- 18.4%

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

      2. associate-/r/ 18.4%

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

      3. associate-+r- 18.4%

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

      4. associate-/r/ 18.4%

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

   9. Simplified 18.4%

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

   10. Taylor expanded in x around inf 25.5%

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

   11. Taylor expanded in lo around inf 27.0%

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

       1. associate--l+ 27.0%

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

       2. div-sub 27.0%

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

   13. Simplified 27.0%

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

   if 1.51e308 < hi

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 8.0%

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

   5. Simplified 8.0%

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

      1. add-sqr-sqrt 4.9%

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

      2. sqrt-unprod 7.9%

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

      3. sqr-neg 7.9%

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

      4. sqrt-unprod 3.0%

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

      5. add-sqr-sqrt 6.3%

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

      6. associate-+r- 6.0%

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

      7. div-sub 6.0%

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

      8. clear-num 6.0%

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

      9. un-div-inv 6.0%

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

   7. Applied egg-rr 6.0%

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

      1. associate-+r- 6.0%

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

      2. +-commutative 6.0%

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

      3. associate-/r/ 6.0%

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

      4. *-commutative 6.0%

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

      5. fma-define 6.0%

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

   9. Applied egg-rr 6.0%

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

       1. associate--l+ 6.0%

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

       2. div-sub 6.0%

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

       3. div-sub 6.0%

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

       4. fma-undefine 6.0%

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

       5. associate-*r/ 0.7%

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

       6. *-commutative 0.7%

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

       7. +-commutative 0.7%

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

       8. div-sub 0.7%

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

   11. Simplified 19.4%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.51 \cdot 10^{+308}:\\ \;\;\;\;\frac{1}{1 - \frac{hi - \left(x + x \cdot \frac{hi}{lo}\right)}{lo}} + \frac{{\left(\frac{\left(x + \frac{hi}{lo} \cdot \left(x - hi\right)\right) - hi}{lo}\right)}^{2}}{-1 - \frac{x - hi}{lo}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\mathsf{fma}\left(hi, \frac{x - hi}{lo} + -1, x\right)}{lo}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 23.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;hi \leq 1.51 \cdot 10^{+308}:\\ \;\;\;\;\frac{1}{1 - \frac{hi - \left(x + x \cdot \frac{hi}{lo}\right)}{lo}} + \frac{{\left(\frac{\left(x + \frac{hi}{lo} \cdot \left(x - hi\right)\right) - hi}{lo}\right)}^{2}}{-1 - \frac{x}{lo}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\mathsf{fma}\left(hi, \frac{x - hi}{lo} + -1, x\right)}{lo}\\ \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (if (<= hi 1.51e+308)
   (+
    (/ 1.0 (- 1.0 (/ (- hi (+ x (* x (/ hi lo)))) lo)))
    (/ (pow (/ (- (+ x (* (/ hi lo) (- x hi))) hi) lo) 2.0) (- -1.0 (/ x lo))))
   (+ 1.0 (/ (fma hi (+ (/ (- x hi) lo) -1.0) x) lo))))

C

double code(double lo, double hi, double x) {
	double tmp;
	if (hi <= 1.51e+308) {
		tmp = (1.0 / (1.0 - ((hi - (x + (x * (hi / lo)))) / lo))) + (pow((((x + ((hi / lo) * (x - hi))) - hi) / lo), 2.0) / (-1.0 - (x / lo)));
	} else {
		tmp = 1.0 + (fma(hi, (((x - hi) / lo) + -1.0), x) / lo);
	}
	return tmp;
}

Julia

function code(lo, hi, x)
	tmp = 0.0
	if (hi <= 1.51e+308)
		tmp = Float64(Float64(1.0 / Float64(1.0 - Float64(Float64(hi - Float64(x + Float64(x * Float64(hi / lo)))) / lo))) + Float64((Float64(Float64(Float64(x + Float64(Float64(hi / lo) * Float64(x - hi))) - hi) / lo) ^ 2.0) / Float64(-1.0 - Float64(x / lo))));
	else
		tmp = Float64(1.0 + Float64(fma(hi, Float64(Float64(Float64(x - hi) / lo) + -1.0), x) / lo));
	end
	return tmp
end

Wolfram

code[lo_, hi_, x_] := If[LessEqual[hi, 1.51e+308], N[(N[(1.0 / N[(1.0 - N[(N[(hi - N[(x + N[(x * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(N[(x + N[(N[(hi / lo), $MachinePrecision] * N[(x - hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - hi), $MachinePrecision] / lo), $MachinePrecision], 2.0], $MachinePrecision] / N[(-1.0 - N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(hi * N[(N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if hi < 1.51e308

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 18.6%

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

   5. Simplified 18.6%

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

      1. flip-+ 18.4%

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

      2. metadata-eval 18.4%

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

      3. div-sub 18.4%

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

   7. Applied egg-rr 18.4%

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

      1. associate-+r- 18.4%

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

      2. associate-/r/ 18.4%

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

      3. associate-+r- 18.4%

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

      4. associate-/r/ 18.4%

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

   9. Simplified 18.4%

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

   10. Taylor expanded in x around inf 25.5%

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

   11. Taylor expanded in hi around 0 25.5%

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

   if 1.51e308 < hi

   1. Initial program 3.1%

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

   3. Taylor expanded in lo around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. associate--l+ 3.1%

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

      3. associate-/l* 8.0%

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

   5. Simplified 8.0%

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

      1. add-sqr-sqrt 4.9%

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

      2. sqrt-unprod 7.9%

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

      3. sqr-neg 7.9%

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

      4. sqrt-unprod 3.0%

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

      5. add-sqr-sqrt 6.3%

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

      6. associate-+r- 6.0%

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

      7. div-sub 6.0%

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

      8. clear-num 6.0%

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

      9. un-div-inv 6.0%

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

   7. Applied egg-rr 6.0%

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

      1. associate-+r- 6.0%

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

      2. +-commutative 6.0%

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

      3. associate-/r/ 6.0%

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

      4. *-commutative 6.0%

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

      5. fma-define 6.0%

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

   9. Applied egg-rr 6.0%

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

       1. associate--l+ 6.0%

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

       2. div-sub 6.0%

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

       3. div-sub 6.0%

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

       4. fma-undefine 6.0%

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

       5. associate-*r/ 0.7%

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

       6. *-commutative 0.7%

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

       7. +-commutative 0.7%

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

       8. div-sub 0.7%

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

   11. Simplified 19.4%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;hi \leq 1.51 \cdot 10^{+308}:\\ \;\;\;\;\frac{1}{1 - \frac{hi - \left(x + x \cdot \frac{hi}{lo}\right)}{lo}} + \frac{{\left(\frac{\left(x + \frac{hi}{lo} \cdot \left(x - hi\right)\right) - hi}{lo}\right)}^{2}}{-1 - \frac{x}{lo}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\mathsf{fma}\left(hi, \frac{x - hi}{lo} + -1, x\right)}{lo}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 15.5%

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

5. Simplified 15.5%

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

   1. add-sqr-sqrt 5.8%

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

   2. sqrt-unprod 14.9%

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

   3. sqr-neg 14.9%

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

   4. sqrt-unprod 9.1%

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

   5. add-sqr-sqrt 14.6%

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

   6. associate-+r- 14.5%

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

   7. div-sub 14.5%

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

   8. clear-num 14.5%

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

   9. un-div-inv 14.5%

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

7. Applied egg-rr 14.5%

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

   1. associate-+r- 14.5%

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

   2. +-commutative 14.5%

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

   3. associate-/r/ 14.5%

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

   4. *-commutative 14.5%

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

   5. fma-define 14.5%

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

9. Applied egg-rr 14.5%

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

    1. associate--l+ 14.5%

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

    2. div-sub 14.5%

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

    3. div-sub 14.5%

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

    4. fma-undefine 14.5%

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

    5. associate-*r/ 0.7%

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

    6. *-commutative 0.7%

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

    7. +-commutative 0.7%

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

    8. div-sub 0.7%

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

11. Simplified 19.1%

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

12. Final simplification 19.1%

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

Alternative 6: 18.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return (1.0 - pow((hi / lo), 2.0)) - (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) ** 2.0d0)) - (hi / lo)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 15.5%

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

5. Simplified 15.5%

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

   1. unsub-neg 15.5%

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

   2. add-sqr-sqrt 9.6%

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

   3. sqrt-unprod 15.2%

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

   4. sqr-neg 15.2%

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

   5. sqrt-unprod 5.5%

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

   6. add-sqr-sqrt 14.6%

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

   7. *-un-lft-identity 14.6%

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

   8. neg-mul-1 14.6%

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

   9. cancel-sign-sub-inv 14.6%

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

7. Applied egg-rr 14.6%

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

   1. *-lft-identity 14.6%

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

   2. associate-+r- 14.5%

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

   3. associate-/r/ 14.5%

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

9. Simplified 14.5%

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

10. Taylor expanded in x around 0 0.0%

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

    1. mul-1-neg 0.0%

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

    2. unsub-neg 0.0%

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

    3. unpow2 0.0%

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

    4. unpow2 0.0%

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

    5. times-frac 19.0%

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

    6. unpow2 19.0%

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

12. Simplified 19.0%

    \[\leadsto \color{blue}{\left(1 - {\left(\frac{hi}{lo}\right)}^{2}\right) - \frac{hi}{lo}} \]
13. Add Preprocessing

Alternative 7: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 15.5%

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

5. Simplified 15.5%

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

6. Taylor expanded in hi around 0 18.8%

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

   1. sub-neg 18.8%

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

   2. +-commutative 18.8%

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

   3. neg-mul-1 18.8%

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

   4. sub-neg 18.8%

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

   5. div-sub 18.8%

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

   6. metadata-eval 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

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

Alternative 8: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. associate--l+ 3.1%

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

   3. associate-/l* 15.5%

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

5. Simplified 15.5%

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

6. Taylor expanded in hi around 0 18.8%

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

   1. sub-neg 18.8%

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

   2. +-commutative 18.8%

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

   3. neg-mul-1 18.8%

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

   4. sub-neg 18.8%

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

   5. div-sub 18.8%

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

   6. metadata-eval 18.8%

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

8. Simplified 18.8%

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

9. Taylor expanded in x around 0 18.8%

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

Alternative 9: 18.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

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

Alternative 10: 18.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 18.7%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024156 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))