xlohi (overflows)

Percentage Accurate: 3.1% → 31.8%

Time: 15.0s

Alternatives: 8

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: 31.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := hi \cdot \frac{\frac{hi}{lo} + 1}{lo}\\ \frac{-1 - {t\_0}^{3}}{-1 + \left(t\_0 - {\left(\frac{hi}{lo}\right)}^{2}\right)} \end{array} \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. *-commutative 0.0%

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

   5. distribute-lft-out-- 0.0%

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

   6. associate-*r* 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. +-commutative 0.0%

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

   12. mul-1-neg 0.0%

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

5. Simplified 18.9%

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

6. Taylor expanded in x around 0 18.9%

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

   1. associate-/l* 18.9%

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

   2. +-commutative 18.9%

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

8. Simplified 18.9%

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

   1. flip3-+ 18.9%

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

   2. frac-2neg 18.9%

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

   3. metadata-eval 18.9%

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

   4. *-commutative 18.9%

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

   5. div-inv 18.9%

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

   6. associate-*l* 18.9%

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

   7. associate-/r/ 18.9%

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

   8. clear-num 18.9%

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

10. Applied egg-rr 18.9%

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

    1. distribute-neg-in 18.9%

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

    2. metadata-eval 18.9%

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

    3. associate-*r/ 18.9%

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

    4. *-commutative 18.9%

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

    5. associate-/l* 18.9%

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

    6. distribute-neg-in 18.9%

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

    7. metadata-eval 18.9%

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

    8. associate-*r/ 18.9%

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

    9. *-commutative 18.9%

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

    10. associate-/l* 18.9%

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

    11. associate-*r/ 18.9%

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

12. Simplified 18.9%

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

13. Taylor expanded in hi around 0 31.5%

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

14. Final simplification 31.5%

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

Alternative 2: 29.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := hi \cdot \frac{\frac{hi}{lo} + 1}{lo}\\ \frac{-1 - {t\_0}^{3}}{-1 + \left(\frac{hi}{lo} - {t\_0}^{2}\right)} \end{array} \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. *-commutative 0.0%

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

   5. distribute-lft-out-- 0.0%

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

   6. associate-*r* 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. +-commutative 0.0%

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

   12. mul-1-neg 0.0%

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

5. Simplified 18.9%

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

6. Taylor expanded in x around 0 18.9%

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

   1. associate-/l* 18.9%

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

   2. +-commutative 18.9%

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

8. Simplified 18.9%

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

   1. flip3-+ 18.9%

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

   2. frac-2neg 18.9%

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

   3. metadata-eval 18.9%

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

   4. *-commutative 18.9%

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

   5. div-inv 18.9%

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

   6. associate-*l* 18.9%

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

   7. associate-/r/ 18.9%

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

   8. clear-num 18.9%

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

10. Applied egg-rr 18.9%

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

    1. distribute-neg-in 18.9%

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

    2. metadata-eval 18.9%

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

    3. associate-*r/ 18.9%

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

    4. *-commutative 18.9%

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

    5. associate-/l* 18.9%

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

    6. distribute-neg-in 18.9%

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

    7. metadata-eval 18.9%

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

    8. associate-*r/ 18.9%

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

    9. *-commutative 18.9%

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

    10. associate-/l* 18.9%

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

    11. associate-*r/ 18.9%

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

12. Simplified 18.9%

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

13. Taylor expanded in hi around 0 29.2%

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

14. Final simplification 29.2%

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

Alternative 3: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. *-commutative 0.0%

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

   5. distribute-lft-out-- 0.0%

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

   6. associate-*r* 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. +-commutative 0.0%

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

   12. mul-1-neg 0.0%

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

5. Simplified 18.9%

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

6. Taylor expanded in x around 0 18.9%

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

   1. associate-/l* 18.9%

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

   2. +-commutative 18.9%

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

8. Simplified 18.9%

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

   1. add-sqr-sqrt 9.3%

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

   2. sqrt-unprod 19.7%

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

   3. pow2 19.7%

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

10. Applied egg-rr 19.7%

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

    1. unpow2 19.7%

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

    2. rem-sqrt-square 19.7%

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

12. Simplified 19.7%

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

13. Final simplification 19.7%

    \[\leadsto hi \cdot \frac{\left|\frac{hi}{lo} + 1\right|}{lo} + 1 \]
14. Add Preprocessing

Alternative 4: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. *-commutative 0.0%

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

   5. distribute-lft-out-- 0.0%

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

   6. associate-*r* 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. +-commutative 0.0%

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

   12. mul-1-neg 0.0%

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

5. Simplified 18.9%

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

   1. clear-num 18.9%

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

   2. associate-/r/ 18.9%

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

7. Applied egg-rr 18.9%

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

   1. *-commutative 18.9%

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

   2. associate-*l/ 18.9%

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

   3. *-un-lft-identity 18.9%

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

   4. associate-*r/ 18.9%

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

9. Applied egg-rr 19.1%

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

    1. associate-*r/ 19.1%

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

    2. *-commutative 19.1%

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

    3. +-commutative 19.1%

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

11. Simplified 19.1%

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

12. Final simplification 19.1%

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

Alternative 5: 18.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ hi \cdot \frac{\frac{hi}{lo} + 1}{lo} + 1 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. *-commutative 0.0%

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

   5. distribute-lft-out-- 0.0%

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

   6. associate-*r* 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. +-commutative 0.0%

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

   12. mul-1-neg 0.0%

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

5. Simplified 18.9%

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

6. Taylor expanded in x around 0 18.9%

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

   1. associate-/l* 18.9%

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

   2. +-commutative 18.9%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

   \[\leadsto hi \cdot \frac{\frac{hi}{lo} + 1}{lo} + 1 \]
10. Add Preprocessing

Alternative 6: 18.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ hi \cdot \frac{-1 - \frac{hi}{lo}}{lo} + 1 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

   1. associate--l+ 0.0%

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

   2. +-commutative 0.0%

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

   3. associate--l+ 0.0%

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

   4. *-commutative 0.0%

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

   5. distribute-lft-out-- 0.0%

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

   6. associate-*r* 0.0%

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

   7. *-commutative 0.0%

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

   8. associate-*r/ 0.0%

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

   9. distribute-lft-out-- 0.0%

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

   10. div-sub 0.0%

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

   11. +-commutative 0.0%

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

   12. mul-1-neg 0.0%

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

5. Simplified 18.9%

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

6. Taylor expanded in x around 0 18.9%

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

   1. associate-/l* 18.9%

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

   2. +-commutative 18.9%

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

8. Simplified 18.9%

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

   1. add-sqr-sqrt 9.3%

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

   2. sqrt-unprod 19.7%

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

   3. pow2 19.7%

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

10. Applied egg-rr 19.7%

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

    1. unpow2 19.7%

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

    2. rem-sqrt-square 19.7%

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

12. Simplified 19.7%

    \[\leadsto 1 + hi \cdot \frac{\color{blue}{\left|\frac{hi}{lo} + 1\right|}}{lo} \]
13. Step-by-step derivation

    1. add-sqr-sqrt 9.3%

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

    2. fabs-sqr 9.3%

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

    3. add-sqr-sqrt 18.9%

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

    4. frac-2neg 18.9%

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

    5. +-commutative 18.9%

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

    6. distribute-neg-in 18.9%

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

    7. metadata-eval 18.9%

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

    8. sub-neg 18.9%

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

    9. associate-*r/ 18.9%

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

14. Applied egg-rr 19.1%

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

    1. associate-/l* 19.1%

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

    2. distribute-neg-frac2 19.1%

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

    3. distribute-neg-frac 19.1%

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

    4. +-commutative 19.1%

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

    5. distribute-neg-in 19.1%

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

    6. metadata-eval 19.1%

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

    7. unsub-neg 19.1%

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

16. Simplified 19.1%

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

17. Final simplification 19.1%

    \[\leadsto hi \cdot \frac{-1 - \frac{hi}{lo}}{lo} + 1 \]
18. Add Preprocessing

Alternative 7: 18.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

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

4. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac2 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

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

Alternative 8: 18.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 18.7%

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

4. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))