xlohi (overflows)

Percentage Accurate: 3.1% → 99.1%

Time: 9.5s

Alternatives: 7

Speedup: 7.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[(x - hi), $MachinePrecision] / N[(N[(hi / lo), $MachinePrecision] + -1.0), $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 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. flip-- 18.9%

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

   2. clear-num 18.9%

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

   3. metadata-eval 18.9%

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

   4. pow2 18.9%

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

7. Applied egg-rr 18.9%

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

8. Taylor expanded in hi around 0 99.1%

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

   1. associate-*l/ 99.0%

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

   2. un-div-inv 99.2%

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

   3. sub-neg 99.2%

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

   4. metadata-eval 99.2%

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

10. Applied egg-rr 99.2%

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

11. Final simplification 99.2%

    \[\leadsto 1 + \frac{\frac{x - hi}{\frac{hi}{lo} + -1}}{lo} \]
12. Add Preprocessing

Alternative 2: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(x - hi), $MachinePrecision] * N[(N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $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 lo around 0 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. times-frac 18.9%

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

   4. distribute-lft-neg-out 18.9%

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

   5. distribute-frac-neg2 18.9%

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

   6. distribute-rgt-in 18.9%

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

   7. +-commutative 18.9%

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

   8. distribute-frac-neg2 18.9%

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

   9. sub-neg 18.9%

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

   10. associate-*l/ 18.9%

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

   11. associate-/l* 18.9%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

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

Alternative 3: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] / N[(N[(hi / lo), $MachinePrecision] + -1.0), $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. Step-by-step derivation

   1. flip-- 18.9%

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

   2. clear-num 18.9%

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

   3. metadata-eval 18.9%

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

   4. pow2 18.9%

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

7. Applied egg-rr 18.9%

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

8. Taylor expanded in hi around 0 99.1%

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

9. Taylor expanded in x around 0 18.7%

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

    1. associate--l+ 18.7%

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

    2. sub-neg 18.7%

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

    3. metadata-eval 18.7%

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

    4. associate-/r* 18.7%

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

    5. sub-neg 18.7%

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

    6. metadata-eval 18.7%

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

    7. associate-/r* 99.1%

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

    8. div-sub 99.1%

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

    9. div-sub 99.1%

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

11. Simplified 99.1%

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

12. Final simplification 99.1%

    \[\leadsto 1 + \frac{\frac{x - hi}{lo}}{\frac{hi}{lo} + -1} \]
13. Add Preprocessing

Alternative 4: 18.9% accurate, 0.6× speedup

Math

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

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(hi * Float64(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[(hi * N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $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. Final simplification 18.9%

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

Alternative 5: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 18.8%

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

4. Final simplification 18.8%

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

Alternative 6: 18.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. 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 7: 18.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. 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 2024041 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))