xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 17.1s

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: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 19.0%

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

4. Taylor expanded in lo around inf 19.0%

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

   1. associate--l+ 19.0%

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

   2. div-sub 19.0%

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

6. Simplified 19.0%

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

7. Taylor expanded in x around 0 19.0%

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

8. Taylor expanded in hi around inf 0.0%

   \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 0.0%

      \[\leadsto \frac{\color{blue}{hi \cdot hi}}{{lo}^{2}} \]

   2. unpow2 0.0%

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

   3. times-frac 19.6%

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

   4. unpow2 19.6%

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

10. Simplified 19.6%

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

Alternative 2: 19.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 19.0%

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

4. Taylor expanded in lo around inf 19.0%

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

   1. associate--l+ 19.0%

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

   2. div-sub 19.0%

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

6. Simplified 19.0%

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

7. Taylor expanded in x around inf 9.2%

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

8. Taylor expanded in lo around 0 19.6%

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

9. Final simplification 19.6%

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

Alternative 3: 19.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 19.0%

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

4. Taylor expanded in lo around inf 19.0%

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

   1. associate--l+ 19.0%

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

   2. div-sub 19.0%

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

6. Simplified 19.0%

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

7. Taylor expanded in x around 0 19.0%

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

   1. *-commutative 19.0%

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

   2. frac-2neg 19.0%

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

   3. distribute-frac-neg2 19.0%

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

   4. add-sqr-sqrt 0.0%

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

   5. sqrt-unprod 18.7%

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

   6. sqr-neg 18.7%

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

   7. sqrt-unprod 19.1%

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

   8. add-sqr-sqrt 19.1%

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

   9. frac-2neg 19.1%

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

   10. cancel-sign-sub-inv 19.1%

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

   11. metadata-eval 19.1%

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

   12. div-inv 19.1%

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

   13. associate-*l* 19.1%

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

   14. *-commutative 19.1%

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

   15. div-inv 19.1%

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

9. Applied egg-rr 19.1%

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

    1. *-commutative 19.1%

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

11. Simplified 19.1%

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

12. Final simplification 19.1%

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

Alternative 4: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(hi / lo), $MachinePrecision] * N[(N[(N[(hi - x), $MachinePrecision] / 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 hi around 0 19.0%

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

4. Taylor expanded in lo around inf 19.0%

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

   1. associate--l+ 19.0%

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

   2. div-sub 19.0%

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

6. Simplified 19.0%

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

7. Taylor expanded in x around 0 19.0%

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

8. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. unsub-neg 3.1%

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

   3. distribute-lft-out 3.1%

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

   4. *-lft-identity 3.1%

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

   5. times-frac 3.1%

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

   6. metadata-eval 3.1%

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

   7. *-rgt-identity 3.1%

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

   8. associate-/l* 15.5%

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

   9. distribute-lft-in 19.0%

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

   10. *-commutative 19.0%

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

   11. associate-*r/ 19.0%

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

   12. neg-mul-1 19.0%

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

10. Simplified 19.0%

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

11. Final simplification 19.0%

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

Alternative 5: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 19.0%

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

4. Taylor expanded in lo around inf 19.0%

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

   1. associate--l+ 19.0%

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

   2. div-sub 19.0%

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

6. Simplified 19.0%

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

7. Taylor expanded in x around 0 19.0%

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

   1. associate-/l* 19.0%

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

9. Simplified 19.0%

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

10. Final simplification 19.0%

    \[\leadsto 1 + hi \cdot \frac{\frac{hi}{lo} + 1}{lo} \]
11. 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(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. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

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. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))