xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 31.8s

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

   1. add-sqr-sqrt 10.0%

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

   2. sqrt-unprod 19.5%

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

   3. pow2 19.5%

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

8. Applied egg-rr 19.5%

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

   1. unpow2 19.5%

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

   2. rem-sqrt-square 19.5%

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

10. Simplified 19.5%

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

11. Taylor expanded in lo around 0 19.5%

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

    1. neg-mul-1 19.5%

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

    2. associate-+r+ 19.5%

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

    3. sub-neg 19.5%

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

    4. associate--l+ 19.5%

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

    5. div-sub 19.5%

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

13. Simplified 19.5%

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

14. Final simplification 19.5%

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

Alternative 2: 19.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

   1. add-sqr-sqrt 10.0%

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

   2. sqrt-unprod 19.5%

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

   3. pow2 19.5%

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

8. Applied egg-rr 19.5%

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

   1. unpow2 19.5%

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

   2. rem-sqrt-square 19.5%

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

10. Simplified 19.5%

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

    1. add-exp-log 19.5%

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

12. Applied egg-rr 19.5%

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

13. Final simplification 19.5%

    \[\leadsto hi \cdot \frac{\left|\frac{hi - x}{lo} + 1\right|}{lo} - \frac{e^{\log \left(x - lo\right)}}{lo} \]
14. Add Preprocessing

Alternative 3: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(hi * N[(N[Abs[N[(N[(N[(hi - x), $MachinePrecision] / 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 hi around 0 18.9%

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

   1. add-sqr-sqrt 10.0%

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

   2. sqrt-unprod 19.5%

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

   3. pow2 19.5%

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

8. Applied egg-rr 19.5%

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

   1. unpow2 19.5%

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

   2. rem-sqrt-square 19.5%

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

10. Simplified 19.5%

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

11. Taylor expanded in x around 0 19.5%

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

    1. associate-/l* 19.5%

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

    2. associate--l+ 19.5%

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

    3. div-sub 19.5%

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

13. Simplified 19.5%

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

14. Final simplification 19.5%

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

Alternative 4: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

7. Final simplification 18.9%

   \[\leadsto \frac{lo - x}{lo} + hi \cdot \frac{\frac{hi - x}{lo} + 1}{lo} \]
8. 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 hi around 0 18.9%

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

7. Taylor expanded in x around 0 18.9%

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

   1. associate-/l* 18.9%

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

9. Simplified 18.9%

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

10. Final simplification 18.9%

    \[\leadsto hi \cdot \frac{\frac{hi}{lo} + 1}{lo} + 1 \]
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. 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. 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 2024118 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))