xlohi (overflows)

Percentage Accurate: 3.1% → 18.8%

Time: 9.4s

Alternatives: 6

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: 18.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. flip-- 18.8%

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

6. Applied egg-rr 18.8%

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

8. Simplified 18.8%

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

9. Taylor expanded in hi around inf 18.8%

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

    1. +-commutative 18.8%

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

11. Simplified 18.8%

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

12. Taylor expanded in lo around inf 18.8%

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

13. Final simplification 18.8%

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

Alternative 2: 18.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = -((1.0d0 - (x / hi)) ** 2.0d0) / (hi / lo)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. flip-- 18.8%

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

6. Applied egg-rr 18.8%

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

8. Simplified 18.8%

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

9. Taylor expanded in hi around inf 18.8%

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

    1. +-commutative 18.8%

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

11. Simplified 18.8%

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

12. Taylor expanded in lo around inf 18.8%

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

    1. mul-1-neg 18.8%

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

    2. associate-/l* 18.8%

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

    3. distribute-frac-neg 18.8%

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

14. Simplified 18.8%

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

15. Final simplification 18.8%

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

Alternative 3: 18.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. flip-- 18.8%

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

6. Applied egg-rr 18.8%

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

8. Simplified 18.8%

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

9. Taylor expanded in lo around -inf 18.8%

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

    1. *-commutative 18.8%

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

    2. associate-*r/ 18.8%

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

    3. div-sub 18.8%

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

    4. associate-/r* 18.8%

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

    5. unpow2 18.8%

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

    6. sub-neg 18.8%

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

    7. mul-1-neg 18.8%

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

    8. +-commutative 18.8%

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

    9. mul-1-neg 18.8%

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

    10. distribute-lft-neg-in 18.8%

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

    11. +-commutative 18.8%

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

    12. mul-1-neg 18.8%

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

    13. sub-neg 18.8%

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

    14. unpow2 18.8%

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

    15. associate-/r* 18.8%

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

    16. div-sub 18.8%

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

11. Simplified 18.8%

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

12. Final simplification 18.8%

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

Alternative 4: 18.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

   1. flip-- 18.8%

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

6. Applied egg-rr 18.8%

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

8. Simplified 18.8%

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

9. Taylor expanded in lo around -inf 18.8%

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

10. Final simplification 18.8%

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

Alternative 5: 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. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

5. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

Alternative 6: 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. Taylor expanded in lo around inf 18.7%

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

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023208 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))