xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 11.9s

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 lo around inf 0.0%

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

5. Simplified 18.9%

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

6. Taylor expanded in x around -inf 18.9%

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

7. Taylor expanded in hi around inf 0.0%

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

   1. unpow2 0.0%

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

   2. unpow2 0.0%

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

   3. times-frac 19.6%

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

   4. unpow2 19.6%

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

   5. neg-mul-1 19.6%

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

9. Simplified 19.6%

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

10. Final simplification 19.6%

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

Alternative 2: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(hi * N[(N[(N[(hi / lo), $MachinePrecision] + 1.0), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] - N[(N[(x - lo), $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 10.3%

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

   1. mul-1-neg 10.3%

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

   2. unsub-neg 10.3%

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

   3. *-commutative 10.3%

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

   4. *-lft-identity 10.3%

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

   5. times-frac 18.9%

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

   6. /-rgt-identity 18.9%

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

6. Simplified 18.9%

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

7. Taylor expanded in lo around -inf 18.9%

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

   1. mul-1-neg 18.9%

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

   2. sub-neg 18.9%

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

   3. mul-1-neg 18.9%

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

   4. distribute-neg-in 18.9%

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

   5. +-commutative 18.9%

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

   6. distribute-neg-frac2 18.9%

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

   7. +-commutative 18.9%

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

   8. distribute-neg-in 18.9%

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

   9. mul-1-neg 18.9%

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

   10. sub-neg 18.9%

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

   11. associate-*r/ 18.9%

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

   12. *-commutative 18.9%

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

   13. associate-/l* 18.9%

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

   14. metadata-eval 18.9%

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

   15. distribute-neg-frac 18.9%

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

   16. fma-neg 18.9%

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

   17. distribute-neg-frac 18.9%

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

   18. metadata-eval 18.9%

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

   19. metadata-eval 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in x around 0 18.9%

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

    1. mul-1-neg 18.9%

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

    2. associate-/l* 18.9%

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

    3. associate-*r/ 18.9%

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

    4. *-commutative 18.9%

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

    5. associate-*r/ 18.9%

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

    6. fma-neg 18.9%

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

    7. metadata-eval 18.9%

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

    8. distribute-lft-neg-in 18.9%

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

    9. fma-define 18.9%

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

    10. +-commutative 18.9%

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

    11. associate-*r/ 18.9%

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

    12. *-commutative 18.9%

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

    13. associate-*r/ 18.9%

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

    14. mul-1-neg 18.9%

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

    15. unsub-neg 18.9%

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

12. Simplified 18.9%

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

13. Final simplification 18.9%

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

Alternative 3: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.9%

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

6. Final simplification 18.9%

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

Alternative 4: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.9%

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

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%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

   \[\leadsto hi \cdot \frac{\frac{hi}{lo} + 1}{lo} + 1 \]
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(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. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024060 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))