xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 18.2s

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, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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. Step-by-step derivation

   1. mul-1-neg 18.9%

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

   2. distribute-rgt-neg-in 18.9%

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

   3. +-commutative 18.9%

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

   4. mul-1-neg 18.9%

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

   5. unsub-neg 18.9%

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

   6. +-commutative 18.9%

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

   7. associate-/l* 18.9%

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

   8. fma-define 18.9%

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

8. Simplified 18.9%

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

9. Taylor expanded in hi around inf 0.0%

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

    1. unpow2 0.0%

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

    2. unpow2 0.0%

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

    3. times-frac 19.4%

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

    4. unpow2 19.4%

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

11. Simplified 19.4%

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

12. Taylor expanded in lo around 0 19.4%

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

13. Final simplification 19.4%

    \[\leadsto x \cdot \left(\frac{{\left(\frac{hi}{lo}\right)}^{2}}{x} - \frac{hi}{{lo}^{2}}\right) \]
14. Add Preprocessing

Alternative 2: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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. Step-by-step derivation

   1. mul-1-neg 18.9%

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

   2. distribute-rgt-neg-in 18.9%

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

   3. +-commutative 18.9%

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

   4. mul-1-neg 18.9%

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

   5. unsub-neg 18.9%

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

   6. +-commutative 18.9%

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

   7. associate-/l* 18.9%

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

   8. fma-define 18.9%

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

8. Simplified 18.9%

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

9. Taylor expanded in hi around inf 0.0%

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

    1. unpow2 0.0%

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

    2. unpow2 0.0%

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

    3. times-frac 19.4%

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

    4. unpow2 19.4%

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

11. Simplified 19.4%

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

12. Taylor expanded in lo around inf 19.4%

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

13. Final simplification 19.4%

    \[\leadsto x \cdot \left(\frac{{\left(\frac{hi}{lo}\right)}^{2}}{x} + \frac{-1 - \frac{hi}{lo}}{lo}\right) \]
14. Add Preprocessing

Alternative 3: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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. Step-by-step derivation

   1. mul-1-neg 18.9%

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

   2. distribute-rgt-neg-in 18.9%

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

   3. +-commutative 18.9%

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

   4. mul-1-neg 18.9%

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

   5. unsub-neg 18.9%

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

   6. +-commutative 18.9%

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

   7. associate-/l* 18.9%

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

   8. fma-define 18.9%

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

8. Simplified 18.9%

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

9. Taylor expanded in hi around inf 0.0%

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

    1. unpow2 0.0%

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

    2. unpow2 0.0%

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

    3. times-frac 19.4%

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

    4. unpow2 19.4%

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

11. Simplified 19.4%

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

12. Taylor expanded in lo around inf 19.4%

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

13. Final simplification 19.4%

    \[\leadsto x \cdot \left(\frac{{\left(\frac{hi}{lo}\right)}^{2}}{x} + \frac{-1}{lo}\right) \]
14. Add Preprocessing

Alternative 4: 19.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf 0.7%

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

   1. +-commutative 0.7%

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

   2. associate--l+ 0.7%

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

   3. +-commutative 0.7%

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

   4. *-rgt-identity 0.7%

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

   5. *-commutative 0.7%

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

   6. associate-/l* 9.3%

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

   7. distribute-lft-out 9.5%

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

5. Simplified 9.5%

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

   1. add-sqr-sqrt 8.8%

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

   2. sqrt-unprod 17.9%

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

   3. pow2 17.9%

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

7. Applied egg-rr 17.9%

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

   1. unpow2 17.9%

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

   2. rem-sqrt-square 17.9%

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

   3. *-commutative 17.9%

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

   4. associate-*r/ 17.9%

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

9. Simplified 17.9%

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

10. Taylor expanded in lo around inf 19.1%

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

11. Final simplification 19.1%

    \[\leadsto \left|\frac{lo}{hi} \cdot \frac{x - lo}{hi}\right| \]
12. 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 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 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-frac 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 2024080 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))