xlohi (overflows)

Percentage Accurate: 3.1% → 99.2%

Time: 10.8s

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: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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 - (1.0d0 / ((lo / (x - hi)) + (hi / (hi - x))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[(1.0 / N[(N[(lo / N[(x - hi), $MachinePrecision]), $MachinePrecision] + N[(hi / N[(hi - x), $MachinePrecision]), $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 3.1%

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

   1. mul-1-neg 3.1%

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

   2. unsub-neg 3.1%

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

   3. +-commutative 3.1%

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

   4. associate-/l* 16.0%

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

   5. fma-define 16.0%

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

5. Simplified 16.0%

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

   1. clear-num 16.0%

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

   2. inv-pow 16.0%

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

7. Applied egg-rr 16.0%

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

   1. unpow-1 16.0%

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

9. Simplified 16.0%

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

10. Taylor expanded in lo around inf 10.4%

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

    1. +-commutative 10.4%

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

    2. mul-1-neg 10.4%

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

    3. unsub-neg 10.4%

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

    4. associate-/r* 98.9%

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

12. Simplified 98.9%

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

13. Taylor expanded in lo around 0 99.3%

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

    1. +-commutative 99.3%

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

    2. mul-1-neg 99.3%

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

    3. sub-neg 99.3%

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

15. Simplified 99.3%

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

16. Final simplification 99.3%

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

Alternative 2: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[(1.0 / N[(lo * N[(N[(1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision] / N[(x - hi), $MachinePrecision]), $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 3.1%

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

   1. mul-1-neg 3.1%

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

   2. unsub-neg 3.1%

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

   3. +-commutative 3.1%

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

   4. associate-/l* 16.0%

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

   5. fma-define 16.0%

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

5. Simplified 16.0%

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

   1. clear-num 16.0%

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

   2. inv-pow 16.0%

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

7. Applied egg-rr 16.0%

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

   1. unpow-1 16.0%

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

9. Simplified 16.0%

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

10. Taylor expanded in lo around inf 10.4%

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

    1. +-commutative 10.4%

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

    2. mul-1-neg 10.4%

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

    3. unsub-neg 10.4%

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

    4. associate-/r* 98.9%

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

12. Simplified 98.9%

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

13. Taylor expanded in lo around inf 10.4%

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

    1. +-commutative 10.4%

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

    2. *-commutative 10.4%

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

    3. neg-mul-1 10.4%

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

    4. unsub-neg 10.4%

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

    5. *-commutative 10.4%

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

    6. associate-/r* 98.9%

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

    7. div-sub 99.0%

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

15. Simplified 99.0%

    \[\leadsto 1 - \frac{1}{\color{blue}{lo \cdot \frac{1 - \frac{hi}{lo}}{x - hi}}} \]
16. Add Preprocessing

Alternative 3: 97.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 - N[(1.0 / N[(lo * N[(N[(1.0 / lo), $MachinePrecision] - N[(1.0 / hi), $MachinePrecision]), $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 3.1%

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

   1. mul-1-neg 3.1%

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

   2. unsub-neg 3.1%

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

   3. +-commutative 3.1%

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

   4. associate-/l* 16.0%

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

   5. fma-define 16.0%

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

5. Simplified 16.0%

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

   1. clear-num 16.0%

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

   2. inv-pow 16.0%

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

7. Applied egg-rr 16.0%

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

   1. unpow-1 16.0%

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

9. Simplified 16.0%

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

10. Taylor expanded in lo around inf 10.4%

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

    1. +-commutative 10.4%

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

    2. mul-1-neg 10.4%

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

    3. unsub-neg 10.4%

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

    4. associate-/r* 98.9%

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

12. Simplified 98.9%

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

13. Taylor expanded in x around 0 97.9%

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

Alternative 4: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(hi * 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 3.1%

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

   1. mul-1-neg 3.1%

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

   2. unsub-neg 3.1%

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

   3. +-commutative 3.1%

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

   4. associate-/l* 16.0%

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

   5. fma-define 16.0%

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

5. Simplified 16.0%

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

   1. add-cube-cbrt 16.0%

      \[\leadsto 1 - \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x\right)}} - hi}{lo} \]

   2. pow3 16.0%

      \[\leadsto 1 - \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x\right)}\right)}^{3}} - hi}{lo} \]

7. Applied egg-rr 16.0%

   \[\leadsto 1 - \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x\right)}\right)}^{3}} - hi}{lo} \]

8. Taylor expanded in hi around 0 19.0%

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

   1. sub-neg 19.0%

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

   2. +-commutative 19.0%

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

   3. mul-1-neg 19.0%

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

   4. sub-neg 19.0%

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

   5. div-sub 19.0%

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

   6. metadata-eval 19.0%

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

10. Simplified 19.0%

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

11. Taylor expanded in x around 0 19.0%

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

    1. sub-neg 19.0%

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

    2. mul-1-neg 19.0%

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

    3. remove-double-neg 19.0%

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

    4. associate-/l* 19.0%

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

13. Simplified 19.0%

    \[\leadsto \color{blue}{1 + hi \cdot \frac{1 + \frac{hi}{lo}}{lo}} \]
14. 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 18.7%

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]
4. 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(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. Add Preprocessing

3. Taylor expanded in hi around inf 18.7%

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

4. Taylor expanded in x around 0 18.7%

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

   1. associate-*r/ 18.7%

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

   2. neg-mul-1 18.7%

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

6. Simplified 18.7%

   \[\leadsto \color{blue}{\frac{-lo}{hi}} \]
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 2024139 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))