xlohi (overflows)

Percentage Accurate: 3.1% → 99.2%

Time: 11.6s

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Simplified 18.9%

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

   1. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{lo}{hi}}\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{hi}, x\right), lo\right)\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{lo \cdot \frac{1}{hi}}\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right)\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{lo}}{\frac{1}{hi}}\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{hi}, x\right), lo\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{lo}\right), \left(\frac{1}{hi}\right)\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{hi}, x\right), lo\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, lo\right), \left(\frac{1}{hi}\right)\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right)\right)\right) \]

   6. /-lowering-/.f64 18.9%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(1, hi\right)\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right)\right)\right) \]

7. Applied egg-rr 18.9%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi - x}{lo} \cdot \color{blue}{\left(\frac{\frac{1}{lo}}{\frac{1}{hi}} + 1\right)}\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{lo}}{\frac{1}{hi}} \cdot \frac{hi - x}{lo} + \color{blue}{1 \cdot \frac{hi - x}{lo}}\right)\right) \]

   3. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{lo}}{1} \cdot hi\right) \cdot \frac{hi - x}{lo} + 1 \cdot \frac{hi - x}{lo}\right)\right) \]

   4. /-rgt-identity N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{lo} \cdot hi\right) \cdot \frac{hi - x}{lo} + 1 \cdot \frac{hi - x}{lo}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1 \cdot hi}{lo} \cdot \frac{hi - x}{lo} + 1 \cdot \frac{hi - x}{lo}\right)\right) \]

   6. *-lft-identity N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi}{lo} \cdot \frac{hi - x}{lo} + 1 \cdot \frac{hi - x}{lo}\right)\right) \]

   7. distribute-rgt-in N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi - x}{lo} \cdot \color{blue}{\left(\frac{hi}{lo} + 1\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi - x}{lo} \cdot \left(1 + \color{blue}{\frac{hi}{lo}}\right)\right)\right) \]

   9. flip-+ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi - x}{lo} \cdot \frac{1 \cdot 1 - \frac{hi}{lo} \cdot \frac{hi}{lo}}{\color{blue}{1 - \frac{hi}{lo}}}\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{hi - x}{lo} \cdot \left(1 \cdot 1 - \frac{hi}{lo} \cdot \frac{hi}{lo}\right)}{\color{blue}{1 - \frac{hi}{lo}}}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{hi - x}{lo} \cdot \left(1 \cdot 1 - \frac{hi}{lo} \cdot \frac{hi}{lo}\right)\right), \color{blue}{\left(1 - \frac{hi}{lo}\right)}\right)\right) \]

9. Applied egg-rr 99.4%

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

10. Taylor expanded in hi around 0

    \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{x}{lo} + \frac{hi}{lo}\right)}, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{hi}{lo} + -1 \cdot \frac{x}{lo}\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]

    2. mul-1-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{hi}{lo} + \left(\mathsf{neg}\left(\frac{x}{lo}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]

    3. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{hi}{lo} - \frac{x}{lo}\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]

    4. div-sub N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{hi - x}{lo}\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(hi - x\right), lo\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]

    6. --lowering--.f64 99.4%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right)\right)\right) \]

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 2: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf

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

5. Simplified 18.9%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 - \left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}}{1 \cdot 1 - \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right) \cdot \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 - \left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right), \color{blue}{\left(1 \cdot 1 - \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right) \cdot \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)\right)}\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)\right), \left(\color{blue}{1 \cdot 1} - \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right) \cdot \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{hi}{lo} + 1\right), \left(\frac{hi - x}{lo}\right)\right)\right), \left(1 \cdot \color{blue}{1} - \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right) \cdot \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(1 + \frac{hi}{lo}\right), \left(\frac{hi - x}{lo}\right)\right)\right), \left(1 \cdot 1 - \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right) \cdot \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{hi}{lo}\right)\right), \left(\frac{hi - x}{lo}\right)\right)\right), \left(1 \cdot 1 - \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right) \cdot \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \left(\frac{hi - x}{lo}\right)\right)\right), \left(1 \cdot 1 - \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right) \cdot \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\left(hi - x\right), lo\right)\right)\right), \left(1 \cdot 1 - \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right) \cdot \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(hi, lo\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right)\right)\right), \left(1 \cdot 1 - \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right) \cdot \left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)\right)\right)\right) \]

7. Applied egg-rr 18.9%

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

8. Taylor expanded in lo around -inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{hi - x}{lo}\right)}\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 + \left(\mathsf{neg}\left(\frac{hi - x}{lo}\right)\right)\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 - \color{blue}{\frac{hi - x}{lo}}\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{hi - x}{lo}\right)}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(hi - x\right), \color{blue}{lo}\right)\right)\right) \]

   5. --lowering--.f64 98.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right)\right)\right) \]

10. Simplified 98.5%

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

11. Final simplification 98.5%

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

Alternative 3: 19.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf

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

5. Simplified 18.9%

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

   1. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{lo}{hi}}\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{hi}, x\right), lo\right)\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{lo \cdot \frac{1}{hi}}\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right)\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{lo}}{\frac{1}{hi}}\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{hi}, x\right), lo\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{lo}\right), \left(\frac{1}{hi}\right)\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{hi}, x\right), lo\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, lo\right), \left(\frac{1}{hi}\right)\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right)\right)\right) \]

   6. /-lowering-/.f64 18.9%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, lo\right), \mathsf{/.f64}\left(1, hi\right)\right), 1\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(hi, x\right), lo\right)\right)\right) \]

7. Applied egg-rr 18.9%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi - x}{lo} \cdot \color{blue}{\left(\frac{\frac{1}{lo}}{\frac{1}{hi}} + 1\right)}\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{lo}}{\frac{1}{hi}} \cdot \frac{hi - x}{lo} + \color{blue}{1 \cdot \frac{hi - x}{lo}}\right)\right) \]

   3. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{\frac{1}{lo}}{1} \cdot hi\right) \cdot \frac{hi - x}{lo} + 1 \cdot \frac{hi - x}{lo}\right)\right) \]

   4. /-rgt-identity N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{lo} \cdot hi\right) \cdot \frac{hi - x}{lo} + 1 \cdot \frac{hi - x}{lo}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1 \cdot hi}{lo} \cdot \frac{hi - x}{lo} + 1 \cdot \frac{hi - x}{lo}\right)\right) \]

   6. *-lft-identity N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi}{lo} \cdot \frac{hi - x}{lo} + 1 \cdot \frac{hi - x}{lo}\right)\right) \]

   7. distribute-rgt-in N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi - x}{lo} \cdot \color{blue}{\left(\frac{hi}{lo} + 1\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi - x}{lo} \cdot \left(1 + \color{blue}{\frac{hi}{lo}}\right)\right)\right) \]

   9. flip-+ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{hi - x}{lo} \cdot \frac{1 \cdot 1 - \frac{hi}{lo} \cdot \frac{hi}{lo}}{\color{blue}{1 - \frac{hi}{lo}}}\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\frac{hi - x}{lo} \cdot \left(1 \cdot 1 - \frac{hi}{lo} \cdot \frac{hi}{lo}\right)}{\color{blue}{1 - \frac{hi}{lo}}}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{hi - x}{lo} \cdot \left(1 \cdot 1 - \frac{hi}{lo} \cdot \frac{hi}{lo}\right)\right), \color{blue}{\left(1 - \frac{hi}{lo}\right)}\right)\right) \]

9. Applied egg-rr 99.4%

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

10. Taylor expanded in hi around inf

    \[\leadsto \color{blue}{\frac{{hi}^{2}}{{lo}^{2}}} \]
11. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \frac{hi \cdot hi}{{\color{blue}{lo}}^{2}} \]

    2. associate-*r/ N/A

       \[\leadsto hi \cdot \color{blue}{\frac{hi}{{lo}^{2}}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(hi, \color{blue}{\left(\frac{hi}{{lo}^{2}}\right)}\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(hi, \left(\frac{hi}{lo \cdot \color{blue}{lo}}\right)\right) \]

    5. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(hi, \left(\frac{\frac{hi}{lo}}{\color{blue}{lo}}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\left(\frac{hi}{lo}\right), \color{blue}{lo}\right)\right) \]

    7. /-lowering-/.f64 19.6%

       \[\leadsto \mathsf{*.f64}\left(hi, \mathsf{/.f64}\left(\mathsf{/.f64}\left(hi, lo\right), lo\right)\right) \]

12. Simplified 19.6%

    \[\leadsto \color{blue}{hi \cdot \frac{\frac{hi}{lo}}{lo}} \]
13. Add Preprocessing

Alternative 4: 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

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x - lo\right), \color{blue}{hi}\right) \]

   2. --lowering--.f64 18.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, lo\right), hi\right) \]

5. Simplified 18.8%

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

Alternative 5: 18.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{lo}{hi} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (- 0.0 (/ lo hi)))

C

double code(double lo, double hi, double x) {
	return 0.0 - (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 = 0.0d0 - (lo / hi)
end function

Java

public static double code(double lo, double hi, double x) {
	return 0.0 - (lo / hi);
}

Python

def code(lo, hi, x):
	return 0.0 - (lo / hi)

Julia

function code(lo, hi, x)
	return Float64(0.0 - Float64(lo / hi))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 0.0 - (lo / hi);
end

Wolfram

code[lo_, hi_, x_] := N[(0.0 - N[(lo / hi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x - lo\right), \color{blue}{hi}\right) \]

   2. --lowering--.f64 18.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, lo\right), hi\right) \]

5. Simplified 18.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot lo\right)}, hi\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(lo\right)\right), hi\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(0 - lo\right), hi\right) \]

   3. --lowering--.f64 18.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, lo\right), hi\right) \]

8. Simplified 18.8%

   \[\leadsto \frac{\color{blue}{0 - lo}}{hi} \]

9. Final simplification 18.8%

   \[\leadsto 0 - \frac{lo}{hi} \]
10. Add Preprocessing

Alternative 6: 18.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{x - lo}{lo}\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\frac{x - lo}{lo}} \]

   3. div-sub N/A

      \[\leadsto 0 - \left(\frac{x}{lo} - \color{blue}{\frac{lo}{lo}}\right) \]

   4. sub-neg N/A

      \[\leadsto 0 - \left(\frac{x}{lo} + \color{blue}{\left(\mathsf{neg}\left(\frac{lo}{lo}\right)\right)}\right) \]

   5. *-inverses N/A

      \[\leadsto 0 - \left(\frac{x}{lo} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto 0 - \left(\frac{x}{lo} + -1\right) \]

   7. +-commutative N/A

      \[\leadsto 0 - \left(-1 + \color{blue}{\frac{x}{lo}}\right) \]

   8. associate--r+ N/A

      \[\leadsto \left(0 - -1\right) - \color{blue}{\frac{x}{lo}} \]

   9. metadata-eval N/A

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

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{x}{lo}\right)}\right) \]

   11. /-lowering-/.f64 18.7%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{lo}\right)\right) \]

5. Simplified 18.7%

   \[\leadsto \color{blue}{1 - \frac{x}{lo}} \]
6. 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

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 18.7%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))