xlohi (overflows)

Percentage Accurate: 3.1% → 40.6%

Time: 25.3s

Alternatives: 5

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: 40.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Simplified 18.8%

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

   1. associate-*r/ N/A

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

   2. div-inv N/A

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

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

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

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

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

   5. +-commutative N/A

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

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

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

   7. /-lowering-/.f64 N/A

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

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

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

   9. /-lowering-/.f64 18.8%

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

7. Applied egg-rr 18.8%

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

   1. associate-*l* N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

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

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

9. Applied egg-rr 38.3%

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

Alternative 2: 19.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] * N[(hi / 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 18.8%

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

   1. associate-*r/ N/A

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

   2. div-inv N/A

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

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

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

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

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

   5. +-commutative N/A

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

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

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

   7. /-lowering-/.f64 N/A

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

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

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

   9. /-lowering-/.f64 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in lo around 0

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

   1. unpow2 N/A

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

   2. times-frac N/A

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

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

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

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

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

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

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

   6. --lowering--.f64 19.3%

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

10. Simplified 19.3%

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

11. Final simplification 19.3%

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

Alternative 3: 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 4: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0

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

   1. +-commutative N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x}{hi}\right), \color{blue}{\left(lo \cdot \left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) - \left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \left(\color{blue}{lo} \cdot \left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) - \left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \mathsf{*.f64}\left(lo, \color{blue}{\left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) - \left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \mathsf{*.f64}\left(lo, \left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, hi\right), \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\left(-1 \cdot \left(lo \cdot \left(-1 \cdot \frac{x}{{hi}^{3}} + \frac{1}{{hi}^{2}}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x}{{hi}^{2}} + \frac{1}{hi}\right)\right)\right)}\right)\right)\right) \]

5. Simplified 18.8%

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

6. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(lo, \color{blue}{\left(-1 \cdot \frac{lo}{{hi}^{2}} - \frac{1}{hi}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \left(-1 \cdot \frac{lo}{{hi}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{hi}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\left(-1 \cdot \frac{lo}{{hi}^{2}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{hi}\right)\right)}\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{lo}{{hi}^{2}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{hi}}\right)\right)\right)\right) \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\left(0 - \frac{lo}{{hi}^{2}}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{hi}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{lo}{{hi}^{2}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{hi}}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(lo, \left({hi}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{\color{blue}{hi}}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(lo, \left(hi \cdot hi\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{hi}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, hi\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{hi}\right)\right)\right)\right) \]

   10. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, hi\right)\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{hi}}\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, hi\right)\right)\right), \left(\frac{-1}{hi}\right)\right)\right) \]

   12. /-lowering-/.f64 18.8%

       \[\leadsto \mathsf{*.f64}\left(lo, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(lo, \mathsf{*.f64}\left(hi, hi\right)\right)\right), \mathsf{/.f64}\left(-1, \color{blue}{hi}\right)\right)\right) \]

8. Simplified 18.8%

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

9. Taylor expanded in lo around 0

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

    1. /-lowering-/.f64 18.8%

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

11. Simplified 18.8%

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

Alternative 5: 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 18.6%

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

Reproduce

herbie shell --seed 2024141 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))