xlohi (overflows)

Percentage Accurate: 3.1% → 19.4%

Time: 8.1s

Alternatives: 8

Speedup: 18.0×

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: 19.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. div-sub N/A

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

   5. sub-neg N/A

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

   6. *-inverses N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. metadata-eval N/A

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

   11. mul-1-neg N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 19.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 19.0%

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

9. Taylor expanded in lo around 0

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

11. Applied rewrites 19.6%

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

12. Taylor expanded in lo around inf

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

14. Applied rewrites 19.6%

    \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{lo \cdot lo}, hi, \frac{\frac{hi}{lo} \cdot \frac{hi}{lo}}{x}\right) - \frac{1}{lo}\right) \cdot x \]
15. Add Preprocessing

Alternative 2: 18.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. div-sub N/A

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

   5. sub-neg N/A

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

   6. *-inverses N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. metadata-eval N/A

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

   11. mul-1-neg N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 19.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 19.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 19.0%

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

Alternative 3: 18.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. div-sub N/A

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

   5. sub-neg N/A

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

   6. *-inverses N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. metadata-eval N/A

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

   11. mul-1-neg N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 19.0%

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

7. Applied rewrites 19.0%

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

8. Final simplification 19.0%

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

Alternative 4: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

5. Applied rewrites 19.0%

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

Alternative 5: 18.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{hi + lo}{lo}, \frac{hi}{lo}, 1\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. div-sub N/A

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

   5. sub-neg N/A

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

   6. *-inverses N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. metadata-eval N/A

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

   11. mul-1-neg N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 19.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 19.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 19.0%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 19.0%

    \[\leadsto \mathsf{fma}\left(\frac{hi + lo}{lo}, \frac{hi}{\color{blue}{lo}}, 1\right) \]
15. Add Preprocessing

Alternative 6: 18.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around 0

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

   1. distribute-rgt-in N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. neg-mul-1 N/A

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

   6. associate-+l+ N/A

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

   7. associate-*r/ N/A

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

   8. mul-1-neg N/A

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

   9. div-add N/A

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

   10. +-commutative N/A

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

   11. sub-neg N/A

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in lo around inf

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

8. Applied rewrites 18.8%

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

Alternative 7: 18.8% accurate, 1.3× 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

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 18.8

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in lo around inf

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

8. Applied rewrites 18.8%

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

Alternative 8: 18.7% accurate, 18.0× 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. Applied rewrites 18.7%

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

Reproduce

herbie shell --seed 2024312 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))