xlohi (overflows)

Percentage Accurate: 3.1% → 18.8%

Time: 7.7s

Alternatives: 4

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: 18.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0 18.8%

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

3. Taylor expanded in lo around inf 18.8%

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

   1. mul-1-neg 18.8%

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

   2. *-commutative 18.8%

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

   3. distribute-rgt-neg-in 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

   7. unpow2 18.8%

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

   8. associate-/r* 18.8%

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

5. Simplified 18.8%

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

   1. *-un-lft-identity 18.8%

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

   2. div-inv 18.8%

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

   3. prod-diff 18.8%

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

   4. times-frac 18.8%

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

   5. *-un-lft-identity 18.8%

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

   6. associate-/l/ 18.8%

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

   7. fma-neg 18.8%

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

   8. *-un-lft-identity 18.8%

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

   9. sub-div 18.8%

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

   10. times-frac 18.8%

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

   11. *-un-lft-identity 18.8%

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

7. Applied egg-rr 18.8%

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

   1. distribute-neg-frac 18.8%

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

   2. metadata-eval 18.8%

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

9. Simplified 18.8%

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

10. Final simplification 18.8%

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

Alternative 2: 18.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0 18.8%

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

3. Taylor expanded in lo around inf 18.8%

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

   1. mul-1-neg 18.8%

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

   2. *-commutative 18.8%

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

   3. distribute-rgt-neg-in 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

   7. unpow2 18.8%

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

   8. associate-/r* 18.8%

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

5. Simplified 18.8%

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

6. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. *-commutative 18.8%

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

   3. unpow2 18.8%

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

   4. associate-/r* 18.8%

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

   5. div-sub 18.8%

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

   6. *-commutative 18.8%

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

   7. distribute-rgt-neg-in 18.8%

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

   8. distribute-neg-frac 18.8%

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

   9. sub-neg 18.8%

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

   10. distribute-neg-in 18.8%

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

   11. metadata-eval 18.8%

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

   12. remove-double-neg 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

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

Alternative 3: 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. Taylor expanded in lo around 0 18.8%

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

3. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

5. Simplified 18.8%

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

6. Final simplification 18.8%

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

Alternative 4: 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. Taylor expanded in lo around inf 18.7%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))