xlohi (overflows)

Percentage Accurate: 3.1% → 18.9%

Time: 15.6s

Alternatives: 6

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: 18.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ hi \cdot \left(\frac{\frac{hi}{lo} + 1}{lo} - \frac{x}{{lo}^{2}}\right) - \frac{x - lo}{lo} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (- (* hi (- (/ (+ (/ hi lo) 1.0) lo) (/ x (pow lo 2.0)))) (/ (- x lo) lo)))

C

double code(double lo, double hi, double x) {
	return (hi * ((((hi / lo) + 1.0) / lo) - (x / pow(lo, 2.0)))) - ((x - 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) + 1.0d0) / lo) - (x / (lo ** 2.0d0)))) - ((x - lo) / lo)
end function

Java

public static double code(double lo, double hi, double x) {
	return (hi * ((((hi / lo) + 1.0) / lo) - (x / Math.pow(lo, 2.0)))) - ((x - lo) / lo);
}

Python

def code(lo, hi, x):
	return (hi * ((((hi / lo) + 1.0) / lo) - (x / math.pow(lo, 2.0)))) - ((x - lo) / lo)

Julia

function code(lo, hi, x)
	return Float64(Float64(hi * Float64(Float64(Float64(Float64(hi / lo) + 1.0) / lo) - Float64(x / (lo ^ 2.0)))) - Float64(Float64(x - lo) / lo))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (hi * ((((hi / lo) + 1.0) / lo) - (x / (lo ^ 2.0)))) - ((x - lo) / lo);
end

Wolfram

code[lo_, hi_, x_] := N[(N[(hi * N[(N[(N[(N[(hi / lo), $MachinePrecision] + 1.0), $MachinePrecision] / lo), $MachinePrecision] - N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x - lo), $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 18.8%

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

4. Taylor expanded in lo around inf 18.8%

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

5. Final simplification 18.8%

   \[\leadsto hi \cdot \left(\frac{\frac{hi}{lo} + 1}{lo} - \frac{x}{{lo}^{2}}\right) - \frac{x - lo}{lo} \]
6. Add Preprocessing

Alternative 2: 18.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left(\frac{hi}{lo} + {\left(\frac{hi}{lo}\right)}^{2}\right) + 1 \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+ (+ (/ hi lo) (pow (/ hi lo) 2.0)) 1.0))

C

double code(double lo, double hi, double x) {
	return ((hi / lo) + pow((hi / lo), 2.0)) + 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 / lo) + ((hi / lo) ** 2.0d0)) + 1.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return ((hi / lo) + Math.pow((hi / lo), 2.0)) + 1.0;
}

Python

def code(lo, hi, x):
	return ((hi / lo) + math.pow((hi / lo), 2.0)) + 1.0

Julia

function code(lo, hi, x)
	return Float64(Float64(Float64(hi / lo) + (Float64(hi / lo) ^ 2.0)) + 1.0)
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = ((hi / lo) + ((hi / lo) ^ 2.0)) + 1.0;
end

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(hi / lo), $MachinePrecision] + N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision]), $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 18.8%

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

4. Taylor expanded in lo around inf 18.8%

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

5. Taylor expanded in x around 0 9.6%

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

   1. distribute-rgt-in 9.6%

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

   2. associate-*l/ 9.6%

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

   3. *-lft-identity 9.6%

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

   4. associate-*l/ 0.0%

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

   5. unpow2 0.0%

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

   6. times-frac 18.8%

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

   7. unpow2 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

   \[\leadsto \left(\frac{hi}{lo} + {\left(\frac{hi}{lo}\right)}^{2}\right) + 1 \]
9. Add Preprocessing

Alternative 3: 18.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.8%

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

4. Taylor expanded in lo around inf 18.8%

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

   1. *-un-lft-identity 18.8%

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

   2. unpow2 18.8%

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

   3. times-frac 18.8%

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

6. Applied egg-rr 18.8%

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

   1. associate-*l/ 18.8%

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

   2. *-lft-identity 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

   \[\leadsto \frac{lo - x}{lo} + hi \cdot \left(\frac{\frac{hi}{lo} + 1}{lo} - \frac{\frac{x}{lo}}{lo}\right) \]
10. 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 18.8%

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

Alternative 5: 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 18.7%

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

   1. div-sub 18.7%

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

   2. sub-neg 18.7%

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

   3. *-inverses 18.7%

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

   4. metadata-eval 18.7%

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

   5. distribute-lft-in 18.7%

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

   6. metadata-eval 18.7%

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

   7. +-commutative 18.7%

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

   8. mul-1-neg 18.7%

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

   9. unsub-neg 18.7%

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

5. Simplified 18.7%

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

Alternative 6: 18.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(lo, hi, x)
	return 1.0
end

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 18.7%

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

Reproduce

herbie shell --seed 2024087 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))