xlohi (overflows)

Percentage Accurate: 3.1% → 19.5%

Time: 16.4s

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: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return pow((hi / lo), 2.0);
}

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) ** 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

   1. *-un-lft-identity 18.9%

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

   2. add-cbrt-cube 9.7%

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

   3. unpow2 9.7%

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

   4. cbrt-prod 9.7%

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

   5. times-frac 9.7%

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

   6. unpow2 9.7%

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

   7. cbrt-prod 18.9%

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

   8. pow2 18.9%

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

8. Applied egg-rr 18.9%

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

9. Taylor expanded in hi around inf 0.0%

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

    1. unpow2 0.0%

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

    2. unpow2 0.0%

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

    3. times-frac 19.6%

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

    4. unpow2 19.6%

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

11. Simplified 19.6%

    \[\leadsto \color{blue}{{\left(\frac{hi}{lo}\right)}^{2}} \]
12. Add Preprocessing

Alternative 2: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(hi * N[(1.0 + N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $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 18.9%

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

4. Taylor expanded in x around 0 19.0%

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

5. Taylor expanded in lo around inf 19.0%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

7. Simplified 19.0%

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

   1. add-exp-log 19.0%

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

   2. metadata-eval 19.0%

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

   3. log1p-define 19.0%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(hi \cdot \frac{1 + \frac{hi - x}{lo}}{lo}\right)}} \]

9. Applied egg-rr 19.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(hi \cdot \frac{1 + \frac{hi - x}{lo}}{lo}\right)}} \]
10. Step-by-step derivation

    1. log1p-undefine 19.0%

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

    2. rem-exp-log 19.0%

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

    3. associate-*r/ 19.0%

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

11. Simplified 19.0%

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

Alternative 3: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.9%

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

4. Taylor expanded in lo around inf 18.9%

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

   1. associate--l+ 18.9%

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

   2. div-sub 18.9%

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

6. Simplified 18.9%

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

7. Taylor expanded in x around 0 19.0%

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

   1. associate-/l* 19.0%

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

9. Simplified 19.0%

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

10. Final simplification 19.0%

    \[\leadsto 1 + hi \cdot \frac{\frac{hi}{lo} + 1}{lo} \]
11. Add Preprocessing

Alternative 4: 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(lo / Float64(-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 18.7%

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

4. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

   \[\leadsto \frac{lo}{-hi} \]
8. 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 18.7%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))