xlohi (overflows)

Percentage Accurate: 3.1% → 19.3%

Time: 23.7s

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: 19.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{x - hi}{lo} \cdot -0.3333333333333333 + 1\right)}^{3} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (pow (+ (* (/ (- x hi) lo) -0.3333333333333333) 1.0) 3.0))

C

double code(double lo, double hi, double x) {
	return pow(((((x - hi) / lo) * -0.3333333333333333) + 1.0), 3.0);
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = ((((x - hi) / lo) * (-0.3333333333333333d0)) + 1.0d0) ** 3.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.pow(((((x - hi) / lo) * -0.3333333333333333) + 1.0), 3.0);
}

Python

def code(lo, hi, x):
	return math.pow(((((x - hi) / lo) * -0.3333333333333333) + 1.0), 3.0)

Julia

function code(lo, hi, x)
	return Float64(Float64(Float64(Float64(x - hi) / lo) * -0.3333333333333333) + 1.0) ^ 3.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = ((((x - hi) / lo) * -0.3333333333333333) + 1.0) ^ 3.0;
end

Wolfram

code[lo_, hi_, x_] := N[Power[N[(N[(N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf 3.1%

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

   1. mul-1-neg 3.1%

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

   2. unsub-neg 3.1%

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

   3. +-commutative 3.1%

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

   4. associate-/l* 14.7%

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

   5. fma-define 14.7%

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

5. Simplified 14.7%

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

   1. add-cube-cbrt 14.7%

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

   2. pow3 14.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{1 - \frac{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x\right) - hi}{lo}}\right)}^{3}} \]

7. Applied egg-rr 14.7%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{1 - \frac{\mathsf{fma}\left(hi, \frac{x - hi}{lo}, x\right) - hi}{lo}}\right)}^{3}} \]

8. Taylor expanded in lo around -inf 19.3%

   \[\leadsto {\color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x - hi}{lo}\right)}}^{3} \]
9. Step-by-step derivation

   1. *-commutative 19.3%

      \[\leadsto {\left(1 + \color{blue}{\frac{x - hi}{lo} \cdot -0.3333333333333333}\right)}^{3} \]

10. Simplified 19.3%

    \[\leadsto {\color{blue}{\left(1 + \frac{x - hi}{lo} \cdot -0.3333333333333333\right)}}^{3} \]

11. Final simplification 19.3%

    \[\leadsto {\left(\frac{x - hi}{lo} \cdot -0.3333333333333333 + 1\right)}^{3} \]
12. Add Preprocessing

Alternative 2: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Taylor expanded in lo around inf 18.8%

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

   1. associate--l+ 18.8%

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

   2. div-sub 18.8%

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

6. Simplified 18.8%

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

7. Taylor expanded in x around 0 18.8%

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

8. Final simplification 18.8%

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

Alternative 3: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate--l+ 18.8%

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

   2. div-sub 18.8%

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

6. Simplified 18.8%

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

   1. div-inv 18.8%

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

8. Applied egg-rr 18.8%

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

9. Taylor expanded in x around 0 18.8%

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

    1. associate-/l* 18.8%

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

11. Simplified 18.8%

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

12. Final simplification 18.8%

    \[\leadsto hi \cdot \frac{\frac{hi}{lo} + 1}{lo} + 1 \]
13. 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: 18.8% 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 18.8%

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

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 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 2024143 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))