xlohi (overflows)

Percentage Accurate: 3.1% → 19.4%

Time: 31.2s

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: 19.4% 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 19.0%

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

4. Taylor expanded in lo around inf 19.0%

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

   1. associate--l+ 19.0%

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

   2. div-sub 19.0%

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

6. Simplified 19.0%

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

   1. add-cbrt-cube 19.0%

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

   2. pow3 19.0%

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

   3. +-commutative 19.0%

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

   4. fma-define 19.0%

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

   5. mul-1-neg 19.0%

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

   6. div-sub 19.0%

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

   7. pow1 19.0%

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

   8. pow1 19.0%

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

   9. pow-div 19.0%

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

   10. metadata-eval 19.0%

       \[\leadsto \sqrt[3]{{\left(\mathsf{fma}\left(hi, \frac{1 + \frac{hi - x}{lo}}{lo}, -\left(\frac{x}{lo} - {lo}^{\color{blue}{0}}\right)\right)\right)}^{3}} \]

   11. metadata-eval 19.0%

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

8. Applied egg-rr 19.0%

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

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 19.5%

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

    4. unpow2 19.5%

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

11. Simplified 19.5%

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

Alternative 2: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(lo + x), $MachinePrecision] / lo), $MachinePrecision] + N[(hi * N[(N[(1.0 + N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $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 19.0%

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

4. Taylor expanded in lo around inf 19.0%

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

   1. associate--l+ 19.0%

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

   2. div-sub 19.0%

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

6. Simplified 19.0%

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

   1. div-inv 19.0%

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

8. Applied egg-rr 19.0%

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

   1. add-sqr-sqrt 19.0%

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

   2. sqrt-unprod 19.0%

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

   3. mul-1-neg 19.0%

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

   4. mul-1-neg 19.0%

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

   5. sqr-neg 19.0%

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

   6. sqrt-unprod 0.0%

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

   7. add-sqr-sqrt 2.1%

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

   8. un-div-inv 2.1%

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

   9. sub-neg 2.1%

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

   10. add-sqr-sqrt 2.1%

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

   11. sqrt-unprod 0.7%

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

   12. sqr-neg 0.7%

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

   13. sqrt-unprod 0.0%

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

   14. add-sqr-sqrt 19.0%

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

10. Applied egg-rr 19.0%

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

11. Final simplification 19.0%

    \[\leadsto \frac{lo + x}{lo} + hi \cdot \frac{1 + \frac{hi - x}{lo}}{lo} \]
12. Add Preprocessing

Alternative 3: 18.9% accurate, 0.5× speedup

Math

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

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(hi * Float64(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[(hi * N[(N[(1.0 + N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $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 19.0%

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

4. Taylor expanded in lo around inf 19.0%

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

   1. associate--l+ 19.0%

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

   2. div-sub 19.0%

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

6. Simplified 19.0%

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

7. Taylor expanded in x around 0 19.0%

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

8. Final simplification 19.0%

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

Alternative 4: 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 19.0%

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

4. Taylor expanded in lo around inf 19.0%

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

   1. associate--l+ 19.0%

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

   2. div-sub 19.0%

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

6. Simplified 19.0%

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

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 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%

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

   1. neg-mul-1 18.8%

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

6. Simplified 18.8%

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

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 2024157 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))