xlohi (overflows)

Percentage Accurate: 3.1% → 99.4%

Time: 7.1s

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: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{-{t_0}^{2}}{t_0 \cdot \frac{lo}{hi} + \frac{lo - x}{hi}} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (/ (- (pow t_0 2.0)) (+ (* t_0 (/ lo hi)) (/ (- lo x) hi)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return -pow(t_0, 2.0) / ((t_0 * (lo / hi)) + ((lo - x) / hi));
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (x - lo) / hi
    code = -(t_0 ** 2.0d0) / ((t_0 * (lo / hi)) + ((lo - x) / hi))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return -Math.pow(t_0, 2.0) / ((t_0 * (lo / hi)) + ((lo - x) / hi));
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	return -math.pow(t_0, 2.0) / ((t_0 * (lo / hi)) + ((lo - x) / hi))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	return Float64(Float64(-(t_0 ^ 2.0)) / Float64(Float64(t_0 * Float64(lo / hi)) + Float64(Float64(lo - x) / hi)))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	tmp = -(t_0 ^ 2.0) / ((t_0 * (lo / hi)) + ((lo - x) / hi));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, N[((-N[Power[t$95$0, 2.0], $MachinePrecision]) / N[(N[(t$95$0 * N[(lo / hi), $MachinePrecision]), $MachinePrecision] + N[(N[(lo - x), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. associate--l+ 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. times-frac 9.6%

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

   6. div-sub 9.6%

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

4. Simplified 9.6%

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

   1. flip-+ 9.6%

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

   2. div-sub 9.6%

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

6. Applied egg-rr 0.0%

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

   1. div-sub 0.0%

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

   2. associate-*l* 99.2%

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

   3. sub-neg 99.2%

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

   4. metadata-eval 99.2%

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

8. Simplified 99.2%

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

9. Taylor expanded in hi around inf 0.0%

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

    1. mul-1-neg 0.0%

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

    2. unpow2 0.0%

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

    3. unpow2 0.0%

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

    4. times-frac 99.2%

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

    5. unpow2 99.2%

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

11. Simplified 99.2%

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

    1. distribute-lft-in 99.5%

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

13. Applied egg-rr 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{t_0 \cdot \frac{lo - x}{hi}}{t_0 \cdot \left(\frac{lo}{hi} + -1\right)} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (/ (* t_0 (/ (- lo x) hi)) (* t_0 (+ (/ lo hi) -1.0)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return (t_0 * ((lo - x) / hi)) / (t_0 * ((lo / hi) + -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
    real(8) :: t_0
    t_0 = (x - lo) / hi
    code = (t_0 * ((lo - x) / hi)) / (t_0 * ((lo / hi) + (-1.0d0)))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return (t_0 * ((lo - x) / hi)) / (t_0 * ((lo / hi) + -1.0));
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	return (t_0 * ((lo - x) / hi)) / (t_0 * ((lo / hi) + -1.0))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	return Float64(Float64(t_0 * Float64(Float64(lo - x) / hi)) / Float64(t_0 * Float64(Float64(lo / hi) + -1.0)))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	tmp = (t_0 * ((lo - x) / hi)) / (t_0 * ((lo / hi) + -1.0));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, N[(N[(t$95$0 * N[(N[(lo - x), $MachinePrecision] / hi), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(lo / hi), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. associate--l+ 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. times-frac 9.6%

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

   6. div-sub 9.6%

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

4. Simplified 9.6%

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

   1. flip-+ 9.6%

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

   2. div-sub 9.6%

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

6. Applied egg-rr 0.0%

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

   1. div-sub 0.0%

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

   2. associate-*l* 99.2%

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

   3. sub-neg 99.2%

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

   4. metadata-eval 99.2%

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

8. Simplified 99.2%

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

9. Taylor expanded in hi around inf 0.0%

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

    1. mul-1-neg 0.0%

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

    2. unpow2 0.0%

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

    3. unpow2 0.0%

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

    4. times-frac 99.2%

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

    5. unpow2 99.2%

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

11. Simplified 99.2%

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

    1. unpow2 99.2%

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

13. Applied egg-rr 99.2%

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

14. Final simplification 99.2%

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

Alternative 3: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in hi around inf 0.0%

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

   1. +-commutative 0.0%

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

   2. associate--l+ 0.0%

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

   3. *-commutative 0.0%

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

   4. unpow2 0.0%

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

   5. times-frac 9.6%

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

   6. div-sub 9.6%

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

4. Simplified 9.6%

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

   1. flip-+ 9.6%

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

   2. div-sub 9.6%

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

6. Applied egg-rr 0.0%

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

   1. div-sub 0.0%

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

   2. associate-*l* 99.2%

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

   3. sub-neg 99.2%

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

   4. metadata-eval 99.2%

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

8. Simplified 99.2%

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

9. Taylor expanded in hi around inf 0.0%

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

    1. mul-1-neg 0.0%

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

    2. unpow2 0.0%

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

    3. unpow2 0.0%

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

    4. times-frac 99.2%

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

    5. unpow2 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in x around 0 3.1%

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

    1. sub-neg 3.1%

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

    2. metadata-eval 3.1%

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

    3. associate-/r* 98.4%

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

14. Simplified 98.4%

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

15. Final simplification 98.4%

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

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(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%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

5. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

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. Taylor expanded in lo around inf 18.7%

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

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023199 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))