xlohi (overflows)

Percentage Accurate: 3.1% → 99.2%

Time: 8.5s

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: 99.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - lo}{hi}\\ \frac{{t_0}^{2} - {\left(\frac{lo}{\frac{hi \cdot hi}{x}}\right)}^{2}}{t_0 \cdot \left(1 - \frac{lo}{hi}\right)} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)))
   (/
    (- (pow t_0 2.0) (pow (/ lo (/ (* hi hi) x)) 2.0))
    (* t_0 (- 1.0 (/ lo hi))))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return (pow(t_0, 2.0) - pow((lo / ((hi * hi) / x)), 2.0)) / (t_0 * (1.0 - (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
    real(8) :: t_0
    t_0 = (x - lo) / hi
    code = ((t_0 ** 2.0d0) - ((lo / ((hi * hi) / x)) ** 2.0d0)) / (t_0 * (1.0d0 - (lo / 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) - Math.pow((lo / ((hi * hi) / x)), 2.0)) / (t_0 * (1.0 - (lo / hi)));
}

Python

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	return (math.pow(t_0, 2.0) - math.pow((lo / ((hi * hi) / x)), 2.0)) / (t_0 * (1.0 - (lo / hi)))

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	return Float64(Float64((t_0 ^ 2.0) - (Float64(lo / Float64(Float64(hi * hi) / x)) ^ 2.0)) / Float64(t_0 * Float64(1.0 - Float64(lo / hi))))
end

MATLAB

function tmp = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	tmp = ((t_0 ^ 2.0) - ((lo / ((hi * hi) / x)) ^ 2.0)) / (t_0 * (1.0 - (lo / hi)));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[Power[N[(lo / N[(N[(hi * hi), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(1.0 - N[(lo / hi), $MachinePrecision]), $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 10.1%

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

   6. div-sub 10.1%

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

4. Simplified 10.1%

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

   1. div-inv 10.1%

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

6. Applied egg-rr 10.1%

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

   1. div-inv 10.1%

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

   2. add-cube-cbrt 10.1%

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

   3. unpow2 10.1%

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

   4. +-commutative 10.1%

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

   5. flip-+ 10.1%

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

8. Applied egg-rr 10.1%

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

   1. times-frac 0.0%

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

   2. *-lft-identity 0.0%

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

   3. times-frac 99.4%

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

   4. /-rgt-identity 99.4%

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

   5. *-lft-identity 99.4%

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

   6. distribute-rgt-out-- 99.2%

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

10. Simplified 99.2%

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

11. Taylor expanded in lo around 0 51.6%

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

    1. unpow2 51.6%

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

    2. associate-/l* 99.2%

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

13. Simplified 99.2%

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

14. Final simplification 99.2%

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

Alternative 2: 19.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left|\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right| \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return fabs((((x - lo) / hi) * (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 = abs((((x - lo) / hi) * (lo / hi)))
end function

Java

public static double code(double lo, double hi, double x) {
	return Math.abs((((x - lo) / hi) * (lo / hi)));
}

Python

def code(lo, hi, x):
	return math.fabs((((x - lo) / hi) * (lo / hi)))

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[Abs[N[(N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision] * N[(lo / 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 10.1%

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

   6. div-sub 10.1%

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

4. Simplified 10.1%

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

   1. add-sqr-sqrt 9.4%

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

   2. sqrt-unprod 17.9%

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

   3. pow2 17.9%

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

   4. fma-def 17.9%

      \[\leadsto \sqrt{{\color{blue}{\left(\mathsf{fma}\left(\frac{x - lo}{hi}, \frac{lo}{hi}, \frac{x - lo}{hi}\right)\right)}}^{2}} \]

6. Applied egg-rr 17.9%

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

   1. unpow2 17.9%

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

   2. rem-sqrt-square 17.9%

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

   3. fma-udef 17.9%

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

   4. *-rgt-identity 17.9%

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

   5. distribute-lft-out 17.9%

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

8. Simplified 17.9%

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

9. Taylor expanded in lo around inf 19.3%

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

10. Final simplification 19.3%

    \[\leadsto \left|\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right| \]

Alternative 3: 18.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return ((x - lo) / hi) - (x * ((lo / hi) / 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) - (x * ((lo / hi) / hi))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around 0 18.8%

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

   1. add-sqr-sqrt 9.3%

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

   2. sqrt-unprod 14.6%

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

   3. sqr-neg 14.6%

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

   4. sqrt-unprod 9.5%

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

   5. unpow2 9.5%

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

   6. add-sqr-sqrt 18.8%

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

   7. neg-mul-1 18.8%

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

   8. times-frac 18.8%

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

4. Applied egg-rr 18.8%

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

5. Taylor expanded in lo around 0 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

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. Taylor expanded in hi around inf 18.8%

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

3. Final simplification 18.8%

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

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(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 hi around inf 18.8%

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

3. Taylor expanded in x around 0 18.8%

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

   1. associate-*r/ 18.8%

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

   2. neg-mul-1 18.8%

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

5. Simplified 18.8%

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

6. Final simplification 18.8%

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

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. Taylor expanded in lo around inf 18.6%

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

3. Final simplification 18.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))