xlohi (overflows)

Percentage Accurate: 3.1% → 99.1%

Time: 12.3s

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: 99.1% accurate, 0.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	return fabs((pow(t_0, 2.0) / (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 = abs(((t_0 ** 2.0d0) / (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 Math.abs((Math.pow(t_0, 2.0) / (t_0 * ((lo / hi) + -1.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, N[Abs[N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[(t$95$0 * N[(N[(lo / hi), $MachinePrecision] + -1.0), $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 8.4%

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

   6. div-sub 8.4%

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

4. Simplified 8.4%

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

   1. add-sqr-sqrt 7.5%

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

   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. +-commutative 17.9%

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

   5. *-lft-identity 17.9%

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

   6. *-commutative 17.9%

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

   7. distribute-rgt-out 17.9%

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

8. Simplified 17.9%

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

   1. +-commutative 17.9%

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

   2. distribute-lft-in 17.9%

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

   3. expm1-log1p-u 7.5%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)\right)} + \frac{x - lo}{hi} \cdot 1\right| \]

   4. *-rgt-identity 7.5%

      \[\leadsto \left|\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)\right) + \color{blue}{\frac{x - lo}{hi}}\right| \]

   5. flip-+ 7.5%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)\right) - \frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)\right) - \frac{x - lo}{hi}}}\right| \]

   6. div-sub 7.5%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)\right) - \frac{x - lo}{hi}} - \frac{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - lo}{hi} \cdot \frac{lo}{hi}\right)\right) - \frac{x - lo}{hi}}}\right| \]

10. Applied egg-rr 37.2%

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

    1. div-sub 37.2%

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

    2. associate-*r/ 98.9%

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

    3. associate-/r/ 98.9%

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

    4. /-rgt-identity 98.9%

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

    5. times-frac 0.0%

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

    6. *-commutative 0.0%

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

    7. *-commutative 0.0%

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

    8. times-frac 98.9%

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

    9. /-rgt-identity 98.9%

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

    10. sub-neg 98.9%

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

    11. metadata-eval 98.9%

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

12. Simplified 98.9%

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

13. Taylor expanded in hi around inf 0.0%

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

    1. associate-*r/ 0.0%

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

    2. unpow2 0.0%

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

    3. neg-mul-1 0.0%

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

    4. unpow2 0.0%

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

    5. distribute-rgt-neg-out 0.0%

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

    6. associate-/r* 3.1%

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

    7. associate-*l/ 67.7%

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

    8. distribute-rgt-neg-out 67.7%

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

    9. distribute-neg-frac 67.7%

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

    10. associate-*r/ 98.9%

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

    11. unpow2 98.9%

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

15. Simplified 98.9%

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

16. Final simplification 98.9%

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

Alternative 2: 19.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return pow(((0.3333333333333333 * (hi / lo)) + 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 = ((0.3333333333333333d0 * (hi / lo)) + 1.0d0) ** 3.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

2. Taylor expanded in lo around inf 10.6%

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

   1. +-commutative 10.6%

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

   2. associate--l+ 10.6%

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

   3. associate-*r/ 10.6%

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

   4. associate-*r/ 10.6%

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

   5. div-sub 10.6%

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

   6. distribute-lft-out-- 10.6%

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

   7. associate-*r/ 10.6%

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

   8. mul-1-neg 10.6%

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

   9. unsub-neg 10.6%

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

4. Simplified 10.6%

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

5. Taylor expanded in x around 0 10.5%

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

   1. add-cube-cbrt 10.5%

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

   2. pow3 10.5%

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

7. Applied egg-rr 10.5%

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

8. Taylor expanded in hi around 0 19.4%

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

9. Final simplification 19.4%

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

Alternative 3: 19.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(lo / hi), $MachinePrecision] * N[(lo / hi), $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 8.4%

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

   6. div-sub 8.4%

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

4. Simplified 8.4%

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

   1. add-sqr-sqrt 7.5%

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

   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. +-commutative 17.9%

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

   5. *-lft-identity 17.9%

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

   6. *-commutative 17.9%

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

   7. distribute-rgt-out 17.9%

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

8. Simplified 17.9%

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

9. Taylor expanded in lo around inf 18.9%

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

10. Taylor expanded in x around 0 19.0%

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

    1. associate-*r/ 18.8%

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

    2. neg-mul-1 18.8%

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

12. Simplified 19.0%

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

13. Final simplification 19.0%

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

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 hi around inf 18.7%

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

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 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 2023275 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))