xlohi (overflows)

Average Accuracy: 3.1% → 99.2%

Time: 13.0s

Precision: binary64

Cost: 22016

\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x - lo}{hi}\\ t_1 := {t_0}^{2}\\ t_2 := \frac{lo}{hi \cdot hi}\\ \frac{{t_0}^{3}}{t_2 \cdot \left(\left(x - lo\right) \cdot \left(\left(x - lo\right) \cdot t_2\right)\right) + \left(t_1 - t_1 \cdot \frac{lo}{hi}\right)} \end{array} \]

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (/ (- x lo) hi)) (t_1 (pow t_0 2.0)) (t_2 (/ lo (* hi hi))))
   (/
    (pow t_0 3.0)
    (+ (* t_2 (* (- x lo) (* (- x lo) t_2))) (- t_1 (* t_1 (/ lo hi)))))))

C

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

↓

double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double t_1 = pow(t_0, 2.0);
	double t_2 = lo / (hi * hi);
	return pow(t_0, 3.0) / ((t_2 * ((x - lo) * ((x - lo) * t_2))) + (t_1 - (t_1 * (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 - lo)
end function

↓

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
    real(8) :: t_1
    real(8) :: t_2
    t_0 = (x - lo) / hi
    t_1 = t_0 ** 2.0d0
    t_2 = lo / (hi * hi)
    code = (t_0 ** 3.0d0) / ((t_2 * ((x - lo) * ((x - lo) * t_2))) + (t_1 - (t_1 * (lo / hi))))
end function

Java

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

↓

public static double code(double lo, double hi, double x) {
	double t_0 = (x - lo) / hi;
	double t_1 = Math.pow(t_0, 2.0);
	double t_2 = lo / (hi * hi);
	return Math.pow(t_0, 3.0) / ((t_2 * ((x - lo) * ((x - lo) * t_2))) + (t_1 - (t_1 * (lo / hi))));
}

Python

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

↓

def code(lo, hi, x):
	t_0 = (x - lo) / hi
	t_1 = math.pow(t_0, 2.0)
	t_2 = lo / (hi * hi)
	return math.pow(t_0, 3.0) / ((t_2 * ((x - lo) * ((x - lo) * t_2))) + (t_1 - (t_1 * (lo / hi))))

Julia

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

↓

function code(lo, hi, x)
	t_0 = Float64(Float64(x - lo) / hi)
	t_1 = t_0 ^ 2.0
	t_2 = Float64(lo / Float64(hi * hi))
	return Float64((t_0 ^ 3.0) / Float64(Float64(t_2 * Float64(Float64(x - lo) * Float64(Float64(x - lo) * t_2))) + Float64(t_1 - Float64(t_1 * Float64(lo / hi)))))
end

MATLAB

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

↓

function tmp = code(lo, hi, x)
	t_0 = (x - lo) / hi;
	t_1 = t_0 ^ 2.0;
	t_2 = lo / (hi * hi);
	tmp = (t_0 ^ 3.0) / ((t_2 * ((x - lo) * ((x - lo) * t_2))) + (t_1 - (t_1 * (lo / hi))));
end

Wolfram

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

↓

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(lo / N[(hi * hi), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[t$95$0, 3.0], $MachinePrecision] / N[(N[(t$95$2 * N[(N[(x - lo), $MachinePrecision] * N[(N[(x - lo), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[(t$95$1 * N[(lo / hi), $MachinePrecision]), $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. Simplified 9.4%

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

   [Start] 0.0	\[ \left(\frac{x}{hi} + \frac{lo \cdot \left(x - lo\right)}{{hi}^{2}}\right) - \frac{lo}{hi} \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \frac{x}{hi}\right)} - \frac{lo}{hi} \]
   associate--l+ [=>] 0.0	\[ \color{blue}{\frac{lo \cdot \left(x - lo\right)}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right)} \]
   *-commutative [=>] 0.0	\[ \frac{\color{blue}{\left(x - lo\right) \cdot lo}}{{hi}^{2}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   unpow2 [=>] 0.0	\[ \frac{\left(x - lo\right) \cdot lo}{\color{blue}{hi \cdot hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   times-frac [=>] 9.4	\[ \color{blue}{\frac{x - lo}{hi} \cdot \frac{lo}{hi}} + \left(\frac{x}{hi} - \frac{lo}{hi}\right) \]
   div-sub [<=] 9.4	\[ \frac{x - lo}{hi} \cdot \frac{lo}{hi} + \color{blue}{\frac{x - lo}{hi}} \]

4. Applied egg-rr 99.2%

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

5. Taylor expanded in hi around inf 0.0%

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

6. Simplified 99.2%

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

   [Start] 0.0	\[ \frac{\frac{{\left(x - lo\right)}^{3}}{{hi}^{3}}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]
   cube-div [<=] 99.2	\[ \frac{\color{blue}{{\left(\frac{x - lo}{hi}\right)}^{3}}}{{\left(\left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\right)}^{2} + \left({\left(\frac{x - lo}{hi}\right)}^{2} - \frac{lo}{hi} \cdot {\left(\frac{x - lo}{hi}\right)}^{2}\right)} \]

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	99.2%
Cost	2752

\[\begin{array}{l} t_0 := \frac{x - lo}{hi}\\ t_1 := \left(x - lo\right) \cdot \frac{lo}{hi \cdot hi}\\ \frac{\left(t_0 + t_1\right) \cdot \left(t_1 + \frac{lo - x}{hi}\right)}{t_0 \cdot \left(\frac{lo}{hi} + -1\right)} \end{array} \]

Alternative 2
Accuracy	18.9%
Cost	704

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

Alternative 3
Accuracy	18.8%
Cost	576

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

Alternative 4
Accuracy	18.8%
Cost	320

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

Alternative 5
Accuracy	18.8%
Cost	256

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

Alternative 6
Accuracy	18.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023125 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))