xlohi (overflows)

Average Error: 62.0 → 52.0

Time: 12.4s

Precision: binary64

Cost: 15168

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

Math

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

↓

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

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (- (/ x (pow hi 2.0)) (/ 1.0 hi))))
   (*
    (+ (/ x hi) (* lo t_0))
    (+ 1.0 (* (/ x hi) (/ 1.0 (+ (* t_0 lo) (/ x 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 / pow(hi, 2.0)) - (1.0 / hi);
	return ((x / hi) + (lo * t_0)) * (1.0 + ((x / hi) * (1.0 / ((t_0 * 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
    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
    t_0 = (x / (hi ** 2.0d0)) - (1.0d0 / hi)
    code = ((x / hi) + (lo * t_0)) * (1.0d0 + ((x / hi) * (1.0d0 / ((t_0 * lo) + (x / 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 / Math.pow(hi, 2.0)) - (1.0 / hi);
	return ((x / hi) + (lo * t_0)) * (1.0 + ((x / hi) * (1.0 / ((t_0 * lo) + (x / hi)))));
}

Python

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

↓

def code(lo, hi, x):
	t_0 = (x / math.pow(hi, 2.0)) - (1.0 / hi)
	return ((x / hi) + (lo * t_0)) * (1.0 + ((x / hi) * (1.0 / ((t_0 * lo) + (x / 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 / (hi ^ 2.0)) - Float64(1.0 / hi))
	return Float64(Float64(Float64(x / hi) + Float64(lo * t_0)) * Float64(1.0 + Float64(Float64(x / hi) * Float64(1.0 / Float64(Float64(t_0 * lo) + Float64(x / hi))))))
end

MATLAB

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

↓

function tmp = code(lo, hi, x)
	t_0 = (x / (hi ^ 2.0)) - (1.0 / hi);
	tmp = ((x / hi) + (lo * t_0)) * (1.0 + ((x / hi) * (1.0 / ((t_0 * lo) + (x / 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 / N[Power[hi, 2.0], $MachinePrecision]), $MachinePrecision] - N[(1.0 / hi), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(x / hi), $MachinePrecision] + N[(lo * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x / hi), $MachinePrecision] * N[(1.0 / N[(N[(t$95$0 * lo), $MachinePrecision] + N[(x / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.0

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

2. Taylor expanded in lo around 0 52.0

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

3. Simplified 52.0

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

   [Start] 52.0	\[ \frac{x}{hi} + -1 \cdot \left(lo \cdot \left(\frac{1}{hi} + -1 \cdot \frac{x}{{hi}^{2}}\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 52.0	\[ \frac{x}{hi} + \color{blue}{\left(lo \cdot \left(\frac{1}{hi} + -1 \cdot \frac{x}{{hi}^{2}}\right)\right) \cdot -1} \]
   rational_best_oopsla_all_46_json_45_simplify-92 [=>] 52.0	\[ \frac{x}{hi} + \color{blue}{\left(-lo \cdot \left(\frac{1}{hi} + -1 \cdot \frac{x}{{hi}^{2}}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 52.0	\[ \frac{x}{hi} + \left(-lo \cdot \left(\frac{1}{hi} + \color{blue}{\frac{x}{{hi}^{2}} \cdot -1}\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-92 [=>] 52.0	\[ \frac{x}{hi} + \left(-lo \cdot \left(\frac{1}{hi} + \color{blue}{\left(-\frac{x}{{hi}^{2}}\right)}\right)\right) \]

4. Applied egg-rr 52.0

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

5. Applied egg-rr 52.0

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

6. Simplified 52.0

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

   [Start] 52.0	\[ \left(\frac{x}{hi} + lo \cdot \left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right)\right) \cdot \left(\left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) \cdot \left(lo \cdot \frac{1}{lo \cdot \left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) + \frac{x}{hi}}\right) + \frac{x}{hi} \cdot \frac{1}{lo \cdot \left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) + \frac{x}{hi}}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 52.0	\[ \left(\frac{x}{hi} + lo \cdot \left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right)\right) \cdot \left(\color{blue}{lo \cdot \left(\left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) \cdot \frac{1}{lo \cdot \left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) + \frac{x}{hi}}\right)} + \frac{x}{hi} \cdot \frac{1}{lo \cdot \left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) + \frac{x}{hi}}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [<=] 52.0	\[ \left(\frac{x}{hi} + lo \cdot \left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right)\right) \cdot \left(lo \cdot \left(\left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) \cdot \frac{1}{\color{blue}{\left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) \cdot lo} + \frac{x}{hi}}\right) + \frac{x}{hi} \cdot \frac{1}{lo \cdot \left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) + \frac{x}{hi}}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [<=] 52.0	\[ \left(\frac{x}{hi} + lo \cdot \left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right)\right) \cdot \left(lo \cdot \left(\left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) \cdot \frac{1}{\left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) \cdot lo + \frac{x}{hi}}\right) + \frac{x}{hi} \cdot \frac{1}{\color{blue}{\left(\frac{x}{{hi}^{2}} - \frac{1}{hi}\right) \cdot lo} + \frac{x}{hi}}\right) \]

7. Taylor expanded in lo around inf 52.0

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

8. Final simplification 52.0

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

Alternatives

Alternative 1
Error	52.0
Cost	1536

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

Alternative 2
Error	52.0
Cost	320

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

Alternative 3
Error	52.0
Cost	256

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

Alternative 4
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))