xlohi (overflows)

Average Error: 96.87% → 73.19%

Time: 13.3s

Precision: binary64

Cost: 2112

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

Math

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

↓

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

FPCore

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

↓

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

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 - hi) / lo;
	return (1.0 + ((t_0 * (1.0 + (hi / lo))) * (t_0 * (-1.0 - (hi / lo))))) / (1.0 + t_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 = (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) / lo
    code = (1.0d0 + ((t_0 * (1.0d0 + (hi / lo))) * (t_0 * ((-1.0d0) - (hi / lo))))) / (1.0d0 + t_0)
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 - hi) / lo;
	return (1.0 + ((t_0 * (1.0 + (hi / lo))) * (t_0 * (-1.0 - (hi / lo))))) / (1.0 + t_0);
}

Python

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

↓

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

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) / lo)
	return Float64(Float64(1.0 + Float64(Float64(t_0 * Float64(1.0 + Float64(hi / lo))) * Float64(t_0 * Float64(-1.0 - Float64(hi / lo))))) / Float64(1.0 + t_0))
end

MATLAB

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

↓

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

Derivation

1. Initial program 96.87

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

2. Taylor expanded in lo around inf 100

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

3. Simplified 81.12

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

   [Start] 100	\[ \left(-1 \cdot \frac{x}{lo} + \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right)\right) - -1 \cdot \frac{hi}{lo} \]
   sub-neg [=>] 100	\[ \color{blue}{\left(-1 \cdot \frac{x}{lo} + \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right)\right) + \left(--1 \cdot \frac{hi}{lo}\right)} \]
   +-commutative [=>] 100	\[ \color{blue}{\left(\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) + -1 \cdot \frac{x}{lo}\right)} + \left(--1 \cdot \frac{hi}{lo}\right) \]
   mul-1-neg [=>] 100	\[ \left(\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) + \color{blue}{\left(-\frac{x}{lo}\right)}\right) + \left(--1 \cdot \frac{hi}{lo}\right) \]
   unsub-neg [=>] 100	\[ \color{blue}{\left(\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) - \frac{x}{lo}\right)} + \left(--1 \cdot \frac{hi}{lo}\right) \]
   associate-+l- [=>] 100	\[ \color{blue}{\left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) - \left(\frac{x}{lo} - \left(--1 \cdot \frac{hi}{lo}\right)\right)} \]
   mul-1-neg [=>] 100	\[ \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) - \left(\frac{x}{lo} - \left(-\color{blue}{\left(-\frac{hi}{lo}\right)}\right)\right) \]
   remove-double-neg [=>] 100	\[ \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) - \left(\frac{x}{lo} - \color{blue}{\frac{hi}{lo}}\right) \]
   div-sub [<=] 100	\[ \left(\frac{hi \cdot \left(-1 \cdot x - -1 \cdot hi\right)}{{lo}^{2}} + 1\right) - \color{blue}{\frac{x - hi}{lo}} \]

4. Applied egg-rr 81.12

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

5. Taylor expanded in lo around inf 73.19

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

6. Final simplification 73.19

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

Alternatives

Alternative 1
Error	80.55%
Cost	448

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

Alternative 2
Error	81.21%
Cost	320

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

Alternative 3
Error	81.21%
Cost	256

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

Alternative 4
Error	81.33%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023121 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))