(/ (- x lo) (- hi lo))

Average Error: 62.0 → 52.0

Time: 8.2s

Precision: binary64

Cost: 59968

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

Math

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

↓

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

FPCore

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

↓

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (fma lo (fma x (pow hi -2.0) (/ 1.0 hi)) (/ x hi))))
   (-
    (/ (pow (/ x hi) 2.0) t_0)
    (/ (pow (fma (/ lo hi) (/ x hi) (/ lo hi)) 2.0) 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 = fma(lo, fma(x, pow(hi, -2.0), (1.0 / hi)), (x / hi));
	return (pow((x / hi), 2.0) / t_0) - (pow(fma((lo / hi), (x / hi), (lo / hi)), 2.0) / t_0);
}

Julia

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

↓

function code(lo, hi, x)
	t_0 = fma(lo, fma(x, (hi ^ -2.0), Float64(1.0 / hi)), Float64(x / hi))
	return Float64(Float64((Float64(x / hi) ^ 2.0) / t_0) - Float64((fma(Float64(lo / hi), Float64(x / hi), Float64(lo / hi)) ^ 2.0) / 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[(lo * N[(x * N[Power[hi, -2.0], $MachinePrecision] + N[(1.0 / hi), $MachinePrecision]), $MachinePrecision] + N[(x / hi), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[(x / hi), $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision] - N[(N[Power[N[(N[(lo / hi), $MachinePrecision] * N[(x / hi), $MachinePrecision] + N[(lo / hi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $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. Applied egg-rr 52.0

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

4. Applied egg-rr 52.0

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

5. Final simplification 52.0

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

Alternatives

Alternative 1
Error	52.0
Cost	1088

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

Alternative 2
Error	52.0
Cost	448

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

Alternative 3
Error	52.0
Cost	320

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

Alternative 4
Error	52.0
Cost	256

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

Alternative 5
Error	52.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022297 
(FPCore (lo hi x)
  :name "(/ (- x lo) (- hi lo))"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))