Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.3

Time: 9.0s

Precision: binary32

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

\[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

↓

\[n0_i + \mathsf{fma}\left(\mathsf{fma}\left(normAngle, \frac{u}{\sin normAngle}, -0.16666666666666666 \cdot \left({u}^{3} \cdot {normAngle}^{2}\right)\right), n1_i, -n0_i \cdot u\right) \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* (* (sin (* (- 1.0 u) normAngle)) (/ 1.0 (sin normAngle))) n0_i)
  (* (* (sin (* u normAngle)) (/ 1.0 (sin normAngle))) n1_i)))

↓

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (fma
   (fma
    normAngle
    (/ u (sin normAngle))
    (* -0.16666666666666666 (* (pow u 3.0) (pow normAngle 2.0))))
   n1_i
   (- (* n0_i u)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((sinf((1.0f - u) * normAngle) * (1.0f / sinf(normAngle))) * n0_i) + ((sinf(u * normAngle) * (1.0f / sinf(normAngle))) * n1_i);
}

↓

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + fmaf(fmaf(normAngle, (u / sinf(normAngle)), (-0.16666666666666666f * (powf(u, 3.0f) * powf(normAngle, 2.0f)))), n1_i, -(n0_i * u));
}

Error

Bits error versus normAngle

Bits error versus u

Bits error versus n0_i

Bits error versus n1_i

Derivation

1. Initial program 0.9

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

2. Taylor expanded in u around 0 0.9

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\left(\frac{u \cdot normAngle}{\sin normAngle} - 0.16666666666666666 \cdot \frac{{u}^{3} \cdot {normAngle}^{3}}{\sin normAngle}\right)} \cdot n1_i \]

3. Simplified 0.4

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\mathsf{fma}\left(\frac{u}{\sin normAngle}, normAngle, \frac{{u}^{3} \cdot {normAngle}^{3}}{\sin normAngle} \cdot -0.16666666666666666\right)} \cdot n1_i \]

4. Taylor expanded in normAngle around 0 0.3

   \[\leadsto \color{blue}{\left(1 - u\right)} \cdot n0_i + \mathsf{fma}\left(\frac{u}{\sin normAngle}, normAngle, \frac{{u}^{3} \cdot {normAngle}^{3}}{\sin normAngle} \cdot -0.16666666666666666\right) \cdot n1_i \]

5. Taylor expanded in u around 0 4.5

   \[\leadsto \color{blue}{\left(\frac{n1_i \cdot \left(u \cdot normAngle\right)}{\sin normAngle} + n0_i\right) - \left(0.16666666666666666 \cdot \frac{n1_i \cdot \left({u}^{3} \cdot {normAngle}^{3}\right)}{\sin normAngle} + u \cdot n0_i\right)} \]

6. Simplified 0.3

   \[\leadsto \color{blue}{n0_i + \mathsf{fma}\left(\mathsf{fma}\left(normAngle, \frac{u}{\sin normAngle}, \frac{{u}^{3} \cdot {normAngle}^{3}}{\sin normAngle} \cdot -0.16666666666666666\right), n1_i, -n0_i \cdot u\right)} \]

7. Taylor expanded in normAngle around 0 0.3

   \[\leadsto n0_i + \mathsf{fma}\left(\mathsf{fma}\left(normAngle, \frac{u}{\sin normAngle}, \color{blue}{-0.16666666666666666 \cdot \left({u}^{3} \cdot {normAngle}^{2}\right)}\right), n1_i, -n0_i \cdot u\right) \]

8. Final simplification 0.3

   \[\leadsto n0_i + \mathsf{fma}\left(\mathsf{fma}\left(normAngle, \frac{u}{\sin normAngle}, -0.16666666666666666 \cdot \left({u}^{3} \cdot {normAngle}^{2}\right)\right), n1_i, -n0_i \cdot u\right) \]

Reproduce

herbie shell --seed 2021275 
(FPCore (normAngle u n0_i n1_i)
  :name "Curve intersection, scale width based on ribbon orientation"
  :precision binary32
  :pre (and (and (and (and (<= 0.0 normAngle) (<= normAngle (/ PI 2.0))) (and (<= -1.0 n0_i) (<= n0_i 1.0))) (and (<= -1.0 n1_i) (<= n1_i 1.0))) (and (<= 2.328306437e-10 u) (<= u 1.0)))
  (+ (* (* (sin (* (- 1.0 u) normAngle)) (/ 1.0 (sin normAngle))) n0_i) (* (* (sin (* u normAngle)) (/ 1.0 (sin normAngle))) n1_i)))