Curve intersection, scale width based on ribbon orientation

Average Error: 0.8 → 0.4

Time: 16.4s

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

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

↓

\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin \left(normAngle - normAngle \cdot u\right)}{\sin normAngle} \cdot n0_i\right)\right) + \left(\mathsf{fma}\left(0.16666666666666666, u \cdot \left(normAngle \cdot normAngle\right), u\right) - 0.16666666666666666 \cdot \left(\left(normAngle \cdot normAngle\right) \cdot {u}^{3}\right)\right) \cdot n1_i \]

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
 (+
  (log1p
   (expm1 (* (/ (sin (- normAngle (* normAngle u))) (sin normAngle)) n0_i)))
  (*
   (-
    (fma 0.16666666666666666 (* u (* normAngle normAngle)) u)
    (* 0.16666666666666666 (* (* normAngle normAngle) (pow u 3.0))))
   n1_i)))

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 log1pf(expm1f((sinf(normAngle - (normAngle * u)) / sinf(normAngle)) * n0_i)) + ((fmaf(0.16666666666666666f, (u * (normAngle * normAngle)), u) - (0.16666666666666666f * ((normAngle * normAngle) * powf(u, 3.0f)))) * n1_i);
}

Error

Bits error versus normAngle

Bits error versus u

Bits error versus n0_i

Bits error versus n1_i

Derivation

1. Initial program 0.8

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

2. Taylor expanded in normAngle around 0 0.5

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\left(\left(0.16666666666666666 \cdot \left(u \cdot {normAngle}^{2}\right) + u\right) - 0.16666666666666666 \cdot \left({u}^{3} \cdot {normAngle}^{2}\right)\right)} \cdot n1_i \]

3. Simplified 0.5

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

4. Applied log1p-expm1-u_binary32 0.5

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i\right)\right)} + \left(\mathsf{fma}\left(0.16666666666666666, u \cdot \left(normAngle \cdot normAngle\right), u\right) - 0.16666666666666666 \cdot \left({u}^{3} \cdot \left(normAngle \cdot normAngle\right)\right)\right) \cdot n1_i \]

5. Simplified 0.4

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\sin \left(normAngle - u \cdot normAngle\right)}{\sin normAngle} \cdot n0_i\right)}\right) + \left(\mathsf{fma}\left(0.16666666666666666, u \cdot \left(normAngle \cdot normAngle\right), u\right) - 0.16666666666666666 \cdot \left({u}^{3} \cdot \left(normAngle \cdot normAngle\right)\right)\right) \cdot n1_i \]

6. Final simplification 0.4

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin \left(normAngle - normAngle \cdot u\right)}{\sin normAngle} \cdot n0_i\right)\right) + \left(\mathsf{fma}\left(0.16666666666666666, u \cdot \left(normAngle \cdot normAngle\right), u\right) - 0.16666666666666666 \cdot \left(\left(normAngle \cdot normAngle\right) \cdot {u}^{3}\right)\right) \cdot n1_i \]

Reproduce

herbie shell --seed 2022068 
(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)))