Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.2

Time: 7.4s

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

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

↓

\[\mathsf{fma}\left(u, \frac{normAngle}{\sin normAngle} \cdot n1_i - \mathsf{fma}\left(n0_i, -0.3333333333333333 \cdot \left(normAngle \cdot normAngle\right), n0_i\right), n0_i\right) \]

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
 (fma
  u
  (-
   (* (/ normAngle (sin normAngle)) n1_i)
   (fma n0_i (* -0.3333333333333333 (* normAngle normAngle)) n0_i))
  n0_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 fmaf(u, (((normAngle / sinf(normAngle)) * n1_i) - fmaf(n0_i, (-0.3333333333333333f * (normAngle * normAngle)), n0_i)), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * Float32(Float32(1.0) / sin(normAngle))) * n0_i) + Float32(Float32(sin(Float32(u * normAngle)) * Float32(Float32(1.0) / sin(normAngle))) * n1_i))
end

↓

function code(normAngle, u, n0_i, n1_i)
	return fma(u, Float32(Float32(Float32(normAngle / sin(normAngle)) * n1_i) - fma(n0_i, Float32(Float32(-0.3333333333333333) * Float32(normAngle * normAngle)), n0_i)), n0_i)
end

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

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

3. Simplified 0.4

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

4. Taylor expanded in u around inf 8.3

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

5. Simplified 0.3

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

6. Taylor expanded in u around 0 3.8

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

7. Simplified 0.2

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

8. Taylor expanded in normAngle around 0 0.2

   \[\leadsto \mathsf{fma}\left(u, \frac{normAngle}{\sin normAngle} \cdot n1_i - \color{blue}{\left(\left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) \cdot {normAngle}^{2} + n0_i\right)}, n0_i\right) \]

9. Simplified 0.2

   \[\leadsto \mathsf{fma}\left(u, \frac{normAngle}{\sin normAngle} \cdot n1_i - \color{blue}{\mathsf{fma}\left(n0_i, -0.3333333333333333 \cdot \left(normAngle \cdot normAngle\right), n0_i\right)}, n0_i\right) \]

10. Final simplification 0.2

    \[\leadsto \mathsf{fma}\left(u, \frac{normAngle}{\sin normAngle} \cdot n1_i - \mathsf{fma}\left(n0_i, -0.3333333333333333 \cdot \left(normAngle \cdot normAngle\right), n0_i\right), n0_i\right) \]

Reproduce

herbie shell --seed 2022192 
(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)))