Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.4

Time: 19.9s

Precision: binary32

Cost: 6752

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

Math

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

↓

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

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 n0_i (- 1.0 u) (* n1_i (* normAngle (/ u (sin normAngle))))))

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(n0_i, (1.0f - u), (n1_i * (normAngle * (u / sinf(normAngle)))));
}

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(n0_i, Float32(Float32(1.0) - u), Float32(n1_i * Float32(normAngle * Float32(u / sin(normAngle)))))
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 normAngle around 0 0.9

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

3. Taylor expanded in u around 0 0.9

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

4. Simplified 0.4

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

   [Start] 0.9	\[ \left(1 - u\right) \cdot n0_i + \frac{u \cdot normAngle}{\sin normAngle} \cdot n1_i \]
   associate-/l* [=>] 0.4	\[ \left(1 - u\right) \cdot n0_i + \color{blue}{\frac{u}{\frac{\sin normAngle}{normAngle}}} \cdot n1_i \]
   associate-/r/ [=>] 0.4	\[ \left(1 - u\right) \cdot n0_i + \color{blue}{\left(\frac{u}{\sin normAngle} \cdot normAngle\right)} \cdot n1_i \]

5. Applied egg-rr 4.4

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

6. Taylor expanded in n0_i around 0 4.4

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

7. Simplified 0.4

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

   [Start] 4.4	\[ \frac{n1_i \cdot \left(u \cdot normAngle\right)}{\sin normAngle} + \left(1 - u\right) \cdot n0_i \]
   +-commutative [=>] 4.4	\[ \color{blue}{\left(1 - u\right) \cdot n0_i + \frac{n1_i \cdot \left(u \cdot normAngle\right)}{\sin normAngle}} \]
   *-commutative [=>] 4.4	\[ \color{blue}{n0_i \cdot \left(1 - u\right)} + \frac{n1_i \cdot \left(u \cdot normAngle\right)}{\sin normAngle} \]
   *-commutative [=>] 4.4	\[ n0_i \cdot \left(1 - u\right) + \frac{\color{blue}{\left(u \cdot normAngle\right) \cdot n1_i}}{\sin normAngle} \]
   associate-*r* [<=] 4.4	\[ n0_i \cdot \left(1 - u\right) + \frac{\color{blue}{u \cdot \left(normAngle \cdot n1_i\right)}}{\sin normAngle} \]
   *-commutative [=>] 4.4	\[ n0_i \cdot \left(1 - u\right) + \frac{\color{blue}{\left(normAngle \cdot n1_i\right) \cdot u}}{\sin normAngle} \]
   associate-*l/ [<=] 3.1	\[ n0_i \cdot \left(1 - u\right) + \color{blue}{\frac{normAngle \cdot n1_i}{\sin normAngle} \cdot u} \]
   fma-udef [<=] 3.1	\[ \color{blue}{\mathsf{fma}\left(n0_i, 1 - u, \frac{normAngle \cdot n1_i}{\sin normAngle} \cdot u\right)} \]
   *-commutative [=>] 3.1	\[ \mathsf{fma}\left(n0_i, 1 - u, \frac{\color{blue}{n1_i \cdot normAngle}}{\sin normAngle} \cdot u\right) \]
   associate-/l* [=>] 0.3	\[ \mathsf{fma}\left(n0_i, 1 - u, \color{blue}{\frac{n1_i}{\frac{\sin normAngle}{normAngle}}} \cdot u\right) \]
   associate-*l/ [=>] 0.3	\[ \mathsf{fma}\left(n0_i, 1 - u, \color{blue}{\frac{n1_i \cdot u}{\frac{\sin normAngle}{normAngle}}}\right) \]
   associate-*r/ [<=] 0.3	\[ \mathsf{fma}\left(n0_i, 1 - u, \color{blue}{n1_i \cdot \frac{u}{\frac{\sin normAngle}{normAngle}}}\right) \]
   associate-/r/ [=>] 0.4	\[ \mathsf{fma}\left(n0_i, 1 - u, n1_i \cdot \color{blue}{\left(\frac{u}{\sin normAngle} \cdot normAngle\right)}\right) \]

8. Final simplification 0.4

   \[\leadsto \mathsf{fma}\left(n0_i, 1 - u, n1_i \cdot \left(normAngle \cdot \frac{u}{\sin normAngle}\right)\right) \]

Alternatives

Alternative 1
Error	0.4
Cost	3616

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

Alternative 2
Error	0.4
Cost	608

\[n0_i + u \cdot \left(\left(n1_i - n0_i\right) - \left(normAngle \cdot normAngle\right) \cdot \left(n1_i \cdot -0.16666666666666666 + n0_i \cdot -0.3333333333333333\right)\right) \]

Alternative 3
Error	0.4
Cost	480

\[n0_i - u \cdot \left(n0_i + n1_i \cdot \left(-1 + \left(normAngle \cdot normAngle\right) \cdot -0.16666666666666666\right)\right) \]

Alternative 4
Error	9.4
Cost	297

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -1.5000000170217692 \cdot 10^{-19} \lor \neg \left(n0_i \leq 2.0000000063421537 \cdot 10^{-28}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 5
Error	4.6
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.4999999445713045 \cdot 10^{-28} \lor \neg \left(n1_i \leq 4.000000094968912 \cdot 10^{-33}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 6
Error	4.6
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.4999999445713045 \cdot 10^{-28} \lor \neg \left(n1_i \leq 4.000000094968912 \cdot 10^{-33}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \]

Alternative 7
Error	12.7
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -1.5000000170217692 \cdot 10^{-19}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 2.0000000063421537 \cdot 10^{-28}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 8
Error	0.6
Cost	224

\[n0_i - u \cdot \left(n0_i - n1_i\right) \]

Alternative 9
Error	16.7
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2023045 
(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)))