Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.4

Time: 19.1s

Precision: binary32

Cost: 3808

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

Math

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

↓

\[\left(n1_i - n0_i\right) \cdot u + \left(n0_i + \mathsf{fma}\left(0.16666666666666666, n1_i, n0_i \cdot 0.3333333333333333\right) \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right)\right) \]

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
 (+
  (* (- n1_i n0_i) u)
  (+
   n0_i
   (*
    (fma 0.16666666666666666 n1_i (* n0_i 0.3333333333333333))
    (* u (* normAngle 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 ((n1_i - n0_i) * u) + (n0_i + (fmaf(0.16666666666666666f, n1_i, (n0_i * 0.3333333333333333f)) * (u * (normAngle * 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 Float32(Float32(Float32(n1_i - n0_i) * u) + Float32(n0_i + Float32(fma(Float32(0.16666666666666666), n1_i, Float32(n0_i * Float32(0.3333333333333333))) * Float32(u * Float32(normAngle * 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. Simplified 8.2

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

   [Start] 0.9	\[ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   *-commutative [=>] 0.9	\[ \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   associate-*l* [=>] 6.1	\[ \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   *-commutative [=>] 6.1	\[ \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]
   associate-*l* [=>] 8.3	\[ \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]
   distribute-lft-out [=>] 8.3	\[ \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]
   +-commutative [<=] 8.3	\[ \frac{1}{\sin normAngle} \cdot \color{blue}{\left(\sin \left(u \cdot normAngle\right) \cdot n1_i + \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} \]
   associate-*l/ [=>] 8.2	\[ \color{blue}{\frac{1 \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i + \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)}{\sin normAngle}} \]
   *-commutative [=>] 8.2	\[ \frac{\color{blue}{\left(\sin \left(u \cdot normAngle\right) \cdot n1_i + \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) \cdot 1}}{\sin normAngle} \]
   associate-/l* [=>] 8.2	\[ \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot n1_i + \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\frac{\sin normAngle}{1}}} \]
   /-rgt-identity [=>] 8.2	\[ \frac{\sin \left(u \cdot normAngle\right) \cdot n1_i + \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\color{blue}{\sin normAngle}} \]

3. Taylor expanded in u around 0 8.4

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

4. Taylor expanded in normAngle around 0 0.4

   \[\leadsto \color{blue}{\left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(\left(-0.16666666666666666 \cdot n0_i + 0.5 \cdot \left(n0_i \cdot u\right)\right) - -0.16666666666666666 \cdot \left(\left(n1_i + -1 \cdot n0_i\right) \cdot u + n0_i\right)\right) \cdot {normAngle}^{2} + n0_i\right)} \]

5. Taylor expanded in u around 0 0.4

   \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\color{blue}{\left(\left(0.5 \cdot n0_i - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right) \cdot u\right)} \cdot {normAngle}^{2} + n0_i\right) \]

6. Simplified 0.4

   \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\color{blue}{\left(u \cdot \left(0.5 \cdot n0_i + 0.16666666666666666 \cdot \left(n1_i + \left(-n0_i\right)\right)\right)\right)} \cdot {normAngle}^{2} + n0_i\right) \]
   Proof

   [Start] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(\left(0.5 \cdot n0_i - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right) \cdot u\right) \cdot {normAngle}^{2} + n0_i\right) \]
   *-commutative [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\color{blue}{\left(u \cdot \left(0.5 \cdot n0_i - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right)\right)} \cdot {normAngle}^{2} + n0_i\right) \]
   *-commutative [<=] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(u \cdot \left(\color{blue}{n0_i \cdot 0.5} - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right)\right) \cdot {normAngle}^{2} + n0_i\right) \]
   cancel-sign-sub-inv [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(u \cdot \color{blue}{\left(n0_i \cdot 0.5 + \left(--0.16666666666666666\right) \cdot \left(n1_i + -1 \cdot n0_i\right)\right)}\right) \cdot {normAngle}^{2} + n0_i\right) \]
   *-commutative [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(u \cdot \left(\color{blue}{0.5 \cdot n0_i} + \left(--0.16666666666666666\right) \cdot \left(n1_i + -1 \cdot n0_i\right)\right)\right) \cdot {normAngle}^{2} + n0_i\right) \]
   metadata-eval [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(u \cdot \left(0.5 \cdot n0_i + \color{blue}{0.16666666666666666} \cdot \left(n1_i + -1 \cdot n0_i\right)\right)\right) \cdot {normAngle}^{2} + n0_i\right) \]
   mul-1-neg [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(u \cdot \left(0.5 \cdot n0_i + 0.16666666666666666 \cdot \left(n1_i + \color{blue}{\left(-n0_i\right)}\right)\right)\right) \cdot {normAngle}^{2} + n0_i\right) \]

7. Taylor expanded in u around 0 0.4

   \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\color{blue}{\left(0.5 \cdot n0_i + 0.16666666666666666 \cdot \left(n1_i - n0_i\right)\right) \cdot \left(u \cdot {normAngle}^{2}\right)} + n0_i\right) \]

8. Simplified 0.4

   \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, n1_i, n0_i \cdot 0.3333333333333333\right) \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right)} + n0_i\right) \]
   Proof

   [Start] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(0.5 \cdot n0_i + 0.16666666666666666 \cdot \left(n1_i - n0_i\right)\right) \cdot \left(u \cdot {normAngle}^{2}\right) + n0_i\right) \]
   associate-*r* [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\color{blue}{\left(\left(0.5 \cdot n0_i + 0.16666666666666666 \cdot \left(n1_i - n0_i\right)\right) \cdot u\right) \cdot {normAngle}^{2}} + n0_i\right) \]
   unpow2 [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(\left(0.5 \cdot n0_i + 0.16666666666666666 \cdot \left(n1_i - n0_i\right)\right) \cdot u\right) \cdot \color{blue}{\left(normAngle \cdot normAngle\right)} + n0_i\right) \]
   associate-*r* [<=] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\color{blue}{\left(0.5 \cdot n0_i + 0.16666666666666666 \cdot \left(n1_i - n0_i\right)\right) \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right)} + n0_i\right) \]
   +-commutative [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\color{blue}{\left(0.16666666666666666 \cdot \left(n1_i - n0_i\right) + 0.5 \cdot n0_i\right)} \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right) + n0_i\right) \]
   distribute-lft-out-- [<=] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(\color{blue}{\left(0.16666666666666666 \cdot n1_i - 0.16666666666666666 \cdot n0_i\right)} + 0.5 \cdot n0_i\right) \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right) + n0_i\right) \]
   cancel-sign-sub-inv [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(\color{blue}{\left(0.16666666666666666 \cdot n1_i + \left(-0.16666666666666666\right) \cdot n0_i\right)} + 0.5 \cdot n0_i\right) \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right) + n0_i\right) \]
   metadata-eval [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(\left(0.16666666666666666 \cdot n1_i + \color{blue}{-0.16666666666666666} \cdot n0_i\right) + 0.5 \cdot n0_i\right) \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right) + n0_i\right) \]
   *-commutative [<=] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(\left(0.16666666666666666 \cdot n1_i + -0.16666666666666666 \cdot n0_i\right) + \color{blue}{n0_i \cdot 0.5}\right) \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right) + n0_i\right) \]
   associate-+r+ [<=] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\color{blue}{\left(0.16666666666666666 \cdot n1_i + \left(-0.16666666666666666 \cdot n0_i + n0_i \cdot 0.5\right)\right)} \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right) + n0_i\right) \]
   *-commutative [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(0.16666666666666666 \cdot n1_i + \left(-0.16666666666666666 \cdot n0_i + \color{blue}{0.5 \cdot n0_i}\right)\right) \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right) + n0_i\right) \]
   fma-def [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, n1_i, -0.16666666666666666 \cdot n0_i + 0.5 \cdot n0_i\right)} \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right) + n0_i\right) \]
   distribute-rgt-out [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\mathsf{fma}\left(0.16666666666666666, n1_i, \color{blue}{n0_i \cdot \left(-0.16666666666666666 + 0.5\right)}\right) \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right) + n0_i\right) \]
   metadata-eval [=>] 0.4	\[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\mathsf{fma}\left(0.16666666666666666, n1_i, n0_i \cdot \color{blue}{0.3333333333333333}\right) \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right) + n0_i\right) \]

9. Final simplification 0.4

   \[\leadsto \left(n1_i - n0_i\right) \cdot u + \left(n0_i + \mathsf{fma}\left(0.16666666666666666, n1_i, n0_i \cdot 0.3333333333333333\right) \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right)\right) \]

Alternatives

Alternative 1
Error	0.4
Cost	544

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

Alternative 2
Error	9.8
Cost	434

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -3.99999987306209 \cdot 10^{-20} \lor \neg \left(n0_i \leq -2.0000000390829628 \cdot 10^{-24}\right) \land \left(n0_i \leq -2.0000000063421537 \cdot 10^{-30} \lor \neg \left(n0_i \leq 9.999999998199587 \cdot 10^{-24}\right)\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 3
Error	4.7
Cost	297

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -6.500000190867716 \cdot 10^{-13} \lor \neg \left(n0_i \leq 9.99999983775159 \cdot 10^{-18}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i + n1_i \cdot u\\ \end{array} \]

Alternative 4
Error	4.7
Cost	297

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -6.500000190867716 \cdot 10^{-13} \lor \neg \left(n0_i \leq 9.99999983775159 \cdot 10^{-18}\right):\\ \;\;\;\;n0_i - n0_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i + n1_i \cdot u\\ \end{array} \]

Alternative 5
Error	0.7
Cost	288

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

Alternative 6
Error	12.6
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -3.99999987306209 \cdot 10^{-20}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 7
Error	0.6
Cost	224

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

Alternative 8
Error	17.0
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2023053 
(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)))