Curve intersection, scale width based on ribbon orientation

Average Error: 2.81% → 1.91%

Time: 18.8s

Precision: binary32

Cost: 3424

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

Math

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

↓

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

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 n1_i (- n0_i (* u 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, n1_i, (n0_i - (u * 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, n1_i, Float32(n0_i - Float32(u * n0_i)))
end

Derivation

1. Initial program 2.81

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

2. Simplified 2.24

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

   [Start] 2.81	\[ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   fma-def [=>] 2.78	\[ \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]
   associate-*r/ [=>] 2.55	\[ \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]
   *-rgt-identity [=>] 2.55	\[ \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]
   associate-*r/ [=>] 2.24	\[ \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]
   *-rgt-identity [=>] 2.24	\[ \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

3. Taylor expanded in normAngle around 0 2.16

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

4. Simplified 2.04

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

   [Start] 2.16	\[ n1_i \cdot u + \left(1 - u\right) \cdot n0_i \]
   *-commutative [=>] 2.16	\[ \color{blue}{u \cdot n1_i} + \left(1 - u\right) \cdot n0_i \]
   fma-def [=>] 2.04	\[ \color{blue}{\mathsf{fma}\left(u, n1_i, \left(1 - u\right) \cdot n0_i\right)} \]

5. Applied egg-rr 1.91

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

6. Taylor expanded in n0_i around 0 2.04

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

7. Simplified 1.91

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

   [Start] 2.04	\[ \mathsf{fma}\left(u, n1_i, \left(1 + -1 \cdot u\right) \cdot n0_i\right) \]
   +-commutative [=>] 2.04	\[ \mathsf{fma}\left(u, n1_i, \color{blue}{\left(-1 \cdot u + 1\right)} \cdot n0_i\right) \]
   mul-1-neg [=>] 2.04	\[ \mathsf{fma}\left(u, n1_i, \left(\color{blue}{\left(-u\right)} + 1\right) \cdot n0_i\right) \]
   distribute-rgt1-in [<=] 1.91	\[ \mathsf{fma}\left(u, n1_i, \color{blue}{n0_i + \left(-u\right) \cdot n0_i}\right) \]
   cancel-sign-sub-inv [<=] 1.91	\[ \mathsf{fma}\left(u, n1_i, \color{blue}{n0_i - u \cdot n0_i}\right) \]

8. Final simplification 1.91

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

Alternatives

Alternative 1
Error	13.82%
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -5.000000097707407 \cdot 10^{-26} \lor \neg \left(n1_i \leq 1.0000000195414814 \cdot 10^{-25}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 2
Error	13.76%
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -5.000000097707407 \cdot 10^{-26} \lor \neg \left(n1_i \leq 1.0000000195414814 \cdot 10^{-25}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \end{array} \]

Alternative 3
Error	29.27%
Cost	296

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 4
Error	39.21%
Cost	232

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 5
Error	1.95%
Cost	224

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

Alternative 6
Error	53.36%
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2023090 
(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)))