Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.2

Time: 14.2s

Precision: binary32

Cost: 6688

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

Math

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

↓

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

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) 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) - 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) - 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 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. Simplified 0.9

   \[\leadsto \color{blue}{\left(n0_i - u \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   Proof
   (-.f32 n0_i (*.f32 u n0_i)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unsub-neg_binary32 (+.f32 n0_i (neg.f32 (*.f32 u n0_i)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (Rewrite<= *-lft-identity_binary32 (*.f32 1 n0_i)) (neg.f32 (*.f32 u n0_i))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 1 n0_i) (Rewrite<= distribute-lft-neg-out_binary32 (*.f32 (neg.f32 u) n0_i))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-in_binary32 (*.f32 n0_i (+.f32 1 (neg.f32 u)))): 44 points increase in error, 9 points decrease in error
   (*.f32 n0_i (Rewrite<= sub-neg_binary32 (-.f32 1 u))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary32 (*.f32 (-.f32 1 u) n0_i)): 0 points increase in error, 0 points decrease in error

4. Taylor expanded in u around 0 3.2

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

5. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \frac{normAngle}{\sin normAngle} \cdot n1_i - n0_i, n0_i\right)} \]
   Proof
   (fma.f32 u (-.f32 (*.f32 (/.f32 normAngle (sin.f32 normAngle)) n1_i) n0_i) n0_i): 0 points increase in error, 0 points decrease in error
   (fma.f32 u (-.f32 (Rewrite<= associate-/r/_binary32 (/.f32 normAngle (/.f32 (sin.f32 normAngle) n1_i))) n0_i) n0_i): 12 points increase in error, 3 points decrease in error
   (fma.f32 u (-.f32 (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 normAngle n1_i) (sin.f32 normAngle))) n0_i) n0_i): 49 points increase in error, 6 points decrease in error
   (fma.f32 u (-.f32 (/.f32 (Rewrite<= *-commutative_binary32 (*.f32 n1_i normAngle)) (sin.f32 normAngle)) n0_i) n0_i): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary32 (+.f32 (*.f32 u (-.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) n0_i)) n0_i)): 12 points increase in error, 2 points decrease in error
   (+.f32 (Rewrite<= *-commutative_binary32 (*.f32 (-.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) n0_i) u)) n0_i): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.4
Cost	3712

\[u \cdot \left(n1_i - n0_i\right) + \left(n0_i + 0.16666666666666666 \cdot \left(n1_i \cdot \left(u \cdot {normAngle}^{2}\right)\right)\right) \]

Alternative 2
Error	0.6
Cost	3360

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

Alternative 3
Error	0.5
Cost	544

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

Alternative 4
Error	10.1
Cost	296

\[\begin{array}{l} t_0 := n0_i \cdot \left(1 - u\right)\\ \mathbf{if}\;n0_i \leq -1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n0_i \leq 9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	4.7
Cost	296

\[\begin{array}{l} t_0 := n0_i \cdot \left(1 - u\right)\\ \mathbf{if}\;n0_i \leq -9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n0_i \leq 4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	4.7
Cost	296

\[\begin{array}{l} t_0 := n0_i - u \cdot n0_i\\ \mathbf{if}\;n0_i \leq -9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n0_i \leq 4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	12.9
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 8
Error	0.7
Cost	224

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

Alternative 9
Error	16.8
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2022312 
(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)))