Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.3

Time: 7.5s

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

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

↓

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

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
 (fma
  u
  (* (/ normAngle (sin normAngle)) n1_i)
  (* (/ n0_i (sin normAngle)) (sin (fma normAngle (- u) 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(u, ((normAngle / sinf(normAngle)) * n1_i), ((n0_i / sinf(normAngle)) * sinf(fmaf(normAngle, -u, 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(u, Float32(Float32(normAngle / sin(normAngle)) * n1_i), Float32(Float32(n0_i / sin(normAngle)) * sin(fma(normAngle, Float32(-u), 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 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 8.1

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

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. Final simplification 0.3

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

Reproduce

herbie shell --seed 2022185 
(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)))