Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.4

Time: 8.5s

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

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

↓

\[e^{\log \left(\frac{\sin \left(normAngle \cdot \left(1 - u\right)\right)}{\sin normAngle}\right)} \cdot n0_i + \left(\mathsf{fma}\left(0.16666666666666666, u \cdot \left(normAngle \cdot normAngle\right), u\right) - 0.16666666666666666 \cdot \left(\left(normAngle \cdot normAngle\right) \cdot {u}^{3}\right)\right) \cdot n1_i \]

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
 (+
  (* (exp (log (/ (sin (* normAngle (- 1.0 u))) (sin normAngle)))) n0_i)
  (*
   (-
    (fma 0.16666666666666666 (* u (* normAngle normAngle)) u)
    (* 0.16666666666666666 (* (* normAngle normAngle) (pow u 3.0))))
   n1_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 (expf(logf((sinf((normAngle * (1.0f - u))) / sinf(normAngle)))) * n0_i) + ((fmaf(0.16666666666666666f, (u * (normAngle * normAngle)), u) - (0.16666666666666666f * ((normAngle * normAngle) * powf(u, 3.0f)))) * n1_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 Float32(Float32(exp(log(Float32(sin(Float32(normAngle * Float32(Float32(1.0) - u))) / sin(normAngle)))) * n0_i) + Float32(Float32(fma(Float32(0.16666666666666666), Float32(u * Float32(normAngle * normAngle)), u) - Float32(Float32(0.16666666666666666) * Float32(Float32(normAngle * normAngle) * (u ^ Float32(3.0))))) * n1_i))
end

Error

Bits error versus normAngle

Bits error versus u

Bits error versus n0_i

Bits error versus n1_i

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 0.5

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\left(\left(0.16666666666666666 \cdot \left(u \cdot {normAngle}^{2}\right) + u\right) - 0.16666666666666666 \cdot \left({u}^{3} \cdot {normAngle}^{2}\right)\right)} \cdot n1_i \]

3. Simplified 0.5

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\left(\mathsf{fma}\left(0.16666666666666666, u \cdot \left(normAngle \cdot normAngle\right), u\right) - 0.16666666666666666 \cdot \left({u}^{3} \cdot \left(normAngle \cdot normAngle\right)\right)\right)} \cdot n1_i \]

4. Applied egg-rr 0.4

   \[\leadsto \color{blue}{e^{\log \left(\frac{\sin \left(normAngle \cdot \left(1 - u\right)\right)}{\sin normAngle}\right)}} \cdot n0_i + \left(\mathsf{fma}\left(0.16666666666666666, u \cdot \left(normAngle \cdot normAngle\right), u\right) - 0.16666666666666666 \cdot \left({u}^{3} \cdot \left(normAngle \cdot normAngle\right)\right)\right) \cdot n1_i \]

5. Final simplification 0.4

   \[\leadsto e^{\log \left(\frac{\sin \left(normAngle \cdot \left(1 - u\right)\right)}{\sin normAngle}\right)} \cdot n0_i + \left(\mathsf{fma}\left(0.16666666666666666, u \cdot \left(normAngle \cdot normAngle\right), u\right) - 0.16666666666666666 \cdot \left(\left(normAngle \cdot normAngle\right) \cdot {u}^{3}\right)\right) \cdot n1_i \]

Reproduce

herbie shell --seed 2022133 
(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)))