Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.3

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

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

↓

\[\left(1 - \mathsf{fma}\left(0.5, u \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right), \frac{\cos normAngle}{\frac{\sin normAngle}{u \cdot normAngle}}\right)\right) \cdot n0_i + \left(\mathsf{fma}\left(0.008333333333333333, \frac{{u}^{5}}{\sin normAngle} \cdot {normAngle}^{5}, \mathsf{fma}\left(\frac{u}{\sin normAngle}, normAngle, \left(\frac{{u}^{3}}{\sin normAngle} \cdot {normAngle}^{3}\right) \cdot -0.16666666666666666\right)\right) + -0.0001984126984126984 \cdot \left(\frac{{u}^{7}}{\sin normAngle} \cdot {normAngle}^{7}\right)\right) \cdot n1_i \]

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
 (+
  (*
   (-
    1.0
    (fma
     0.5
     (* u (* u (* normAngle normAngle)))
     (/ (cos normAngle) (/ (sin normAngle) (* u normAngle)))))
   n0_i)
  (*
   (+
    (fma
     0.008333333333333333
     (* (/ (pow u 5.0) (sin normAngle)) (pow normAngle 5.0))
     (fma
      (/ u (sin normAngle))
      normAngle
      (*
       (* (/ (pow u 3.0) (sin normAngle)) (pow normAngle 3.0))
       -0.16666666666666666)))
    (*
     -0.0001984126984126984
     (* (/ (pow u 7.0) (sin normAngle)) (pow normAngle 7.0))))
   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 ((1.0f - fmaf(0.5f, (u * (u * (normAngle * normAngle))), (cosf(normAngle) / (sinf(normAngle) / (u * normAngle))))) * n0_i) + ((fmaf(0.008333333333333333f, ((powf(u, 5.0f) / sinf(normAngle)) * powf(normAngle, 5.0f)), fmaf((u / sinf(normAngle)), normAngle, (((powf(u, 3.0f) / sinf(normAngle)) * powf(normAngle, 3.0f)) * -0.16666666666666666f))) + (-0.0001984126984126984f * ((powf(u, 7.0f) / sinf(normAngle)) * powf(normAngle, 7.0f)))) * 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(Float32(Float32(1.0) - fma(Float32(0.5), Float32(u * Float32(u * Float32(normAngle * normAngle))), Float32(cos(normAngle) / Float32(sin(normAngle) / Float32(u * normAngle))))) * n0_i) + Float32(Float32(fma(Float32(0.008333333333333333), Float32(Float32((u ^ Float32(5.0)) / sin(normAngle)) * (normAngle ^ Float32(5.0))), fma(Float32(u / sin(normAngle)), normAngle, Float32(Float32(Float32((u ^ Float32(3.0)) / sin(normAngle)) * (normAngle ^ Float32(3.0))) * Float32(-0.16666666666666666)))) + Float32(Float32(-0.0001984126984126984) * Float32(Float32((u ^ Float32(7.0)) / sin(normAngle)) * (normAngle ^ Float32(7.0))))) * 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 u around 0 0.8

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

3. Simplified 0.8

   \[\leadsto \color{blue}{\left(1 - \mathsf{fma}\left(0.5, u \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right), \frac{\cos normAngle}{\frac{\sin normAngle}{u \cdot normAngle}}\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

4. Taylor expanded in u around 0 0.7

   \[\leadsto \left(1 - \mathsf{fma}\left(0.5, u \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right), \frac{\cos normAngle}{\frac{\sin normAngle}{u \cdot normAngle}}\right)\right) \cdot n0_i + \color{blue}{\left(\left(0.008333333333333333 \cdot \frac{{u}^{5} \cdot {normAngle}^{5}}{\sin normAngle} + \frac{u \cdot normAngle}{\sin normAngle}\right) - \left(0.0001984126984126984 \cdot \frac{{u}^{7} \cdot {normAngle}^{7}}{\sin normAngle} + 0.16666666666666666 \cdot \frac{{u}^{3} \cdot {normAngle}^{3}}{\sin normAngle}\right)\right)} \cdot n1_i \]

5. Simplified 0.3

   \[\leadsto \left(1 - \mathsf{fma}\left(0.5, u \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right), \frac{\cos normAngle}{\frac{\sin normAngle}{u \cdot normAngle}}\right)\right) \cdot n0_i + \color{blue}{\left(\mathsf{fma}\left(0.008333333333333333, \frac{{u}^{5}}{\sin normAngle} \cdot {normAngle}^{5}, \mathsf{fma}\left(\frac{u}{\sin normAngle}, normAngle, \left(\frac{{u}^{3}}{\sin normAngle} \cdot {normAngle}^{3}\right) \cdot -0.16666666666666666\right)\right) + -0.0001984126984126984 \cdot \left(\frac{{u}^{7}}{\sin normAngle} \cdot {normAngle}^{7}\right)\right)} \cdot n1_i \]

6. Final simplification 0.3

   \[\leadsto \left(1 - \mathsf{fma}\left(0.5, u \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right), \frac{\cos normAngle}{\frac{\sin normAngle}{u \cdot normAngle}}\right)\right) \cdot n0_i + \left(\mathsf{fma}\left(0.008333333333333333, \frac{{u}^{5}}{\sin normAngle} \cdot {normAngle}^{5}, \mathsf{fma}\left(\frac{u}{\sin normAngle}, normAngle, \left(\frac{{u}^{3}}{\sin normAngle} \cdot {normAngle}^{3}\right) \cdot -0.16666666666666666\right)\right) + -0.0001984126984126984 \cdot \left(\frac{{u}^{7}}{\sin normAngle} \cdot {normAngle}^{7}\right)\right) \cdot n1_i \]

Reproduce

herbie shell --seed 2022162 
(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)))