Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.4

Time: 17.6s

Precision: binary32

Cost: 13728

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

Math

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

↓

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

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
  (* normAngle normAngle)
  (*
   -0.16666666666666666
   (+ (* n1_i (- (pow u 3.0) u)) (* n0_i (+ u (+ (pow (- 1.0 u) 3.0) -1.0)))))
  (fma (- 1.0 u) n0_i (* n1_i u))))

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((normAngle * normAngle), (-0.16666666666666666f * ((n1_i * (powf(u, 3.0f) - u)) + (n0_i * (u + (powf((1.0f - u), 3.0f) + -1.0f))))), fmaf((1.0f - u), n0_i, (n1_i * u)));
}

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(Float32(normAngle * normAngle), Float32(Float32(-0.16666666666666666) * Float32(Float32(n1_i * Float32((u ^ Float32(3.0)) - u)) + Float32(n0_i * Float32(u + Float32((Float32(Float32(1.0) - u) ^ Float32(3.0)) + Float32(-1.0)))))), fma(Float32(Float32(1.0) - u), n0_i, Float32(n1_i * u)))
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 0.4

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

3. Simplified 0.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, -0.16666666666666666 \cdot \left(n1_i \cdot \left({u}^{3} - u\right) + n0_i \cdot \left(\left({\left(1 - u\right)}^{3} + -1\right) + u\right)\right), \mathsf{fma}\left(1 - u, n0_i, u \cdot n1_i\right)\right)} \]
   Proof
   (fma.f32 (*.f32 normAngle normAngle) (*.f32 -1/6 (+.f32 (*.f32 n1_i (-.f32 (pow.f32 u 3) u)) (*.f32 n0_i (+.f32 (+.f32 (pow.f32 (-.f32 1 u) 3) -1) u)))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (Rewrite<= unpow2_binary32 (pow.f32 normAngle 2)) (*.f32 -1/6 (+.f32 (*.f32 n1_i (-.f32 (pow.f32 u 3) u)) (*.f32 n0_i (+.f32 (+.f32 (pow.f32 (-.f32 1 u) 3) -1) u)))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (*.f32 -1/6 (+.f32 (Rewrite<= distribute-lft-out--_binary32 (-.f32 (*.f32 n1_i (pow.f32 u 3)) (*.f32 n1_i u))) (*.f32 n0_i (+.f32 (+.f32 (pow.f32 (-.f32 1 u) 3) -1) u)))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (*.f32 -1/6 (+.f32 (-.f32 (*.f32 n1_i (pow.f32 u 3)) (*.f32 n1_i u)) (*.f32 n0_i (+.f32 (+.f32 (pow.f32 (-.f32 1 u) 3) (Rewrite<= metadata-eval (neg.f32 1))) u)))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (*.f32 -1/6 (+.f32 (-.f32 (*.f32 n1_i (pow.f32 u 3)) (*.f32 n1_i u)) (*.f32 n0_i (+.f32 (Rewrite<= sub-neg_binary32 (-.f32 (pow.f32 (-.f32 1 u) 3) 1)) u)))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (*.f32 -1/6 (+.f32 (-.f32 (*.f32 n1_i (pow.f32 u 3)) (*.f32 n1_i u)) (*.f32 n0_i (Rewrite<= associate--r-_binary32 (-.f32 (pow.f32 (-.f32 1 u) 3) (-.f32 1 u)))))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (*.f32 -1/6 (+.f32 (-.f32 (*.f32 n1_i (pow.f32 u 3)) (*.f32 n1_i u)) (Rewrite<= distribute-rgt-out--_binary32 (-.f32 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i) (*.f32 (-.f32 1 u) n0_i))))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (Rewrite<= distribute-lft-out_binary32 (+.f32 (*.f32 -1/6 (-.f32 (*.f32 n1_i (pow.f32 u 3)) (*.f32 n1_i u))) (*.f32 -1/6 (-.f32 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i) (*.f32 (-.f32 1 u) n0_i))))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (+.f32 (Rewrite<= distribute-lft-out--_binary32 (-.f32 (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3))) (*.f32 -1/6 (*.f32 n1_i u)))) (*.f32 -1/6 (-.f32 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i) (*.f32 (-.f32 1 u) n0_i)))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (+.f32 (-.f32 (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3))) (*.f32 -1/6 (*.f32 n1_i u))) (Rewrite<= distribute-lft-out--_binary32 (-.f32 (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i)) (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i))))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (Rewrite=> +-commutative_binary32 (+.f32 (-.f32 (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i)) (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i))) (-.f32 (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3))) (*.f32 -1/6 (*.f32 n1_i u))))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (+.f32 (-.f32 (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i)) (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i))) (Rewrite=> sub-neg_binary32 (+.f32 (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3))) (neg.f32 (*.f32 -1/6 (*.f32 n1_i u)))))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (+.f32 (-.f32 (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i)) (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i))) (Rewrite=> +-commutative_binary32 (+.f32 (neg.f32 (*.f32 -1/6 (*.f32 n1_i u))) (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3)))))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (Rewrite=> associate-+r+_binary32 (+.f32 (+.f32 (-.f32 (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i)) (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i))) (neg.f32 (*.f32 -1/6 (*.f32 n1_i u)))) (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3))))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (+.f32 (Rewrite<= sub-neg_binary32 (-.f32 (-.f32 (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i)) (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i))) (*.f32 -1/6 (*.f32 n1_i u)))) (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3)))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (+.f32 (Rewrite<= associate--r+_binary32 (-.f32 (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i)) (+.f32 (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i)) (*.f32 -1/6 (*.f32 n1_i u))))) (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3)))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3))) (-.f32 (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i)) (+.f32 (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i)) (*.f32 -1/6 (*.f32 n1_i u)))))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3))) (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i))) (+.f32 (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i)) (*.f32 -1/6 (*.f32 n1_i u))))) (fma.f32 (-.f32 1 u) n0_i (*.f32 u n1_i))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (-.f32 (+.f32 (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3))) (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i))) (+.f32 (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i)) (*.f32 -1/6 (*.f32 n1_i u)))) (fma.f32 (-.f32 1 u) n0_i (Rewrite<= *-commutative_binary32 (*.f32 n1_i u)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (-.f32 (+.f32 (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3))) (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i))) (+.f32 (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i)) (*.f32 -1/6 (*.f32 n1_i u)))) (Rewrite<= fma-def_binary32 (+.f32 (*.f32 (-.f32 1 u) n0_i) (*.f32 n1_i u)))): 15 points increase in error, 5 points decrease in error
   (fma.f32 (pow.f32 normAngle 2) (-.f32 (+.f32 (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3))) (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i))) (+.f32 (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i)) (*.f32 -1/6 (*.f32 n1_i u)))) (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 n1_i u) (*.f32 (-.f32 1 u) n0_i)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary32 (+.f32 (*.f32 (pow.f32 normAngle 2) (-.f32 (+.f32 (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3))) (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i))) (+.f32 (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i)) (*.f32 -1/6 (*.f32 n1_i u))))) (+.f32 (*.f32 n1_i u) (*.f32 (-.f32 1 u) n0_i)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (Rewrite<= *-commutative_binary32 (*.f32 (-.f32 (+.f32 (*.f32 -1/6 (*.f32 n1_i (pow.f32 u 3))) (*.f32 -1/6 (*.f32 (pow.f32 (-.f32 1 u) 3) n0_i))) (+.f32 (*.f32 -1/6 (*.f32 (-.f32 1 u) n0_i)) (*.f32 -1/6 (*.f32 n1_i u)))) (pow.f32 normAngle 2))) (+.f32 (*.f32 n1_i u) (*.f32 (-.f32 1 u) n0_i))): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	10880

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

Alternative 2
Error	0.4
Cost	7456

\[\left(n1_i \cdot u + n0_i \cdot \left(1 - u\right)\right) + {normAngle}^{2} \cdot \left(\left(u \cdot -0.5 + 0.5\right) \cdot \left(u \cdot n0_i\right) + \left(n1_i - n0_i\right) \cdot \left(0.16666666666666666 \cdot \left(u - {u}^{3}\right)\right)\right) \]

Alternative 3
Error	0.4
Cost	4096

\[\left(n1_i \cdot u + n0_i \cdot \left(1 - u\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\left(n1_i \cdot 0.16666666666666666 - n0_i \cdot -0.3333333333333333\right) - n0_i \cdot \left(u \cdot 0.5\right)\right)\right) \]

Alternative 4
Error	0.5
Cost	3776

\[\left(n1_i \cdot u + n0_i \cdot \left(1 - u\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n1_i \cdot 0.16666666666666666\right)\right) \]

Alternative 5
Error	0.6
Cost	3360

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

Alternative 6
Error	9.1
Cost	296

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 1.700000037930475 \cdot 10^{-14}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 7
Error	4.6
Cost	296

\[\begin{array}{l} t_0 := n0_i + n1_i \cdot u\\ \mathbf{if}\;n1_i \leq -9.999999887266023 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	12.4
Cost	232

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.000000045813705 \cdot 10^{-18}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 1.700000037930475 \cdot 10^{-14}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 9
Error	0.6
Cost	224

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

Alternative 10
Error	16.9
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2022308 
(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)))