Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.2

Time: 14.8s

Precision: binary32

Cost: 13280

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

Math

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

↓

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

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
  (-
   (/ n1_i (/ (sin normAngle) normAngle))
   (* normAngle (/ (* n0_i (cos normAngle)) (sin normAngle))))
  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, ((n1_i / (sinf(normAngle) / normAngle)) - (normAngle * ((n0_i * cosf(normAngle)) / sinf(normAngle)))), 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(n1_i / Float32(sin(normAngle) / normAngle)) - Float32(normAngle * Float32(Float32(n0_i * cos(normAngle)) / sin(normAngle)))), 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. Simplified 8.2

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(normAngle - u \cdot normAngle\right), n0_i, \sin \left(u \cdot normAngle\right) \cdot n1_i\right)}{\sin normAngle}} \]
   Proof
   (/.f32 (fma.f32 (sin.f32 (-.f32 normAngle (*.f32 u normAngle))) n0_i (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle)): 0 points increase in error, 0 points decrease in error
   (/.f32 (fma.f32 (sin.f32 (-.f32 (Rewrite<= *-lft-identity_binary32 (*.f32 1 normAngle)) (*.f32 u normAngle))) n0_i (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle)): 0 points increase in error, 0 points decrease in error
   (/.f32 (fma.f32 (sin.f32 (Rewrite=> distribute-rgt-out--_binary32 (*.f32 normAngle (-.f32 1 u)))) n0_i (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle)): 14 points increase in error, 3 points decrease in error
   (/.f32 (fma.f32 (sin.f32 (Rewrite<= *-commutative_binary32 (*.f32 (-.f32 1 u) normAngle))) n0_i (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle)): 0 points increase in error, 0 points decrease in error
   (/.f32 (Rewrite<= fma-def_binary32 (+.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i) (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i))) (sin.f32 normAngle)): 3 points increase in error, 7 points decrease in error
   (/.f32 (Rewrite<= cancel-sign-sub_binary32 (-.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i) (*.f32 (neg.f32 (sin.f32 (*.f32 u normAngle))) n1_i))) (sin.f32 normAngle)): 0 points increase in error, 0 points decrease in error
   (/.f32 (-.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i) (Rewrite=> distribute-lft-neg-out_binary32 (neg.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)))) (sin.f32 normAngle)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> div-sub_binary32 (-.f32 (/.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i) (sin.f32 normAngle)) (/.f32 (neg.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle)))): 11 points increase in error, 15 points decrease in error
   (-.f32 (Rewrite<= associate-*l/_binary32 (*.f32 (/.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (sin.f32 normAngle)) n0_i)) (/.f32 (neg.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle))): 27 points increase in error, 71 points decrease in error
   (-.f32 (*.f32 (/.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) 1)) (sin.f32 normAngle)) n0_i) (/.f32 (neg.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 (Rewrite<= associate-*r/_binary32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle)))) n0_i) (/.f32 (neg.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle))): 29 points increase in error, 12 points decrease in error
   (-.f32 (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i) (Rewrite<= distribute-neg-frac_binary32 (neg.f32 (/.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i) (sin.f32 normAngle))))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i) (neg.f32 (Rewrite<= associate-*l/_binary32 (*.f32 (/.f32 (sin.f32 (*.f32 u normAngle)) (sin.f32 normAngle)) n1_i)))): 7 points increase in error, 73 points decrease in error
   (-.f32 (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i) (neg.f32 (*.f32 (/.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 (sin.f32 (*.f32 u normAngle)) 1)) (sin.f32 normAngle)) n1_i))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i) (neg.f32 (*.f32 (Rewrite<= associate-*r/_binary32 (*.f32 (sin.f32 (*.f32 u normAngle)) (/.f32 1 (sin.f32 normAngle)))) n1_i))): 20 points increase in error, 7 points decrease in error
   (-.f32 (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i) (Rewrite<= distribute-lft-neg-out_binary32 (*.f32 (neg.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) (/.f32 1 (sin.f32 normAngle)))) n1_i))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> cancel-sign-sub_binary32 (+.f32 (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i) (*.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) (/.f32 1 (sin.f32 normAngle))) n1_i))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in u around 0 3.8

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

4. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \frac{n0_i \cdot \cos normAngle}{\sin normAngle} \cdot normAngle, n0_i\right)} \]
   Proof
   (fma.f32 u (-.f32 (/.f32 n1_i (/.f32 (sin.f32 normAngle) normAngle)) (*.f32 (/.f32 (*.f32 n0_i (cos.f32 normAngle)) (sin.f32 normAngle)) normAngle)) n0_i): 0 points increase in error, 0 points decrease in error
   (fma.f32 u (-.f32 (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle))) (*.f32 (/.f32 (*.f32 n0_i (cos.f32 normAngle)) (sin.f32 normAngle)) normAngle)) n0_i): 54 points increase in error, 2 points decrease in error
   (fma.f32 u (-.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (*.f32 (/.f32 (Rewrite<= *-commutative_binary32 (*.f32 (cos.f32 normAngle) n0_i)) (sin.f32 normAngle)) normAngle)) n0_i): 0 points increase in error, 0 points decrease in error
   (fma.f32 u (-.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (Rewrite<= associate-/r/_binary32 (/.f32 (*.f32 (cos.f32 normAngle) n0_i) (/.f32 (sin.f32 normAngle) normAngle)))) n0_i): 0 points increase in error, 0 points decrease in error
   (fma.f32 u (-.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 (*.f32 (cos.f32 normAngle) n0_i) normAngle) (sin.f32 normAngle)))) n0_i): 33 points increase in error, 10 points decrease in error
   (fma.f32 u (-.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (/.f32 (Rewrite<= associate-*r*_binary32 (*.f32 (cos.f32 normAngle) (*.f32 n0_i normAngle))) (sin.f32 normAngle))) n0_i): 0 points increase in error, 0 points decrease in error
   (fma.f32 u (Rewrite<= unsub-neg_binary32 (+.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (neg.f32 (/.f32 (*.f32 (cos.f32 normAngle) (*.f32 n0_i normAngle)) (sin.f32 normAngle))))) n0_i): 0 points increase in error, 0 points decrease in error
   (fma.f32 u (+.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (/.f32 (*.f32 (cos.f32 normAngle) (*.f32 n0_i normAngle)) (sin.f32 normAngle))))) 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)) (*.f32 -1 (/.f32 (*.f32 (cos.f32 normAngle) (*.f32 n0_i normAngle)) (sin.f32 normAngle))))) n0_i)): 11 points increase in error, 9 points decrease in error
   (Rewrite<= +-commutative_binary32 (+.f32 n0_i (*.f32 u (+.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (*.f32 -1 (/.f32 (*.f32 (cos.f32 normAngle) (*.f32 n0_i normAngle)) (sin.f32 normAngle))))))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	6688

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

Alternative 2
Error	0.4
Cost	3968

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

Alternative 3
Error	0.4
Cost	544

\[u \cdot \left(n1_i - n0_i\right) + \left(n0_i + u \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n1_i \cdot 0.16666666666666666\right)\right)\right) \]

Alternative 4
Error	9.6
Cost	296

\[\begin{array}{l} t_0 := n0_i - u \cdot n0_i\\ \mathbf{if}\;n0_i \leq -7.99999974612418 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n0_i \leq 6.000000019026461 \cdot 10^{-29}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	4.5
Cost	296

\[\begin{array}{l} t_0 := n0_i + u \cdot n1_i\\ \mathbf{if}\;n1_i \leq -5.000000097707407 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	12.6
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -7.99999974612418 \cdot 10^{-19}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 7
Error	0.7
Cost	224

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

Alternative 8
Error	16.8
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2022297 
(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)))