Curve intersection, scale width based on ribbon orientation

Average Error: 0.8 → 0.5

Time: 17.6s

Precision: binary32

Cost: 608

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

Math

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

↓

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

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
 (+
  (* n1_i u)
  (+
   (* normAngle (* (* normAngle (* n1_i u)) 0.16666666666666666))
   (* (- 1.0 u) 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 (n1_i * u) + ((normAngle * ((normAngle * (n1_i * u)) * 0.16666666666666666f)) + ((1.0f - u) * n0_i));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = ((sin(((1.0e0 - u) * normangle)) * (1.0e0 / sin(normangle))) * n0_i) + ((sin((u * normangle)) * (1.0e0 / sin(normangle))) * n1_i)
end function

↓

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = (n1_i * u) + ((normangle * ((normangle * (n1_i * u)) * 0.16666666666666666e0)) + ((1.0e0 - u) * n0_i))
end function

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(n1_i * u) + Float32(Float32(normAngle * Float32(Float32(normAngle * Float32(n1_i * u)) * Float32(0.16666666666666666))) + Float32(Float32(Float32(1.0) - u) * n0_i)))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = ((sin(((single(1.0) - u) * normAngle)) * (single(1.0) / sin(normAngle))) * n0_i) + ((sin((u * normAngle)) * (single(1.0) / sin(normAngle))) * n1_i);
end

↓

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n1_i * u) + ((normAngle * ((normAngle * (n1_i * u)) * single(0.16666666666666666))) + ((single(1.0) - u) * n0_i));
end

Derivation

1. Initial program 0.8

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

2. Simplified 8.2

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

   [Start] 0.8	\[ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   *-commutative [=>] 0.8	\[ \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   associate-*l* [=>] 6.0	\[ \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   *-commutative [=>] 6.0	\[ \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]
   associate-*l* [=>] 8.2	\[ \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]
   distribute-lft-out [=>] 8.2	\[ \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Taylor expanded in u around 0 8.2

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

4. Taylor expanded in normAngle around 0 8.8

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

5. Taylor expanded in normAngle around 0 1.0

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

6. Taylor expanded in n1_i around inf 0.5

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

7. Simplified 0.5

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

   [Start] 0.5	\[ n1_i \cdot u + \left(0.16666666666666666 \cdot \left(n1_i \cdot \left(u \cdot {normAngle}^{2}\right)\right) + \left(1 - u\right) \cdot n0_i\right) \]
   *-commutative [=>] 0.5	\[ n1_i \cdot u + \left(\color{blue}{\left(n1_i \cdot \left(u \cdot {normAngle}^{2}\right)\right) \cdot 0.16666666666666666} + \left(1 - u\right) \cdot n0_i\right) \]
   associate-*r* [=>] 0.5	\[ n1_i \cdot u + \left(\color{blue}{\left(\left(n1_i \cdot u\right) \cdot {normAngle}^{2}\right)} \cdot 0.16666666666666666 + \left(1 - u\right) \cdot n0_i\right) \]
   unpow2 [=>] 0.5	\[ n1_i \cdot u + \left(\left(\left(n1_i \cdot u\right) \cdot \color{blue}{\left(normAngle \cdot normAngle\right)}\right) \cdot 0.16666666666666666 + \left(1 - u\right) \cdot n0_i\right) \]
   *-commutative [<=] 0.5	\[ n1_i \cdot u + \left(\color{blue}{\left(\left(normAngle \cdot normAngle\right) \cdot \left(n1_i \cdot u\right)\right)} \cdot 0.16666666666666666 + \left(1 - u\right) \cdot n0_i\right) \]
   associate-*l* [=>] 0.5	\[ n1_i \cdot u + \left(\color{blue}{\left(normAngle \cdot \left(normAngle \cdot \left(n1_i \cdot u\right)\right)\right)} \cdot 0.16666666666666666 + \left(1 - u\right) \cdot n0_i\right) \]
   associate-*l* [=>] 0.5	\[ n1_i \cdot u + \left(\color{blue}{normAngle \cdot \left(\left(normAngle \cdot \left(n1_i \cdot u\right)\right) \cdot 0.16666666666666666\right)} + \left(1 - u\right) \cdot n0_i\right) \]
   *-commutative [=>] 0.5	\[ n1_i \cdot u + \left(normAngle \cdot \left(\left(normAngle \cdot \color{blue}{\left(u \cdot n1_i\right)}\right) \cdot 0.16666666666666666\right) + \left(1 - u\right) \cdot n0_i\right) \]

8. Final simplification 0.5

   \[\leadsto n1_i \cdot u + \left(normAngle \cdot \left(\left(normAngle \cdot \left(n1_i \cdot u\right)\right) \cdot 0.16666666666666666\right) + \left(1 - u\right) \cdot n0_i\right) \]

Alternatives

Alternative 1
Error	4.4
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.0000000390829628 \cdot 10^{-25} \lor \neg \left(n1_i \leq 5.000000097707407 \cdot 10^{-25}\right):\\ \;\;\;\;n1_i \cdot u + n0_i\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \end{array} \]

Alternative 2
Error	4.4
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.0000000390829628 \cdot 10^{-24} \lor \neg \left(n1_i \leq 5.000000097707407 \cdot 10^{-25}\right):\\ \;\;\;\;n1_i \cdot u + n0_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \end{array} \]

Alternative 3
Error	9.5
Cost	296

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 4
Error	12.8
Cost	232

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 5
Error	0.6
Cost	224

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

Alternative 6
Error	16.8
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2023054 
(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)))