Curve intersection, scale width based on ribbon orientation

Average Error: 0.8 → 0.6

Time: 18.9s

Precision: binary32

Cost: 224

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

Math

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

↓

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

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
 (- n0_i (* (- 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 n0_i - ((n0_i - n1_i) * u);
}

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 = n0_i - ((n0_i - n1_i) * u)
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(n0_i - Float32(Float32(n0_i - n1_i) * u))
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 = n0_i - ((n0_i - n1_i) * u);
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 8.1

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

   [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	\[ \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]
   +-commutative [<=] 8.2	\[ \frac{1}{\sin normAngle} \cdot \color{blue}{\left(\sin \left(u \cdot normAngle\right) \cdot n1_i + \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} \]
   associate-*l/ [=>] 8.2	\[ \color{blue}{\frac{1 \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i + \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)}{\sin normAngle}} \]
   *-commutative [=>] 8.2	\[ \frac{\color{blue}{\left(\sin \left(u \cdot normAngle\right) \cdot n1_i + \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) \cdot 1}}{\sin normAngle} \]
   associate-/l* [=>] 8.2	\[ \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot n1_i + \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\frac{\sin normAngle}{1}}} \]
   /-rgt-identity [=>] 8.2	\[ \frac{\sin \left(u \cdot normAngle\right) \cdot n1_i + \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\color{blue}{\sin normAngle}} \]

3. Taylor expanded in normAngle around 0 8.7

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

4. Simplified 8.7

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

   [Start] 8.7	\[ \frac{\left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \cdot normAngle}{\sin normAngle} \]
   fma-def [=>] 8.7	\[ \frac{\color{blue}{\mathsf{fma}\left(n1_i, u, \left(1 - u\right) \cdot n0_i\right)} \cdot normAngle}{\sin normAngle} \]

5. Taylor expanded in u around -inf 8.7

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

6. Simplified 8.7

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

   [Start] 8.7	\[ \frac{\left(-1 \cdot \left(u \cdot \left(-1 \cdot n1_i + n0_i\right)\right) + n0_i\right) \cdot normAngle}{\sin normAngle} \]
   +-commutative [=>] 8.7	\[ \frac{\color{blue}{\left(n0_i + -1 \cdot \left(u \cdot \left(-1 \cdot n1_i + n0_i\right)\right)\right)} \cdot normAngle}{\sin normAngle} \]
   mul-1-neg [=>] 8.7	\[ \frac{\left(n0_i + \color{blue}{\left(-u \cdot \left(-1 \cdot n1_i + n0_i\right)\right)}\right) \cdot normAngle}{\sin normAngle} \]
   unsub-neg [=>] 8.7	\[ \frac{\color{blue}{\left(n0_i - u \cdot \left(-1 \cdot n1_i + n0_i\right)\right)} \cdot normAngle}{\sin normAngle} \]
   +-commutative [=>] 8.7	\[ \frac{\left(n0_i - u \cdot \color{blue}{\left(n0_i + -1 \cdot n1_i\right)}\right) \cdot normAngle}{\sin normAngle} \]
   mul-1-neg [=>] 8.7	\[ \frac{\left(n0_i - u \cdot \left(n0_i + \color{blue}{\left(-n1_i\right)}\right)\right) \cdot normAngle}{\sin normAngle} \]
   unsub-neg [=>] 8.7	\[ \frac{\left(n0_i - u \cdot \color{blue}{\left(n0_i - n1_i\right)}\right) \cdot normAngle}{\sin normAngle} \]

7. Taylor expanded in normAngle around 0 0.6

   \[\leadsto \color{blue}{n0_i - \left(n0_i - n1_i\right) \cdot u} \]

8. Final simplification 0.6

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

Alternatives

Alternative 1
Error	4.4
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.5000000601271012 \cdot 10^{-26} \lor \neg \left(n1_i \leq 3.500000191654701 \cdot 10^{-27}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 2
Error	4.4
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.5000000601271012 \cdot 10^{-26} \lor \neg \left(n1_i \leq 3.500000191654701 \cdot 10^{-27}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \]

Alternative 3
Error	9.4
Cost	296

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.499999990010493 \cdot 10^{-12}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 4
Error	12.7
Cost	232

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.499999990010493 \cdot 10^{-12}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 5
Error	16.9
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2023027 
(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)))