Curve intersection, scale width based on ribbon orientation

Average Accuracy: 97.2% → 98.6%

Time: 26.1s

Precision: binary32

Cost: 480

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

Math

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

↓

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

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
  (*
   (- (+ n1_i (* (* n1_i 0.16666666666666666) (* normAngle normAngle))) n0_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 + (((n1_i + ((n1_i * 0.16666666666666666f) * (normAngle * normAngle))) - n0_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 + (((n1_i + ((n1_i * 0.16666666666666666e0) * (normangle * normangle))) - n0_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(Float32(n1_i + Float32(Float32(n1_i * Float32(0.16666666666666666)) * Float32(normAngle * normAngle))) - n0_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 + (((n1_i + ((n1_i * single(0.16666666666666666)) * (normAngle * normAngle))) - n0_i) * u);
end

Derivation

1. Initial program 97.2%

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

2. Simplified 97.7%

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

   [Start] 97.2	\[ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   fma-def [=>] 97.2	\[ \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]
   associate-*r/ [=>] 97.4	\[ \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]
   *-rgt-identity [=>] 97.4	\[ \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]
   associate-*r/ [=>] 97.7	\[ \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]
   *-rgt-identity [=>] 97.7	\[ \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

3. Taylor expanded in normAngle around 0 97.4%

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

4. Taylor expanded in u around 0 90.4%

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

5. Taylor expanded in normAngle around 0 98.6%

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

6. Simplified 98.6%

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

   [Start] 98.6	\[ \left(\left(n1_i + 0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right) + -1 \cdot n0_i\right) \cdot u + n0_i \]
   associate-*r* [=>] 98.6	\[ \left(\left(n1_i + \color{blue}{\left(0.16666666666666666 \cdot n1_i\right) \cdot {normAngle}^{2}}\right) + -1 \cdot n0_i\right) \cdot u + n0_i \]
   unpow2 [=>] 98.6	\[ \left(\left(n1_i + \left(0.16666666666666666 \cdot n1_i\right) \cdot \color{blue}{\left(normAngle \cdot normAngle\right)}\right) + -1 \cdot n0_i\right) \cdot u + n0_i \]

7. Final simplification 98.6%

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

Alternatives

Alternative 1
Accuracy	86.5%
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.0000000063421537 \cdot 10^{-30} \lor \neg \left(n1_i \leq 4.0000000781659255 \cdot 10^{-25}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 2
Accuracy	86.6%
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.0000000063421537 \cdot 10^{-30} \lor \neg \left(n1_i \leq 4.0000000781659255 \cdot 10^{-25}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \]

Alternative 3
Accuracy	70.4%
Cost	296

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 4
Accuracy	59.3%
Cost	232

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 5
Accuracy	97.9%
Cost	224

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

Alternative 6
Accuracy	46.4%
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2023151 
(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)))