Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.5

Time: 28.3s

Precision: binary32

Cost: 3648

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

Math

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

↓

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

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
 (+
  n0_i
  (*
   u
   (- (- n1_i n0_i) (* n1_i (* (pow normAngle 2.0) -0.16666666666666666))))))

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 + (u * ((n1_i - n0_i) - (n1_i * (powf(normAngle, 2.0f) * -0.16666666666666666f))));
}

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 + (u * ((n1_i - n0_i) - (n1_i * ((normangle ** 2.0e0) * (-0.16666666666666666e0)))))
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(u * Float32(Float32(n1_i - n0_i) - Float32(n1_i * Float32((normAngle ^ Float32(2.0)) * Float32(-0.16666666666666666))))))
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 + (u * ((n1_i - n0_i) - (n1_i * ((normAngle ^ single(2.0)) * single(-0.16666666666666666)))));
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

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

   [Start] 0.9	\[ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   rational.json-simplify-1 [=>] 0.9	\[ \color{blue}{\left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i} \]
   rational.json-simplify-2 [=>] 0.9	\[ \color{blue}{n1_i \cdot \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i \]
   rational.json-simplify-2 [=>] 0.9	\[ n1_i \cdot \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i \]
   rational.json-simplify-43 [=>] 4.4	\[ \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i \]
   rational.json-simplify-2 [=>] 4.4	\[ \frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right) + \color{blue}{n0_i \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} \]
   rational.json-simplify-2 [=>] 4.4	\[ \frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right) + n0_i \cdot \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \]
   rational.json-simplify-43 [=>] 8.2	\[ \frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} \]
   rational.json-simplify-2 [=>] 8.2	\[ \frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right) + \color{blue}{\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) \cdot \frac{1}{\sin normAngle}} \]
   rational.json-simplify-47 [=>] 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 normAngle around 0 0.4

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

4. Simplified 0.4

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

   [Start] 0.4	\[ \left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]
   rational.json-simplify-41 [=>] 0.4	\[ \color{blue}{n1_i \cdot u + \left(\left(1 - u\right) \cdot n0_i + \left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \cdot {normAngle}^{2}\right)} \]

5. Taylor expanded in u around 0 0.5

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

6. Simplified 0.5

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

   [Start] 0.5	\[ n1_i \cdot u + \left(\left(1 - u\right) \cdot n0_i + \left(-0.16666666666666666 \cdot \left(\left(-3 \cdot n0_i - \left(n1_i + -1 \cdot n0_i\right)\right) \cdot u\right)\right) \cdot {normAngle}^{2}\right) \]
   rational.json-simplify-43 [=>] 0.5	\[ n1_i \cdot u + \left(\left(1 - u\right) \cdot n0_i + \color{blue}{\left(\left(-3 \cdot n0_i - \left(n1_i + -1 \cdot n0_i\right)\right) \cdot \left(u \cdot -0.16666666666666666\right)\right)} \cdot {normAngle}^{2}\right) \]
   rational.json-simplify-46 [=>] 0.5	\[ n1_i \cdot u + \left(\left(1 - u\right) \cdot n0_i + \left(\color{blue}{\left(\left(-3 \cdot n0_i - n1_i\right) - -1 \cdot n0_i\right)} \cdot \left(u \cdot -0.16666666666666666\right)\right) \cdot {normAngle}^{2}\right) \]
   rational.json-simplify-42 [=>] 0.5	\[ n1_i \cdot u + \left(\left(1 - u\right) \cdot n0_i + \left(\color{blue}{\left(\left(-3 \cdot n0_i - -1 \cdot n0_i\right) - n1_i\right)} \cdot \left(u \cdot -0.16666666666666666\right)\right) \cdot {normAngle}^{2}\right) \]
   rational.json-simplify-2 [=>] 0.5	\[ n1_i \cdot u + \left(\left(1 - u\right) \cdot n0_i + \left(\left(\left(\color{blue}{n0_i \cdot -3} - -1 \cdot n0_i\right) - n1_i\right) \cdot \left(u \cdot -0.16666666666666666\right)\right) \cdot {normAngle}^{2}\right) \]
   rational.json-simplify-48 [=>] 0.5	\[ n1_i \cdot u + \left(\left(1 - u\right) \cdot n0_i + \left(\left(\color{blue}{n0_i \cdot \left(-3 - -1\right)} - n1_i\right) \cdot \left(u \cdot -0.16666666666666666\right)\right) \cdot {normAngle}^{2}\right) \]
   metadata-eval [=>] 0.5	\[ n1_i \cdot u + \left(\left(1 - u\right) \cdot n0_i + \left(\left(n0_i \cdot \color{blue}{-2} - n1_i\right) \cdot \left(u \cdot -0.16666666666666666\right)\right) \cdot {normAngle}^{2}\right) \]

7. Taylor expanded in n0_i around 0 0.5

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

8. Simplified 0.5

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

   [Start] 0.5	\[ n1_i \cdot u + \left(\left(1 - u\right) \cdot n0_i + \left(0.16666666666666666 \cdot \left(n1_i \cdot u\right)\right) \cdot {normAngle}^{2}\right) \]
   rational.json-simplify-2 [=>] 0.5	\[ n1_i \cdot u + \left(\left(1 - u\right) \cdot n0_i + \left(0.16666666666666666 \cdot \color{blue}{\left(u \cdot n1_i\right)}\right) \cdot {normAngle}^{2}\right) \]
   rational.json-simplify-43 [=>] 0.5	\[ n1_i \cdot u + \left(\left(1 - u\right) \cdot n0_i + \color{blue}{\left(u \cdot \left(n1_i \cdot 0.16666666666666666\right)\right)} \cdot {normAngle}^{2}\right) \]

9. Taylor expanded in u around -inf 0.5

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

10. Simplified 0.5

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

    [Start] 0.5	\[ -1 \cdot \left(\left(-1 \cdot n1_i + \left(n0_i + -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)\right) \cdot u\right) + n0_i \]
    rational.json-simplify-1 [=>] 0.5	\[ \color{blue}{n0_i + -1 \cdot \left(\left(-1 \cdot n1_i + \left(n0_i + -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)\right) \cdot u\right)} \]
    rational.json-simplify-2 [=>] 0.5	\[ n0_i + -1 \cdot \color{blue}{\left(u \cdot \left(-1 \cdot n1_i + \left(n0_i + -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)\right)\right)} \]
    rational.json-simplify-43 [=>] 0.5	\[ n0_i + \color{blue}{u \cdot \left(\left(-1 \cdot n1_i + \left(n0_i + -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)\right) \cdot -1\right)} \]
    rational.json-simplify-9 [=>] 0.5	\[ n0_i + u \cdot \color{blue}{\left(-\left(-1 \cdot n1_i + \left(n0_i + -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)\right)\right)} \]
    rational.json-simplify-12 [=>] 0.5	\[ n0_i + u \cdot \color{blue}{\left(0 - \left(-1 \cdot n1_i + \left(n0_i + -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)\right)\right)} \]
    rational.json-simplify-45 [<=] 0.5	\[ n0_i + u \cdot \color{blue}{\left(\left(0 - -1 \cdot n1_i\right) - \left(n0_i + -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)\right)} \]
    rational.json-simplify-2 [=>] 0.5	\[ n0_i + u \cdot \left(\left(0 - \color{blue}{n1_i \cdot -1}\right) - \left(n0_i + -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)\right) \]
    rational.json-simplify-9 [=>] 0.5	\[ n0_i + u \cdot \left(\left(0 - \color{blue}{\left(-n1_i\right)}\right) - \left(n0_i + -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)\right) \]
    rational.json-simplify-12 [=>] 0.5	\[ n0_i + u \cdot \left(\left(0 - \color{blue}{\left(0 - n1_i\right)}\right) - \left(n0_i + -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)\right) \]
    rational.json-simplify-44 [=>] 0.5	\[ n0_i + u \cdot \left(\color{blue}{\left(n1_i - \left(0 - 0\right)\right)} - \left(n0_i + -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)\right) \]
    metadata-eval [=>] 0.5	\[ n0_i + u \cdot \left(\left(n1_i - \color{blue}{0}\right) - \left(n0_i + -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)\right) \]
    rational.json-simplify-5 [=>] 0.5	\[ n0_i + u \cdot \left(\color{blue}{n1_i} - \left(n0_i + -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)\right) \]
    rational.json-simplify-46 [=>] 0.5	\[ n0_i + u \cdot \color{blue}{\left(\left(n1_i - n0_i\right) - -0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)} \]
    rational.json-simplify-43 [=>] 0.5	\[ n0_i + u \cdot \left(\left(n1_i - n0_i\right) - \color{blue}{n1_i \cdot \left({normAngle}^{2} \cdot -0.16666666666666666\right)}\right) \]

11. Final simplification 0.5

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

Alternatives

Alternative 1
Error	4.5
Cost	328

\[\begin{array}{l} t_0 := n1_i \cdot u + n0_i\\ \mathbf{if}\;n1_i \leq -4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 5.000000097707407 \cdot 10^{-26}:\\ \;\;\;\;n0_i + u \cdot \left(-n0_i\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	9.6
Cost	296

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 3
Error	4.5
Cost	296

\[\begin{array}{l} t_0 := n1_i \cdot u + n0_i\\ \mathbf{if}\;n1_i \leq -4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 5.000000097707407 \cdot 10^{-26}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.7
Cost	256

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

Alternative 5
Error	12.8
Cost	232

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -7.99999985961336 \cdot 10^{-13}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 6
Error	17.0
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2023077 
(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)))