Curve intersection, scale width based on ribbon orientation

Average Error: 97.3% → 99.1%

Time: 20.5s

Precision: binary32

Cost: 3552.00

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

Math

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

↓

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

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
 (- n0_i (* u (- n0_i (* (/ normAngle (sin normAngle)) n1_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 n0_i - (u * (n0_i - ((normAngle / sinf(normAngle)) * n1_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 = n0_i - (u * (n0_i - ((normangle / sin(normangle)) * n1_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(n0_i - Float32(u * Float32(n0_i - Float32(Float32(normAngle / sin(normAngle)) * n1_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 = n0_i - (u * (n0_i - ((normAngle / sin(normAngle)) * n1_i)));
end

Derivation

1. Initial program 97.3

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

2. Taylor expanded in normAngle around 0 97.2

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

3. Taylor expanded in u around 0 97.4

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

4. Taylor expanded in u around -inf 89.9

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

5. Simplified 99.1

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

   [Start] 89.9	\[ n0_i + -1 \cdot \left(u \cdot \left(n0_i + -1 \cdot \frac{n1_i \cdot normAngle}{\sin normAngle}\right)\right) \]
   mul-1-neg [=>] 89.9	\[ n0_i + \color{blue}{\left(-u \cdot \left(n0_i + -1 \cdot \frac{n1_i \cdot normAngle}{\sin normAngle}\right)\right)} \]
   unsub-neg [=>] 89.9	\[ \color{blue}{n0_i - u \cdot \left(n0_i + -1 \cdot \frac{n1_i \cdot normAngle}{\sin normAngle}\right)} \]
   mul-1-neg [=>] 89.9	\[ n0_i - u \cdot \left(n0_i + \color{blue}{\left(-\frac{n1_i \cdot normAngle}{\sin normAngle}\right)}\right) \]
   unsub-neg [=>] 89.9	\[ n0_i - u \cdot \color{blue}{\left(n0_i - \frac{n1_i \cdot normAngle}{\sin normAngle}\right)} \]
   *-commutative [=>] 89.9	\[ n0_i - u \cdot \left(n0_i - \frac{\color{blue}{normAngle \cdot n1_i}}{\sin normAngle}\right) \]
   associate-/l* [=>] 99.0	\[ n0_i - u \cdot \left(n0_i - \color{blue}{\frac{normAngle}{\frac{\sin normAngle}{n1_i}}}\right) \]
   associate-/r/ [=>] 99.1	\[ n0_i - u \cdot \left(n0_i - \color{blue}{\frac{normAngle}{\sin normAngle} \cdot n1_i}\right) \]

6. Final simplification 99.1

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

Alternatives

Alternative 1
Error	98.8%
Cost	480.00

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

Alternative 2
Error	86.4%
Cost	297.00

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.9999999774532045 \cdot 10^{-26} \lor \neg \left(n1_i \leq 1.0000000195414814 \cdot 10^{-24}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 3
Error	86.5%
Cost	297.00

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.9999999774532045 \cdot 10^{-26} \lor \neg \left(n1_i \leq 1.0000000195414814 \cdot 10^{-24}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \]

Alternative 4
Error	70.5%
Cost	296.00

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 5
Error	60.4%
Cost	232.00

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 6
Error	98.1%
Cost	224.00

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

Alternative 7
Error	47.2%
Cost	32.00

\[n0_i \]

Reproduce

herbie shell --seed 2023121 
(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)))