Curve intersection, scale width based on ribbon orientation

Average Accuracy: 97.2% → 99.2%

Time: 21.2s

Precision: binary32

Cost: 7296

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

Math

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

↓

\[\left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \left(0.16666666666666666 \cdot \left(\left(1 - u\right) - {\left(1 - u\right)}^{3}\right)\right) - u\right)\right) \cdot n0_i + \frac{u}{\frac{\sin normAngle}{normAngle}} \cdot n1_i \]

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
 (+
  (*
   (+
    1.0
    (-
     (*
      (* normAngle normAngle)
      (* 0.16666666666666666 (- (- 1.0 u) (pow (- 1.0 u) 3.0))))
     u))
   n0_i)
  (* (/ u (/ (sin normAngle) 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 ((1.0f + (((normAngle * normAngle) * (0.16666666666666666f * ((1.0f - u) - powf((1.0f - u), 3.0f)))) - u)) * n0_i) + ((u / (sinf(normAngle) / 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 = ((1.0e0 + (((normangle * normangle) * (0.16666666666666666e0 * ((1.0e0 - u) - ((1.0e0 - u) ** 3.0e0)))) - u)) * n0_i) + ((u / (sin(normangle) / 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(Float32(Float32(Float32(1.0) + Float32(Float32(Float32(normAngle * normAngle) * Float32(Float32(0.16666666666666666) * Float32(Float32(Float32(1.0) - u) - (Float32(Float32(1.0) - u) ^ Float32(3.0))))) - u)) * n0_i) + Float32(Float32(u / Float32(sin(normAngle) / 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 = ((single(1.0) + (((normAngle * normAngle) * (single(0.16666666666666666) * ((single(1.0) - u) - ((single(1.0) - u) ^ single(3.0))))) - u)) * n0_i) + ((u / (sin(normAngle) / normAngle)) * n1_i);
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. Taylor expanded in normAngle around 0 97.4%

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

3. Simplified 97.4%

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

   [Start] 97.4	\[ \left(\left(1 + \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right) \cdot {normAngle}^{2}\right) - u\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   associate--l+ [=>] 97.4	\[ \color{blue}{\left(1 + \left(\left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right) \cdot {normAngle}^{2} - u\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   *-commutative [=>] 97.4	\[ \left(1 + \left(\color{blue}{{normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right)} - u\right)\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   unpow2 [=>] 97.4	\[ \left(1 + \left(\color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right) - u\right)\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   distribute-lft-out-- [=>] 97.4	\[ \left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right)\right)} - u\right)\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

4. Taylor expanded in u around 0 97.6%

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

5. Simplified 99.2%

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

   [Start] 97.6	\[ \left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right)\right) - u\right)\right) \cdot n0_i + \frac{u \cdot normAngle}{\sin normAngle} \cdot n1_i \]
   associate-/l* [=>] 99.2	\[ \left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right)\right) - u\right)\right) \cdot n0_i + \color{blue}{\frac{u}{\frac{\sin normAngle}{normAngle}}} \cdot n1_i \]

6. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	99.1%
Cost	3872

\[\frac{u}{\frac{\sin normAngle}{normAngle}} \cdot n1_i - n0_i \cdot \left(-1 + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot -0.3333333333333333\right)\right)\right) \]

Alternative 2
Accuracy	98.9%
Cost	3616

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

Alternative 3
Accuracy	98.3%
Cost	3360

\[\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \]

Alternative 4
Accuracy	98.2%
Cost	1184

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

Alternative 5
Accuracy	71.2%
Cost	297

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -3.99999987306209 \cdot 10^{-21} \lor \neg \left(n0_i \leq 1.4999999523982838 \cdot 10^{-22}\right):\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 6
Accuracy	86.4%
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -9.999999887266023 \cdot 10^{-27} \lor \neg \left(n1_i \leq 1.0000000031710769 \cdot 10^{-29}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \end{array} \]

Alternative 7
Accuracy	86.1%
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -9.999999887266023 \cdot 10^{-27} \lor \neg \left(n1_i \leq 4.999999999099794 \cdot 10^{-24}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \end{array} \]

Alternative 8
Accuracy	98.0%
Cost	288

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

Alternative 9
Accuracy	61.5%
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -3.99999987306209 \cdot 10^{-21}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 1.4999999523982838 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 10
Accuracy	98.2%
Cost	224

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

Alternative 11
Accuracy	47.2%
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2023122 
(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)))