Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.3

Time: 25.6s

Precision: binary32

Cost: 3584

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

Math

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

↓

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

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 (* (+ (* normAngle (/ n1_i (sin 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 + (((normAngle * (n1_i / sinf(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 + (((normangle * (n1_i / sin(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(normAngle * Float32(n1_i / sin(normAngle))) + Float32(-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 + (((normAngle * (n1_i / sin(normAngle))) + -n0_i) * u);
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 4.6

   \[\leadsto \color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \left(\frac{1}{\sin normAngle} \cdot n0_i\right) + \frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]
   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-2 [=>] 0.9	\[ \color{blue}{n0_i \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   rational.json-simplify-43 [=>] 0.9	\[ \color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \left(\frac{1}{\sin normAngle} \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   rational.json-simplify-2 [=>] 0.9	\[ \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \left(\frac{1}{\sin normAngle} \cdot n0_i\right) + \color{blue}{n1_i \cdot \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} \]
   rational.json-simplify-2 [=>] 0.9	\[ \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \left(\frac{1}{\sin normAngle} \cdot n0_i\right) + n1_i \cdot \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \]
   rational.json-simplify-43 [=>] 4.6	\[ \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \left(\frac{1}{\sin normAngle} \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Taylor expanded in normAngle around 0 4.6

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

4. Taylor expanded in u around 0 3.2

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

5. Simplified 0.3

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

   [Start] 3.2	\[ \left(\frac{n1_i \cdot normAngle}{\sin normAngle} + -1 \cdot n0_i\right) \cdot u + n0_i \]
   rational.json-simplify-1 [=>] 3.2	\[ \color{blue}{n0_i + \left(\frac{n1_i \cdot normAngle}{\sin normAngle} + -1 \cdot n0_i\right) \cdot u} \]
   rational.json-simplify-49 [=>] 0.3	\[ n0_i + \left(\color{blue}{normAngle \cdot \frac{n1_i}{\sin normAngle}} + -1 \cdot n0_i\right) \cdot u \]
   rational.json-simplify-2 [=>] 0.3	\[ n0_i + \left(normAngle \cdot \frac{n1_i}{\sin normAngle} + \color{blue}{n0_i \cdot -1}\right) \cdot u \]
   rational.json-simplify-9 [=>] 0.3	\[ n0_i + \left(normAngle \cdot \frac{n1_i}{\sin normAngle} + \color{blue}{\left(-n0_i\right)}\right) \cdot u \]

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	576

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

Alternative 2
Error	4.5
Cost	328

\[\begin{array}{l} t_0 := n1_i \cdot u + n0_i\\ \mathbf{if}\;n1_i \leq -5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 2.500000068113003 \cdot 10^{-28}:\\ \;\;\;\;n0_i + u \cdot \left(-n0_i\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	9.2
Cost	296

\[\begin{array}{l} t_0 := \left(1 - u\right) \cdot n0_i\\ \mathbf{if}\;n0_i \leq -1.2999999839887283 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n0_i \leq 6.000000117248888 \cdot 10^{-25}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	4.5
Cost	296

\[\begin{array}{l} t_0 := n1_i \cdot u + n0_i\\ \mathbf{if}\;n1_i \leq -5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 2.500000068113003 \cdot 10^{-28}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	12.4
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -1.2999999839887283 \cdot 10^{-22}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 6.000000117248888 \cdot 10^{-25}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 6
Error	0.6
Cost	224

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

Alternative 7
Error	16.9
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2023068 
(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)))