Curve intersection, scale width based on ribbon orientation

Average Error: 0.8 → 0.3

Time: 19.1s

Precision: binary32

Cost: 3776

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

Math

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

↓

\[n0_i + u \cdot \left(\left(n1_i - n0_i\right) + {normAngle}^{2} \cdot \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right)\right) \]

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)
    (*
     (pow normAngle 2.0)
     (+ (* n0_i 0.3333333333333333) (* n1_i 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) + (powf(normAngle, 2.0f) * ((n0_i * 0.3333333333333333f) + (n1_i * 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) + ((normangle ** 2.0e0) * ((n0_i * 0.3333333333333333e0) + (n1_i * 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((normAngle ^ Float32(2.0)) * Float32(Float32(n0_i * Float32(0.3333333333333333)) + Float32(n1_i * 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) + ((normAngle ^ single(2.0)) * ((n0_i * single(0.3333333333333333)) + (n1_i * single(0.16666666666666666))))));
end

Derivation

1. Initial program 0.8

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

2. Simplified 8.2

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

   [Start] 0.8	\[ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   rational.json-simplify-50 [=>] 0.8	\[ \color{blue}{n0_i \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   rational.json-simplify-27 [<=] 0.8	\[ n0_i \cdot \color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\frac{\sin normAngle}{1}}} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   rational.json-simplify-26 [=>] 0.8	\[ n0_i \cdot \frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\color{blue}{\sin normAngle}} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   rational.json-simplify-48 [=>] 6.0	\[ \color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\sin normAngle}} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   rational.json-simplify-50 [=>] 6.0	\[ \frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\sin normAngle} + \color{blue}{n1_i \cdot \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} \]
   rational.json-simplify-24 [=>] 6.0	\[ \frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\sin normAngle} + \color{blue}{\sin \left(u \cdot normAngle\right) \cdot \left(n1_i \cdot \frac{1}{\sin normAngle}\right)} \]
   rational.json-simplify-48 [=>] 6.0	\[ \frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\sin normAngle} + \sin \left(u \cdot normAngle\right) \cdot \color{blue}{\frac{1 \cdot n1_i}{\sin normAngle}} \]
   rational.json-simplify-48 [=>] 8.2	\[ \frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\sin normAngle} + \color{blue}{\frac{\left(1 \cdot n1_i\right) \cdot \sin \left(u \cdot normAngle\right)}{\sin normAngle}} \]
   rational.json-simplify-45 [=>] 8.2	\[ \color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \left(1 \cdot n1_i\right) \cdot \sin \left(u \cdot normAngle\right)}{\sin normAngle}} \]

3. Taylor expanded in u around 0 3.9

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

4. Simplified 3.9

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

   [Start] 3.9	\[ n0_i + u \cdot \left(\frac{n1_i \cdot normAngle}{\sin normAngle} + -1 \cdot \frac{\cos normAngle \cdot \left(n0_i \cdot normAngle\right)}{\sin normAngle}\right) \]
   rational.json-simplify-48 [=>] 3.9	\[ n0_i + u \cdot \left(\frac{n1_i \cdot normAngle}{\sin normAngle} + \color{blue}{\frac{\left(\cos normAngle \cdot \left(n0_i \cdot normAngle\right)\right) \cdot -1}{\sin normAngle}}\right) \]
   rational.json-simplify-50 [<=] 3.9	\[ n0_i + u \cdot \left(\frac{n1_i \cdot normAngle}{\sin normAngle} + \frac{\color{blue}{-1 \cdot \left(\cos normAngle \cdot \left(n0_i \cdot normAngle\right)\right)}}{\sin normAngle}\right) \]
   rational.json-simplify-45 [=>] 3.9	\[ n0_i + u \cdot \color{blue}{\frac{n1_i \cdot normAngle + -1 \cdot \left(\cos normAngle \cdot \left(n0_i \cdot normAngle\right)\right)}{\sin normAngle}} \]
   rational.json-simplify-24 [=>] 3.9	\[ n0_i + u \cdot \frac{n1_i \cdot normAngle + \color{blue}{\cos normAngle \cdot \left(-1 \cdot \left(n0_i \cdot normAngle\right)\right)}}{\sin normAngle} \]
   rational.json-simplify-24 [=>] 3.9	\[ n0_i + u \cdot \frac{n1_i \cdot normAngle + \cos normAngle \cdot \color{blue}{\left(n0_i \cdot \left(-1 \cdot normAngle\right)\right)}}{\sin normAngle} \]
   rational.json-simplify-50 [=>] 3.9	\[ n0_i + u \cdot \frac{n1_i \cdot normAngle + \cos normAngle \cdot \left(n0_i \cdot \color{blue}{\left(normAngle \cdot -1\right)}\right)}{\sin normAngle} \]
   rational.json-simplify-38 [=>] 3.9	\[ n0_i + u \cdot \frac{n1_i \cdot normAngle + \cos normAngle \cdot \left(n0_i \cdot \color{blue}{\left(-normAngle\right)}\right)}{\sin normAngle} \]

5. Taylor expanded in normAngle around 0 0.3

   \[\leadsto n0_i + u \cdot \color{blue}{\left(\left(0.5 \cdot n0_i - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right) \cdot {normAngle}^{2} + \left(n1_i + -1 \cdot n0_i\right)\right)} \]

6. Simplified 0.3

   \[\leadsto n0_i + u \cdot \color{blue}{\left(\left(n1_i + \left(-n0_i\right)\right) + \left(n0_i \cdot 0.5 - \left(n1_i + \left(-n0_i\right)\right) \cdot -0.16666666666666666\right) \cdot {normAngle}^{2}\right)} \]
   Proof

   [Start] 0.3	\[ n0_i + u \cdot \left(\left(0.5 \cdot n0_i - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right) \cdot {normAngle}^{2} + \left(n1_i + -1 \cdot n0_i\right)\right) \]
   rational.json-simplify-1 [=>] 0.3	\[ n0_i + u \cdot \color{blue}{\left(\left(n1_i + -1 \cdot n0_i\right) + \left(0.5 \cdot n0_i - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right) \cdot {normAngle}^{2}\right)} \]
   rational.json-simplify-50 [=>] 0.3	\[ n0_i + u \cdot \left(\left(n1_i + \color{blue}{n0_i \cdot -1}\right) + \left(0.5 \cdot n0_i - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right) \cdot {normAngle}^{2}\right) \]
   rational.json-simplify-31 [<=] 0.3	\[ n0_i + u \cdot \left(\left(n1_i + \color{blue}{\left(-n0_i\right)}\right) + \left(0.5 \cdot n0_i - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right) \cdot {normAngle}^{2}\right) \]
   rational.json-simplify-50 [=>] 0.3	\[ n0_i + u \cdot \left(\left(n1_i + \left(-n0_i\right)\right) + \left(\color{blue}{n0_i \cdot 0.5} - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right) \cdot {normAngle}^{2}\right) \]
   rational.json-simplify-50 [=>] 0.3	\[ n0_i + u \cdot \left(\left(n1_i + \left(-n0_i\right)\right) + \left(n0_i \cdot 0.5 - \color{blue}{\left(n1_i + -1 \cdot n0_i\right) \cdot -0.16666666666666666}\right) \cdot {normAngle}^{2}\right) \]
   rational.json-simplify-50 [=>] 0.3	\[ n0_i + u \cdot \left(\left(n1_i + \left(-n0_i\right)\right) + \left(n0_i \cdot 0.5 - \left(n1_i + \color{blue}{n0_i \cdot -1}\right) \cdot -0.16666666666666666\right) \cdot {normAngle}^{2}\right) \]
   rational.json-simplify-31 [<=] 0.3	\[ n0_i + u \cdot \left(\left(n1_i + \left(-n0_i\right)\right) + \left(n0_i \cdot 0.5 - \left(n1_i + \color{blue}{\left(-n0_i\right)}\right) \cdot -0.16666666666666666\right) \cdot {normAngle}^{2}\right) \]

7. Applied egg-rr 0.3

   \[\leadsto n0_i + \color{blue}{\left(0 - u \cdot \left({normAngle}^{2} \cdot \left(n1_i \cdot -0.16666666666666666 + n0_i \cdot -0.3333333333333333\right) + \left(n0_i - n1_i\right)\right)\right)} \]

8. Simplified 0.3

   \[\leadsto n0_i + \color{blue}{\left(\left(n0_i - n1_i\right) + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot n1_i + n0_i \cdot -0.3333333333333333\right)\right) \cdot \left(-u\right)} \]
   Proof

   [Start] 0.3	\[ n0_i + \left(0 - u \cdot \left({normAngle}^{2} \cdot \left(n1_i \cdot -0.16666666666666666 + n0_i \cdot -0.3333333333333333\right) + \left(n0_i - n1_i\right)\right)\right) \]
   rational.json-simplify-7 [=>] 0.3	\[ n0_i + \color{blue}{\left(-u \cdot \left({normAngle}^{2} \cdot \left(n1_i \cdot -0.16666666666666666 + n0_i \cdot -0.3333333333333333\right) + \left(n0_i - n1_i\right)\right)\right)} \]
   rational.json-simplify-17 [=>] 0.3	\[ n0_i + \color{blue}{\frac{u \cdot \left({normAngle}^{2} \cdot \left(n1_i \cdot -0.16666666666666666 + n0_i \cdot -0.3333333333333333\right) + \left(n0_i - n1_i\right)\right)}{-1}} \]
   rational.json-simplify-43 [=>] 0.3	\[ n0_i + \color{blue}{\left({normAngle}^{2} \cdot \left(n1_i \cdot -0.16666666666666666 + n0_i \cdot -0.3333333333333333\right) + \left(n0_i - n1_i\right)\right) \cdot \frac{u}{-1}} \]
   rational.json-simplify-1 [=>] 0.3	\[ n0_i + \color{blue}{\left(\left(n0_i - n1_i\right) + {normAngle}^{2} \cdot \left(n1_i \cdot -0.16666666666666666 + n0_i \cdot -0.3333333333333333\right)\right)} \cdot \frac{u}{-1} \]
   rational.json-simplify-50 [<=] 0.3	\[ n0_i + \left(\left(n0_i - n1_i\right) + {normAngle}^{2} \cdot \left(\color{blue}{-0.16666666666666666 \cdot n1_i} + n0_i \cdot -0.3333333333333333\right)\right) \cdot \frac{u}{-1} \]
   rational.json-simplify-17 [<=] 0.3	\[ n0_i + \left(\left(n0_i - n1_i\right) + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot n1_i + n0_i \cdot -0.3333333333333333\right)\right) \cdot \color{blue}{\left(-u\right)} \]

9. Applied egg-rr 0.3

   \[\leadsto n0_i + \color{blue}{\left(u \cdot \left(\left(n1_i - n0_i\right) + {normAngle}^{2} \cdot \left(n1_i \cdot 0.16666666666666666 + n0_i \cdot 0.3333333333333333\right)\right) + 0\right)} \]

10. Simplified 0.3

    \[\leadsto n0_i + \color{blue}{u \cdot \left(\left(n1_i - n0_i\right) + {normAngle}^{2} \cdot \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right)\right)} \]
    Proof

    [Start] 0.3	\[ n0_i + \left(u \cdot \left(\left(n1_i - n0_i\right) + {normAngle}^{2} \cdot \left(n1_i \cdot 0.16666666666666666 + n0_i \cdot 0.3333333333333333\right)\right) + 0\right) \]
    rational.json-simplify-8 [=>] 0.3	\[ n0_i + \color{blue}{u \cdot \left(\left(n1_i - n0_i\right) + {normAngle}^{2} \cdot \left(n1_i \cdot 0.16666666666666666 + n0_i \cdot 0.3333333333333333\right)\right)} \]
    rational.json-simplify-1 [=>] 0.3	\[ n0_i + u \cdot \left(\left(n1_i - n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right)}\right) \]

11. Final simplification 0.3

    \[\leadsto n0_i + u \cdot \left(\left(n1_i - n0_i\right) + {normAngle}^{2} \cdot \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right)\right) \]

Alternatives

Alternative 1
Error	0.5
Cost	3680

\[n0_i + u \cdot \left(\left(n1_i + \left(-n0_i\right)\right) + \left(n0_i \cdot 0.3333333333333333\right) \cdot {normAngle}^{2}\right) \]

Alternative 2
Error	9.0
Cost	296

\[\begin{array}{l} t_0 := \left(1 - u\right) \cdot n0_i\\ \mathbf{if}\;n0_i \leq -2.0000000544904023 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n0_i \leq 2.3500000705743845 \cdot 10^{-23}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	4.3
Cost	296

\[\begin{array}{l} t_0 := u \cdot n1_i + n0_i\\ \mathbf{if}\;n1_i \leq -3.000000058624444 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 2.0000000390829628 \cdot 10^{-25}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	4.2
Cost	296

\[\begin{array}{l} t_0 := u \cdot n1_i + n0_i\\ \mathbf{if}\;n1_i \leq -3.000000058624444 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 2.0000000390829628 \cdot 10^{-25}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	12.4
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -4.0000001089808046 \cdot 10^{-27}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 5.999999809593135 \cdot 10^{-21}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 6
Error	0.5
Cost	224

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

Alternative 7
Error	16.7
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2023073 
(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)))