Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.4

Time: 1.2min

Precision: binary32

Cost: 10816

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

Math

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

↓

\[\begin{array}{l} t_0 := u \cdot n1_i + n0_i \cdot \left(1 - u\right)\\ t_0 + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3} + n0_i \cdot {\left(1 - u\right)}^{3}\right) - -0.16666666666666666 \cdot t_0\right) \end{array} \]

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
 (let* ((t_0 (+ (* u n1_i) (* n0_i (- 1.0 u)))))
   (+
    t_0
    (*
     (pow normAngle 2.0)
     (-
      (*
       -0.16666666666666666
       (+ (* n1_i (pow u 3.0)) (* n0_i (pow (- 1.0 u) 3.0))))
      (* -0.16666666666666666 t_0))))))

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) {
	float t_0 = (u * n1_i) + (n0_i * (1.0f - u));
	return t_0 + (powf(normAngle, 2.0f) * ((-0.16666666666666666f * ((n1_i * powf(u, 3.0f)) + (n0_i * powf((1.0f - u), 3.0f)))) - (-0.16666666666666666f * t_0)));
}

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
    real(4) :: t_0
    t_0 = (u * n1_i) + (n0_i * (1.0e0 - u))
    code = t_0 + ((normangle ** 2.0e0) * (((-0.16666666666666666e0) * ((n1_i * (u ** 3.0e0)) + (n0_i * ((1.0e0 - u) ** 3.0e0)))) - ((-0.16666666666666666e0) * t_0)))
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)
	t_0 = Float32(Float32(u * n1_i) + Float32(n0_i * Float32(Float32(1.0) - u)))
	return Float32(t_0 + Float32((normAngle ^ Float32(2.0)) * Float32(Float32(Float32(-0.16666666666666666) * Float32(Float32(n1_i * (u ^ Float32(3.0))) + Float32(n0_i * (Float32(Float32(1.0) - u) ^ Float32(3.0))))) - Float32(Float32(-0.16666666666666666) * t_0))))
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)
	t_0 = (u * n1_i) + (n0_i * (single(1.0) - u));
	tmp = t_0 + ((normAngle ^ single(2.0)) * ((single(-0.16666666666666666) * ((n1_i * (u ^ single(3.0))) + (n0_i * ((single(1.0) - u) ^ single(3.0))))) - (single(-0.16666666666666666) * t_0)));
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

   \[\leadsto \color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i}{\sin normAngle}} \]
   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_best-simplify-1 [=>] 0.9	\[ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{n1_i \cdot \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} \]
   rational_best-simplify-50 [=>] 4.5	\[ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]
   rational_best-simplify-1 [=>] 4.5	\[ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right) \cdot \frac{1}{\sin normAngle}} \]
   rational_best-simplify-1 [=>] 4.5	\[ \color{blue}{n0_i \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right) \cdot \frac{1}{\sin normAngle} \]
   rational_best-simplify-50 [=>] 8.3	\[ \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right) \cdot \frac{1}{\sin normAngle} \]
   rational_best-simplify-1 [=>] 8.3	\[ \color{blue}{\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) \cdot \frac{1}{\sin normAngle}} + \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right) \cdot \frac{1}{\sin normAngle} \]
   rational_best-simplify-55 [=>] 8.2	\[ \color{blue}{1 \cdot \frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\sin normAngle}} + \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right) \cdot \frac{1}{\sin normAngle} \]
   rational_best-simplify-1 [=>] 8.2	\[ \color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\sin normAngle} \cdot 1} + \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right) \cdot \frac{1}{\sin normAngle} \]
   rational_best-simplify-7 [=>] 8.2	\[ \color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\sin normAngle}} + \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right) \cdot \frac{1}{\sin normAngle} \]
   rational_best-simplify-55 [=>] 8.2	\[ \frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\sin normAngle} + \color{blue}{1 \cdot \frac{\sin \left(u \cdot normAngle\right) \cdot n1_i}{\sin normAngle}} \]
   rational_best-simplify-1 [=>] 8.2	\[ \frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\sin normAngle} + \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot n1_i}{\sin normAngle} \cdot 1} \]
   rational_best-simplify-7 [=>] 8.2	\[ \frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}{\sin normAngle} + \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot n1_i}{\sin normAngle}} \]
   rational_best-simplify-64 [=>] 8.2	\[ \color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i}{\sin normAngle}} \]

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

   \[\leadsto \color{blue}{\left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3} + n0_i \cdot {\left(1 - u\right)}^{3}\right) - -0.16666666666666666 \cdot \left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right)\right)} \]
   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_best-simplify-3 [=>] 0.4	\[ \color{blue}{\left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) + \left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \cdot {normAngle}^{2}} \]
   rational_best-simplify-1 [=>] 0.4	\[ \left(\color{blue}{u \cdot n1_i} + \left(1 - u\right) \cdot n0_i\right) + \left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \cdot {normAngle}^{2} \]
   rational_best-simplify-1 [=>] 0.4	\[ \left(u \cdot n1_i + \color{blue}{n0_i \cdot \left(1 - u\right)}\right) + \left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \cdot {normAngle}^{2} \]
   rational_best-simplify-1 [=>] 0.4	\[ \left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) + \color{blue}{{normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right)} \]
   rational_best-simplify-1 [=>] 0.4	\[ \left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) + {normAngle}^{2} \cdot \left(\left(\color{blue}{\left(n1_i \cdot {u}^{3}\right) \cdot -0.16666666666666666} + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \]
   rational_best-simplify-1 [=>] 0.4	\[ \left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) + {normAngle}^{2} \cdot \left(\left(\left(n1_i \cdot {u}^{3}\right) \cdot -0.16666666666666666 + \color{blue}{\left({\left(1 - u\right)}^{3} \cdot n0_i\right) \cdot -0.16666666666666666}\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \]
   rational_best-simplify-63 [=>] 0.4	\[ \left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) + {normAngle}^{2} \cdot \left(\color{blue}{-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3} + {\left(1 - u\right)}^{3} \cdot n0_i\right)} - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \]
   rational_best-simplify-1 [=>] 0.4	\[ \left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3} + \color{blue}{n0_i \cdot {\left(1 - u\right)}^{3}}\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \]
   rational_best-simplify-1 [=>] 0.4	\[ \left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3} + n0_i \cdot {\left(1 - u\right)}^{3}\right) - -0.16666666666666666 \cdot \left(\color{blue}{u \cdot n1_i} + \left(1 - u\right) \cdot n0_i\right)\right) \]
   rational_best-simplify-1 [=>] 0.4	\[ \left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3} + n0_i \cdot {\left(1 - u\right)}^{3}\right) - -0.16666666666666666 \cdot \left(u \cdot n1_i + \color{blue}{n0_i \cdot \left(1 - u\right)}\right)\right) \]

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	4000

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

Alternative 2
Error	0.4
Cost	3968

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

Alternative 3
Error	0.5
Cost	3776

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

Alternative 4
Error	4.3
Cost	328

\[\begin{array}{l} t_0 := u \cdot n1_i + n0_i\\ \mathbf{if}\;n1_i \leq -1.0000000031710769 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;n0_i + n0_i \cdot \left(-u\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	9.5
Cost	296

\[\begin{array}{l} t_0 := \left(1 - u\right) \cdot n0_i\\ \mathbf{if}\;n0_i \leq -1.5000000170217692 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n0_i \leq 3.2000000871846437 \cdot 10^{-26}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	4.4
Cost	296

\[\begin{array}{l} t_0 := u \cdot n1_i + n0_i\\ \mathbf{if}\;n1_i \leq -1.0000000031710769 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	12.7
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -2.000000033724767 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 3.2000000871846437 \cdot 10^{-26}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 8
Error	0.6
Cost	224

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

Alternative 9
Error	17.0
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2023099 
(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)))