Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.2

Time: 8.1s

Precision: binary32

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

Math

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

↓

\[\left(n0_i + n0_i \cdot \left(\left(u \cdot normAngle\right) \cdot \left(\left(u \cdot normAngle\right) \cdot -0.5 - \frac{\cos normAngle}{\sin normAngle}\right)\right)\right) + \left(\frac{normAngle}{\frac{\sin normAngle}{u}} + -0.16666666666666666 \cdot \frac{{\left(u \cdot normAngle\right)}^{3}}{\sin normAngle}\right) \cdot n1_i \]

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
 (+
  (+
   n0_i
   (*
    n0_i
    (*
     (* u normAngle)
     (- (* (* u normAngle) -0.5) (/ (cos normAngle) (sin normAngle))))))
  (*
   (+
    (/ normAngle (/ (sin normAngle) u))
    (* -0.16666666666666666 (/ (pow (* u normAngle) 3.0) (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 + (n0_i * ((u * normAngle) * (((u * normAngle) * -0.5f) - (cosf(normAngle) / sinf(normAngle)))))) + (((normAngle / (sinf(normAngle) / u)) + (-0.16666666666666666f * (powf((u * normAngle), 3.0f) / 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 + (n0_i * ((u * normangle) * (((u * normangle) * (-0.5e0)) - (cos(normangle) / sin(normangle)))))) + (((normangle / (sin(normangle) / u)) + ((-0.16666666666666666e0) * (((u * normangle) ** 3.0e0) / 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(Float32(n0_i + Float32(n0_i * Float32(Float32(u * normAngle) * Float32(Float32(Float32(u * normAngle) * Float32(-0.5)) - Float32(cos(normAngle) / sin(normAngle)))))) + Float32(Float32(Float32(normAngle / Float32(sin(normAngle) / u)) + Float32(Float32(-0.16666666666666666) * Float32((Float32(u * normAngle) ^ Float32(3.0)) / 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 + (n0_i * ((u * normAngle) * (((u * normAngle) * single(-0.5)) - (cos(normAngle) / sin(normAngle)))))) + (((normAngle / (sin(normAngle) / u)) + (single(-0.16666666666666666) * (((u * normAngle) ^ single(3.0)) / sin(normAngle)))) * n1_i);
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. Taylor expanded in u around 0 0.8

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

3. Simplified 0.4

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

4. Taylor expanded in u around 0 0.3

   \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{\cos normAngle \cdot \left(u \cdot normAngle\right)}{\sin normAngle} + -0.5 \cdot \left({u}^{2} \cdot {normAngle}^{2}\right)\right)\right)} \cdot n0_i + \mathsf{fma}\left(\frac{u}{\sin normAngle}, normAngle, \frac{-0.16666666666666666}{\sin normAngle} \cdot \left({u}^{3} \cdot {normAngle}^{3}\right)\right) \cdot n1_i \]

5. Simplified 0.2

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.5, \left(u \cdot normAngle\right) \cdot \left(u \cdot normAngle\right), 1\right) - \frac{\cos normAngle}{\frac{\sin normAngle}{normAngle}} \cdot u\right)} \cdot n0_i + \mathsf{fma}\left(\frac{u}{\sin normAngle}, normAngle, \frac{-0.16666666666666666}{\sin normAngle} \cdot \left({u}^{3} \cdot {normAngle}^{3}\right)\right) \cdot n1_i \]

6. Taylor expanded in u around 0 1.9

   \[\leadsto \color{blue}{\left(-0.5 \cdot \left({u}^{2} \cdot \left(n0_i \cdot {normAngle}^{2}\right)\right) + \left(n0_i + -1 \cdot \frac{\cos normAngle \cdot \left(u \cdot \left(n0_i \cdot normAngle\right)\right)}{\sin normAngle}\right)\right)} + \mathsf{fma}\left(\frac{u}{\sin normAngle}, normAngle, \frac{-0.16666666666666666}{\sin normAngle} \cdot \left({u}^{3} \cdot {normAngle}^{3}\right)\right) \cdot n1_i \]

7. Simplified 0.2

   \[\leadsto \color{blue}{\left(n0_i + n0_i \cdot \left(\left(u \cdot normAngle\right) \cdot \left(\left(u \cdot normAngle\right) \cdot -0.5 - \frac{\cos normAngle}{\sin normAngle}\right)\right)\right)} + \mathsf{fma}\left(\frac{u}{\sin normAngle}, normAngle, \frac{-0.16666666666666666}{\sin normAngle} \cdot \left({u}^{3} \cdot {normAngle}^{3}\right)\right) \cdot n1_i \]

8. Applied egg-rr 0.2

   \[\leadsto \left(n0_i + n0_i \cdot \left(\left(u \cdot normAngle\right) \cdot \left(\left(u \cdot normAngle\right) \cdot -0.5 - \frac{\cos normAngle}{\sin normAngle}\right)\right)\right) + \color{blue}{\left(\frac{normAngle}{\frac{\sin normAngle}{u}} + -0.16666666666666666 \cdot \frac{{\left(u \cdot normAngle\right)}^{3}}{\sin normAngle}\right)} \cdot n1_i \]

9. Final simplification 0.2

   \[\leadsto \left(n0_i + n0_i \cdot \left(\left(u \cdot normAngle\right) \cdot \left(\left(u \cdot normAngle\right) \cdot -0.5 - \frac{\cos normAngle}{\sin normAngle}\right)\right)\right) + \left(\frac{normAngle}{\frac{\sin normAngle}{u}} + -0.16666666666666666 \cdot \frac{{\left(u \cdot normAngle\right)}^{3}}{\sin normAngle}\right) \cdot n1_i \]

Reproduce

herbie shell --seed 2022210 
(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)))