Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.3

Time: 16.5s

Precision: binary32

Cost: 7584

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

Math

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

↓

\[\left(n0_i + \left(\left(\left(n0_i \cdot -0.16666666666666666 + n0_i \cdot 0.5\right) + n1_i \cdot 0.16666666666666666\right) \cdot \left(u \cdot {normAngle}^{2}\right) - u \cdot \left({normAngle}^{4} \cdot \left(n0_i \cdot -0.022222222222222223 + \frac{n1_i}{-51.42857142857143}\right)\right)\right)\right) + u \cdot \left(n1_i - n0_i\right) \]

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
   (-
    (*
     (+
      (+ (* n0_i -0.16666666666666666) (* n0_i 0.5))
      (* n1_i 0.16666666666666666))
     (* u (pow normAngle 2.0)))
    (*
     u
     (*
      (pow normAngle 4.0)
      (+ (* n0_i -0.022222222222222223) (/ n1_i -51.42857142857143))))))
  (* u (- n1_i n0_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 * -0.16666666666666666f) + (n0_i * 0.5f)) + (n1_i * 0.16666666666666666f)) * (u * powf(normAngle, 2.0f))) - (u * (powf(normAngle, 4.0f) * ((n0_i * -0.022222222222222223f) + (n1_i / -51.42857142857143f)))))) + (u * (n1_i - n0_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 * (-0.16666666666666666e0)) + (n0_i * 0.5e0)) + (n1_i * 0.16666666666666666e0)) * (u * (normangle ** 2.0e0))) - (u * ((normangle ** 4.0e0) * ((n0_i * (-0.022222222222222223e0)) + (n1_i / (-51.42857142857143e0))))))) + (u * (n1_i - n0_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(Float32(Float32(Float32(Float32(n0_i * Float32(-0.16666666666666666)) + Float32(n0_i * Float32(0.5))) + Float32(n1_i * Float32(0.16666666666666666))) * Float32(u * (normAngle ^ Float32(2.0)))) - Float32(u * Float32((normAngle ^ Float32(4.0)) * Float32(Float32(n0_i * Float32(-0.022222222222222223)) + Float32(n1_i / Float32(-51.42857142857143))))))) + Float32(u * Float32(n1_i - n0_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 * single(-0.16666666666666666)) + (n0_i * single(0.5))) + (n1_i * single(0.16666666666666666))) * (u * (normAngle ^ single(2.0)))) - (u * ((normAngle ^ single(4.0)) * ((n0_i * single(-0.022222222222222223)) + (n1_i / single(-51.42857142857143))))))) + (u * (n1_i - n0_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. Simplified 8.2

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(normAngle - u \cdot normAngle\right), n0_i, \sin \left(u \cdot normAngle\right) \cdot n1_i\right)}{\sin normAngle}} \]
   Proof
   (/.f32 (fma.f32 (sin.f32 (-.f32 normAngle (*.f32 u normAngle))) n0_i (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle)): 0 points increase in error, 0 points decrease in error
   (/.f32 (fma.f32 (sin.f32 (-.f32 (Rewrite<= *-lft-identity_binary32 (*.f32 1 normAngle)) (*.f32 u normAngle))) n0_i (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle)): 0 points increase in error, 0 points decrease in error
   (/.f32 (fma.f32 (sin.f32 (Rewrite=> distribute-rgt-out--_binary32 (*.f32 normAngle (-.f32 1 u)))) n0_i (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle)): 7 points increase in error, 2 points decrease in error
   (/.f32 (fma.f32 (sin.f32 (Rewrite<= *-commutative_binary32 (*.f32 (-.f32 1 u) normAngle))) n0_i (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle)): 0 points increase in error, 0 points decrease in error
   (/.f32 (Rewrite<= fma-def_binary32 (+.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i) (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i))) (sin.f32 normAngle)): 3 points increase in error, 4 points decrease in error
   (/.f32 (Rewrite<= cancel-sign-sub_binary32 (-.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i) (*.f32 (neg.f32 (sin.f32 (*.f32 u normAngle))) n1_i))) (sin.f32 normAngle)): 0 points increase in error, 0 points decrease in error
   (/.f32 (-.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i) (Rewrite=> distribute-lft-neg-out_binary32 (neg.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)))) (sin.f32 normAngle)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> div-sub_binary32 (-.f32 (/.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i) (sin.f32 normAngle)) (/.f32 (neg.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle)))): 13 points increase in error, 7 points decrease in error
   (-.f32 (Rewrite<= associate-*l/_binary32 (*.f32 (/.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (sin.f32 normAngle)) n0_i)) (/.f32 (neg.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle))): 28 points increase in error, 95 points decrease in error
   (-.f32 (*.f32 (/.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) 1)) (sin.f32 normAngle)) n0_i) (/.f32 (neg.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 (Rewrite<= associate-*r/_binary32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle)))) n0_i) (/.f32 (neg.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (sin.f32 normAngle))): 28 points increase in error, 14 points decrease in error
   (-.f32 (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i) (Rewrite<= distribute-neg-frac_binary32 (neg.f32 (/.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i) (sin.f32 normAngle))))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i) (neg.f32 (Rewrite<= associate-*l/_binary32 (*.f32 (/.f32 (sin.f32 (*.f32 u normAngle)) (sin.f32 normAngle)) n1_i)))): 5 points increase in error, 80 points decrease in error
   (-.f32 (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i) (neg.f32 (*.f32 (/.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 (sin.f32 (*.f32 u normAngle)) 1)) (sin.f32 normAngle)) n1_i))): 0 points increase in error, 0 points decrease in error
   (-.f32 (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i) (neg.f32 (*.f32 (Rewrite<= associate-*r/_binary32 (*.f32 (sin.f32 (*.f32 u normAngle)) (/.f32 1 (sin.f32 normAngle)))) n1_i))): 10 points increase in error, 6 points decrease in error
   (-.f32 (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i) (Rewrite<= distribute-lft-neg-out_binary32 (*.f32 (neg.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) (/.f32 1 (sin.f32 normAngle)))) n1_i))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> cancel-sign-sub_binary32 (+.f32 (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i) (*.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) (/.f32 1 (sin.f32 normAngle))) n1_i))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in u around 0 3.8

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

4. Simplified 3.8

   \[\leadsto \color{blue}{n0_i + u \cdot \left(\frac{n1_i \cdot normAngle}{\sin normAngle} + \frac{-\cos normAngle \cdot \left(n0_i \cdot normAngle\right)}{\sin normAngle}\right)} \]
   Proof
   (+.f32 n0_i (*.f32 u (+.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (/.f32 (neg.f32 (*.f32 (cos.f32 normAngle) (*.f32 n0_i normAngle))) (sin.f32 normAngle))))): 0 points increase in error, 0 points decrease in error
   (+.f32 n0_i (*.f32 u (+.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (/.f32 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (*.f32 (cos.f32 normAngle) (*.f32 n0_i normAngle)))) (sin.f32 normAngle))))): 0 points increase in error, 0 points decrease in error
   (+.f32 n0_i (*.f32 u (+.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (Rewrite<= associate-*r/_binary32 (*.f32 -1 (/.f32 (*.f32 (cos.f32 normAngle) (*.f32 n0_i normAngle)) (sin.f32 normAngle))))))): 0 points increase in error, 0 points decrease in error

5. Taylor expanded in normAngle around 0 0.3

   \[\leadsto \color{blue}{\left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(n0_i + \left(\left(-1 \cdot \left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) - -0.16666666666666666 \cdot n1_i\right) \cdot \left(u \cdot {normAngle}^{2}\right) + u \cdot \left({normAngle}^{4} \cdot \left(-1 \cdot \left(0.041666666666666664 \cdot n0_i - \left(0.008333333333333333 \cdot n0_i + -0.16666666666666666 \cdot \left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right)\right)\right) - \left(0.008333333333333333 \cdot n1_i + -0.027777777777777776 \cdot n1_i\right)\right)\right)\right)\right)} \]

6. Taylor expanded in n0_i around 0 0.3

   \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(n0_i + \left(\left(-1 \cdot \left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) - -0.16666666666666666 \cdot n1_i\right) \cdot \left(u \cdot {normAngle}^{2}\right) + u \cdot \left({normAngle}^{4} \cdot \left(-1 \cdot \color{blue}{\left(-0.022222222222222223 \cdot n0_i\right)} - \left(0.008333333333333333 \cdot n1_i + -0.027777777777777776 \cdot n1_i\right)\right)\right)\right)\right) \]

7. Taylor expanded in n1_i around 0 0.3

   \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(n0_i + \left(\left(-1 \cdot \left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) - -0.16666666666666666 \cdot n1_i\right) \cdot \left(u \cdot {normAngle}^{2}\right) + u \cdot \left({normAngle}^{4} \cdot \left(-1 \cdot \left(-0.022222222222222223 \cdot n0_i\right) - \color{blue}{-0.019444444444444445 \cdot n1_i}\right)\right)\right)\right) \]

8. Simplified 0.3

   \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(n0_i + \left(\left(-1 \cdot \left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) - -0.16666666666666666 \cdot n1_i\right) \cdot \left(u \cdot {normAngle}^{2}\right) + u \cdot \left({normAngle}^{4} \cdot \left(-1 \cdot \left(-0.022222222222222223 \cdot n0_i\right) - \color{blue}{\frac{n1_i}{-51.42857142857143}}\right)\right)\right)\right) \]
   Proof
   (/.f32 n1_i -360/7): 0 points increase in error, 0 points decrease in error
   (/.f32 n1_i (Rewrite<= metadata-eval (/.f32 1 -7/360))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 n1_i -7/360) 1)): 14 points increase in error, 0 points decrease in error
   (Rewrite=> /-rgt-identity_binary32 (*.f32 n1_i -7/360)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> *-commutative_binary32 (*.f32 -7/360 n1_i)): 0 points increase in error, 0 points decrease in error

9. Final simplification 0.3

   \[\leadsto \left(n0_i + \left(\left(\left(n0_i \cdot -0.16666666666666666 + n0_i \cdot 0.5\right) + n1_i \cdot 0.16666666666666666\right) \cdot \left(u \cdot {normAngle}^{2}\right) - u \cdot \left({normAngle}^{4} \cdot \left(n0_i \cdot -0.022222222222222223 + \frac{n1_i}{-51.42857142857143}\right)\right)\right)\right) + u \cdot \left(n1_i - n0_i\right) \]

Alternatives

Alternative 1
Error	0.4
Cost	3808

\[n0_i + \mathsf{fma}\left(u, n1_i - n0_i, u \cdot \left(\left(n1_i \cdot 0.16666666666666666 - n0_i \cdot -0.3333333333333333\right) \cdot \left(normAngle \cdot normAngle\right)\right)\right) \]

Alternative 2
Error	0.3
Cost	3808

\[\mathsf{fma}\left(n1_i - n0_i, u, n0_i\right) + \left(n1_i \cdot 0.16666666666666666 - n0_i \cdot -0.3333333333333333\right) \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right) \]

Alternative 3
Error	0.6
Cost	3360

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

Alternative 4
Error	9.4
Cost	296

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 5
Error	4.5
Cost	296

\[\begin{array}{l} t_0 := n0_i + n1_i \cdot u\\ \mathbf{if}\;n1_i \leq -2.0000000390829628 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 4.0000000781659255 \cdot 10^{-24}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	4.5
Cost	296

\[\begin{array}{l} t_0 := n0_i + n1_i \cdot u\\ \mathbf{if}\;n1_i \leq -2.0000000390829628 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 4.0000000781659255 \cdot 10^{-24}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	12.9
Cost	232

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 8
Error	0.7
Cost	224

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

Alternative 9
Error	17.0
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2022330 
(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)))