Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.3

Time: 15.9s

Precision: binary32

Cost: 608

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

Math

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

↓

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

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)
    (*
     (* normAngle normAngle)
     (+ (* n1_i 0.16666666666666666) (* n0_i 0.3333333333333333)))))))

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) + ((normAngle * normAngle) * ((n1_i * 0.16666666666666666f) + (n0_i * 0.3333333333333333f)))));
}

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 * normangle) * ((n1_i * 0.16666666666666666e0) + (n0_i * 0.3333333333333333e0)))))
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(Float32(normAngle * normAngle) * Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.3333333333333333)))))))
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 * normAngle) * ((n1_i * single(0.16666666666666666)) + (n0_i * single(0.3333333333333333))))));
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 8.3

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(normAngle - u \cdot normAngle\right), n0_i, \sin \left(u \cdot normAngle\right) \cdot n1_i\right)}{\sin normAngle}} \]
   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)): 6 points increase in error, 1 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)): 7 points increase in error, 2 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)))): 9 points increase in error, 8 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))): 27 points increase in error, 98 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))): 29 points increase in error, 17 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)))): 4 points increase in error, 86 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))): 19 points increase in error, 4 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. Taylor expanded in normAngle around 0 0.3

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

5. Simplified 0.3

   \[\leadsto n0_i + \color{blue}{u \cdot \left(\left(n1_i - n0_i\right) + \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(n0_i, 0.3333333333333333, n1_i \cdot 0.16666666666666666\right)\right)} \]
   Proof
   (*.f32 u (+.f32 (-.f32 n1_i n0_i) (*.f32 (*.f32 normAngle normAngle) (fma.f32 n0_i 1/3 (*.f32 n1_i 1/6))))): 0 points increase in error, 0 points decrease in error
   (*.f32 u (+.f32 (Rewrite<= unsub-neg_binary32 (+.f32 n1_i (neg.f32 n0_i))) (*.f32 (*.f32 normAngle normAngle) (fma.f32 n0_i 1/3 (*.f32 n1_i 1/6))))): 0 points increase in error, 0 points decrease in error
   (*.f32 u (+.f32 (+.f32 n1_i (Rewrite<= mul-1-neg_binary32 (*.f32 -1 n0_i))) (*.f32 (*.f32 normAngle normAngle) (fma.f32 n0_i 1/3 (*.f32 n1_i 1/6))))): 0 points increase in error, 0 points decrease in error
   (*.f32 u (+.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 (Rewrite<= unpow2_binary32 (pow.f32 normAngle 2)) (fma.f32 n0_i 1/3 (*.f32 n1_i 1/6))))): 0 points increase in error, 0 points decrease in error
   (*.f32 u (+.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 (pow.f32 normAngle 2) (fma.f32 n0_i (Rewrite<= metadata-eval (neg.f32 -1/3)) (*.f32 n1_i 1/6))))): 0 points increase in error, 0 points decrease in error
   (*.f32 u (+.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 (pow.f32 normAngle 2) (fma.f32 n0_i (neg.f32 (Rewrite<= metadata-eval (-.f32 -1/2 -1/6))) (*.f32 n1_i 1/6))))): 2 points increase in error, 0 points decrease in error
   (*.f32 u (+.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 (pow.f32 normAngle 2) (fma.f32 n0_i (neg.f32 (-.f32 -1/2 -1/6)) (*.f32 n1_i (Rewrite<= metadata-eval (neg.f32 -1/6))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 u (+.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 (pow.f32 normAngle 2) (fma.f32 n0_i (neg.f32 (-.f32 -1/2 -1/6)) (Rewrite<= distribute-rgt-neg-in_binary32 (neg.f32 (*.f32 n1_i -1/6))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 u (+.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 (pow.f32 normAngle 2) (fma.f32 n0_i (neg.f32 (-.f32 -1/2 -1/6)) (neg.f32 (Rewrite<= *-commutative_binary32 (*.f32 -1/6 n1_i))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 u (+.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 (pow.f32 normAngle 2) (Rewrite<= fma-neg_binary32 (-.f32 (*.f32 n0_i (neg.f32 (-.f32 -1/2 -1/6))) (*.f32 -1/6 n1_i)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 u (+.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 (pow.f32 normAngle 2) (-.f32 (Rewrite<= distribute-rgt-neg-in_binary32 (neg.f32 (*.f32 n0_i (-.f32 -1/2 -1/6)))) (*.f32 -1/6 n1_i))))): 0 points increase in error, 0 points decrease in error
   (*.f32 u (+.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 (pow.f32 normAngle 2) (-.f32 (neg.f32 (Rewrite<= distribute-rgt-out--_binary32 (-.f32 (*.f32 -1/2 n0_i) (*.f32 -1/6 n0_i)))) (*.f32 -1/6 n1_i))))): 0 points increase in error, 2 points decrease in error
   (*.f32 u (+.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 (pow.f32 normAngle 2) (-.f32 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (-.f32 (*.f32 -1/2 n0_i) (*.f32 -1/6 n0_i)))) (*.f32 -1/6 n1_i))))): 0 points increase in error, 0 points decrease in error
   (*.f32 u (+.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (Rewrite<= *-commutative_binary32 (*.f32 (-.f32 (*.f32 -1 (-.f32 (*.f32 -1/2 n0_i) (*.f32 -1/6 n0_i))) (*.f32 -1/6 n1_i)) (pow.f32 normAngle 2))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-out_binary32 (+.f32 (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) u) (*.f32 (*.f32 (-.f32 (*.f32 -1 (-.f32 (*.f32 -1/2 n0_i) (*.f32 -1/6 n0_i))) (*.f32 -1/6 n1_i)) (pow.f32 normAngle 2)) u))): 1 points increase in error, 2 points decrease in error
   (+.f32 (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) u) (Rewrite<= associate-*r*_binary32 (*.f32 (-.f32 (*.f32 -1 (-.f32 (*.f32 -1/2 n0_i) (*.f32 -1/6 n0_i))) (*.f32 -1/6 n1_i)) (*.f32 (pow.f32 normAngle 2) u)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) u) (*.f32 (-.f32 (*.f32 -1 (-.f32 (*.f32 -1/2 n0_i) (*.f32 -1/6 n0_i))) (*.f32 -1/6 n1_i)) (Rewrite<= *-commutative_binary32 (*.f32 u (pow.f32 normAngle 2))))): 0 points increase in error, 0 points decrease in error

6. Applied egg-rr 0.3

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

7. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	544

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

Alternative 2
Error	0.4
Cost	480

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

Alternative 3
Error	10.1
Cost	296

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 4
Error	13.5
Cost	232

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 5
Error	0.6
Cost	224

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

Alternative 6
Error	5.9
Cost	160

\[n0_i + u \cdot n1_i \]

Alternative 7
Error	17.0
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2022306 
(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)))