Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.4

Time: 16.8s

Precision: binary32

Cost: 7456

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

Math

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

↓

\[\left(n0_i \cdot \left(u \cdot \left(0.5 + u \cdot -0.5\right)\right) + \left(n1_i - n0_i\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot {normAngle}^{2} + \left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) \]

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 (+ 0.5 (* u -0.5))))
    (* (- n1_i n0_i) (* -0.16666666666666666 (- (pow u 3.0) u))))
   (pow normAngle 2.0))
  (+ (* u n1_i) (* n0_i (- 1.0 u)))))

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 * (0.5f + (u * -0.5f)))) + ((n1_i - n0_i) * (-0.16666666666666666f * (powf(u, 3.0f) - u)))) * powf(normAngle, 2.0f)) + ((u * n1_i) + (n0_i * (1.0f - u)));
}

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 * (0.5e0 + (u * (-0.5e0))))) + ((n1_i - n0_i) * ((-0.16666666666666666e0) * ((u ** 3.0e0) - u)))) * (normangle ** 2.0e0)) + ((u * n1_i) + (n0_i * (1.0e0 - u)))
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(Float32(Float32(n0_i * Float32(u * Float32(Float32(0.5) + Float32(u * Float32(-0.5))))) + Float32(Float32(n1_i - n0_i) * Float32(Float32(-0.16666666666666666) * Float32((u ^ Float32(3.0)) - u)))) * (normAngle ^ Float32(2.0))) + Float32(Float32(u * n1_i) + Float32(n0_i * Float32(Float32(1.0) - u))))
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 * (single(0.5) + (u * single(-0.5))))) + ((n1_i - n0_i) * (single(-0.16666666666666666) * ((u ^ single(3.0)) - u)))) * (normAngle ^ single(2.0))) + ((u * n1_i) + (n0_i * (single(1.0) - u)));
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 8.4

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(normAngle - u \cdot normAngle\right), n0_i, \sin \left(u \cdot normAngle\right) \cdot n1_i\right)}{\sin normAngle}} \]
   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)): 2 points increase in error, 5 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, 3 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)))): 11 points increase in error, 9 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))): 30 points increase in error, 88 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))): 22 points increase in error, 11 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, 82 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))): 17 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 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. Taylor expanded in u around 0 0.4

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

5. Simplified 0.4

   \[\leadsto \color{blue}{\left(n0_i \cdot \left(u \cdot \left(0.5 + u \cdot -0.5\right)\right) + \left(n1_i - n0_i\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right)} \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]
   Proof
   (+.f32 (*.f32 n0_i (*.f32 u (+.f32 1/2 (*.f32 u -1/2)))) (*.f32 (-.f32 n1_i n0_i) (*.f32 -1/6 (-.f32 (pow.f32 u 3) u)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 n0_i (*.f32 u (+.f32 1/2 (Rewrite<= *-commutative_binary32 (*.f32 -1/2 u))))) (*.f32 (-.f32 n1_i n0_i) (*.f32 -1/6 (-.f32 (pow.f32 u 3) u)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 n0_i (Rewrite<= distribute-rgt-out_binary32 (+.f32 (*.f32 1/2 u) (*.f32 (*.f32 -1/2 u) u)))) (*.f32 (-.f32 n1_i n0_i) (*.f32 -1/6 (-.f32 (pow.f32 u 3) u)))): 8 points increase in error, 16 points decrease in error
   (+.f32 (*.f32 n0_i (+.f32 (*.f32 1/2 u) (Rewrite<= associate-*r*_binary32 (*.f32 -1/2 (*.f32 u u))))) (*.f32 (-.f32 n1_i n0_i) (*.f32 -1/6 (-.f32 (pow.f32 u 3) u)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 n0_i (+.f32 (*.f32 1/2 u) (*.f32 -1/2 (Rewrite<= unpow2_binary32 (pow.f32 u 2))))) (*.f32 (-.f32 n1_i n0_i) (*.f32 -1/6 (-.f32 (pow.f32 u 3) u)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 n0_i (Rewrite=> +-commutative_binary32 (+.f32 (*.f32 -1/2 (pow.f32 u 2)) (*.f32 1/2 u)))) (*.f32 (-.f32 n1_i n0_i) (*.f32 -1/6 (-.f32 (pow.f32 u 3) u)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (Rewrite<= distribute-rgt-out_binary32 (+.f32 (*.f32 (*.f32 -1/2 (pow.f32 u 2)) n0_i) (*.f32 (*.f32 1/2 u) n0_i))) (*.f32 (-.f32 n1_i n0_i) (*.f32 -1/6 (-.f32 (pow.f32 u 3) u)))): 15 points increase in error, 16 points decrease in error
   (+.f32 (+.f32 (Rewrite<= associate-*r*_binary32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i))) (*.f32 (*.f32 1/2 u) n0_i)) (*.f32 (-.f32 n1_i n0_i) (*.f32 -1/6 (-.f32 (pow.f32 u 3) u)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (Rewrite<= associate-*r*_binary32 (*.f32 1/2 (*.f32 u n0_i)))) (*.f32 (-.f32 n1_i n0_i) (*.f32 -1/6 (-.f32 (pow.f32 u 3) u)))): 0 points increase in error, 1 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (Rewrite<= unsub-neg_binary32 (+.f32 n1_i (neg.f32 n0_i))) (*.f32 -1/6 (-.f32 (pow.f32 u 3) u)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (+.f32 n1_i (Rewrite<= mul-1-neg_binary32 (*.f32 -1 n0_i))) (*.f32 -1/6 (-.f32 (pow.f32 u 3) u)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (Rewrite<= distribute-lft-out--_binary32 (-.f32 (*.f32 -1/6 (pow.f32 u 3)) (*.f32 -1/6 u))))): 11 points increase in error, 11 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (Rewrite=> cancel-sign-sub-inv_binary32 (+.f32 (*.f32 -1/6 (pow.f32 u 3)) (*.f32 (neg.f32 -1/6) u))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (+.f32 (*.f32 -1/6 (pow.f32 u 3)) (*.f32 (Rewrite=> metadata-eval 1/6) u)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (Rewrite=> +-commutative_binary32 (+.f32 (*.f32 1/6 u) (*.f32 -1/6 (pow.f32 u 3)))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (Rewrite<= distribute-lft-out_binary32 (+.f32 (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 1/6 u)) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3)))))): 17 points increase in error, 14 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) 1/6) u)) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 33 points increase in error, 30 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (Rewrite<= metadata-eval (neg.f32 -1/6))) u) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (Rewrite<= distribute-rgt-neg-in_binary32 (neg.f32 (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) -1/6))) u) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (Rewrite=> distribute-lft-neg-in_binary32 (*.f32 (neg.f32 (+.f32 n1_i (*.f32 -1 n0_i))) -1/6)) u) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (Rewrite=> associate-*l*_binary32 (*.f32 (neg.f32 (+.f32 n1_i (*.f32 -1 n0_i))) (*.f32 -1/6 u))) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 30 points increase in error, 33 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (Rewrite=> distribute-neg-in_binary32 (+.f32 (neg.f32 n1_i) (neg.f32 (*.f32 -1 n0_i)))) (*.f32 -1/6 u)) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 n1_i)) (neg.f32 (*.f32 -1 n0_i))) (*.f32 -1/6 u)) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1 n1_i) (neg.f32 (Rewrite=> mul-1-neg_binary32 (neg.f32 n0_i)))) (*.f32 -1/6 u)) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1 n1_i) (Rewrite=> remove-double-neg_binary32 n0_i)) (*.f32 -1/6 u)) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (+.f32 (*.f32 -1 n1_i) n0_i) -1/6) u)) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 33 points increase in error, 30 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (Rewrite<= *-commutative_binary32 (*.f32 -1/6 (+.f32 (*.f32 -1 n1_i) n0_i))) u) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (*.f32 -1/6 (Rewrite=> +-commutative_binary32 (+.f32 n0_i (*.f32 -1 n1_i)))) u) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (Rewrite=> distribute-rgt-in_binary32 (+.f32 (*.f32 n0_i -1/6) (*.f32 (*.f32 -1 n1_i) -1/6))) u) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 18 points increase in error, 14 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (Rewrite<= *-commutative_binary32 (*.f32 -1/6 n0_i)) (*.f32 (*.f32 -1 n1_i) -1/6)) u) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (Rewrite<= *-commutative_binary32 (*.f32 -1/6 (*.f32 -1 n1_i)))) u) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (Rewrite=> associate-*r*_binary32 (*.f32 (*.f32 -1/6 -1) n1_i))) u) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 (Rewrite=> metadata-eval 1/6) n1_i)) u) (*.f32 (+.f32 n1_i (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (*.f32 (+.f32 (Rewrite<= remove-double-neg_binary32 (neg.f32 (neg.f32 n1_i))) (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (*.f32 (+.f32 (neg.f32 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 n1_i))) (*.f32 -1 n0_i)) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (*.f32 (+.f32 (neg.f32 (*.f32 -1 n1_i)) (Rewrite=> mul-1-neg_binary32 (neg.f32 n0_i))) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (*.f32 (Rewrite<= distribute-neg-in_binary32 (neg.f32 (+.f32 (*.f32 -1 n1_i) n0_i))) (*.f32 -1/6 (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (Rewrite=> distribute-lft-neg-out_binary32 (neg.f32 (*.f32 (+.f32 (*.f32 -1 n1_i) n0_i) (*.f32 -1/6 (pow.f32 u 3))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (neg.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (+.f32 (*.f32 -1 n1_i) n0_i) -1/6) (pow.f32 u 3)))))): 1 points increase in error, 1 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (neg.f32 (*.f32 (Rewrite<= *-commutative_binary32 (*.f32 -1/6 (+.f32 (*.f32 -1 n1_i) n0_i))) (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (neg.f32 (*.f32 (*.f32 -1/6 (Rewrite=> +-commutative_binary32 (+.f32 n0_i (*.f32 -1 n1_i)))) (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (neg.f32 (*.f32 (Rewrite=> distribute-rgt-in_binary32 (+.f32 (*.f32 n0_i -1/6) (*.f32 (*.f32 -1 n1_i) -1/6))) (pow.f32 u 3))))): 1 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (neg.f32 (*.f32 (+.f32 (Rewrite<= *-commutative_binary32 (*.f32 -1/6 n0_i)) (*.f32 (*.f32 -1 n1_i) -1/6)) (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (neg.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (Rewrite<= *-commutative_binary32 (*.f32 -1/6 (*.f32 -1 n1_i)))) (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (neg.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (Rewrite=> associate-*r*_binary32 (*.f32 (*.f32 -1/6 -1) n1_i))) (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (+.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (neg.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 (Rewrite=> metadata-eval 1/6) n1_i)) (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (Rewrite=> unsub-neg_binary32 (-.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) u)) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3)))): 6 points increase in error, 10 points decrease in error
   (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (+.f32 (Rewrite=> *-commutative_binary32 (*.f32 n0_i -1/6)) (*.f32 1/6 n1_i)) u)) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 0 points increase in error, 0 points decrease in error
   (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (+.f32 (*.f32 n0_i -1/6) (*.f32 (Rewrite<= metadata-eval (*.f32 -1/6 -1)) n1_i)) u)) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 0 points increase in error, 0 points decrease in error
   (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (+.f32 (*.f32 n0_i -1/6) (Rewrite<= associate-*r*_binary32 (*.f32 -1/6 (*.f32 -1 n1_i)))) u)) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 0 points increase in error, 0 points decrease in error
   (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (+.f32 (*.f32 n0_i -1/6) (Rewrite=> *-commutative_binary32 (*.f32 (*.f32 -1 n1_i) -1/6))) u)) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 0 points increase in error, 0 points decrease in error
   (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (Rewrite<= distribute-rgt-in_binary32 (*.f32 -1/6 (+.f32 n0_i (*.f32 -1 n1_i)))) u)) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 14 points increase in error, 17 points decrease in error
   (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (*.f32 -1/6 (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 -1 n1_i) n0_i))) u)) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 0 points increase in error, 0 points decrease in error
   (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (Rewrite=> associate-*l*_binary32 (*.f32 -1/6 (*.f32 (+.f32 (*.f32 -1 n1_i) n0_i) u)))) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 39 points increase in error, 34 points decrease in error
   (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 -1/6 (*.f32 (+.f32 (*.f32 -1 n1_i) (Rewrite<= remove-double-neg_binary32 (neg.f32 (neg.f32 n0_i)))) u))) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 0 points increase in error, 0 points decrease in error
   (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 -1/6 (*.f32 (+.f32 (*.f32 -1 n1_i) (neg.f32 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 n0_i)))) u))) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 0 points increase in error, 0 points decrease in error
   (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 -1/6 (*.f32 (+.f32 (Rewrite=> mul-1-neg_binary32 (neg.f32 n1_i)) (neg.f32 (*.f32 -1 n0_i))) u))) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 0 points increase in error, 0 points decrease in error
   (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 -1/6 (*.f32 (Rewrite<= distribute-neg-in_binary32 (neg.f32 (+.f32 n1_i (*.f32 -1 n0_i)))) u))) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 0 points increase in error, 0 points decrease in error
   (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 -1/6 (neg.f32 (+.f32 n1_i (*.f32 -1 n0_i)))) u))) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 34 points increase in error, 39 points decrease in error
   (-.f32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 1/2 (*.f32 u n0_i))) (*.f32 (Rewrite<= distribute-rgt-neg-in_binary32 (neg.f32 (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i))))) u)) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 0 points increase in error, 0 points decrease in error
   (-.f32 (Rewrite<= associate-+r+_binary32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (+.f32 (*.f32 1/2 (*.f32 u n0_i)) (*.f32 (neg.f32 (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u)))) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 26 points increase in error, 27 points decrease in error
   (-.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (+.f32 (*.f32 1/2 (Rewrite=> *-commutative_binary32 (*.f32 n0_i u))) (*.f32 (neg.f32 (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 0 points increase in error, 0 points decrease in error
   (-.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (+.f32 (Rewrite=> associate-*r*_binary32 (*.f32 (*.f32 1/2 n0_i) u)) (*.f32 (neg.f32 (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 1 points increase in error, 0 points decrease in error
   (-.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (Rewrite<= distribute-rgt-in_binary32 (*.f32 u (+.f32 (*.f32 1/2 n0_i) (neg.f32 (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))))))) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 48 points increase in error, 31 points decrease in error
   (-.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 u (Rewrite<= sub-neg_binary32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i))))))) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 0 points increase in error, 0 points decrease in error
   (-.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (Rewrite<= *-commutative_binary32 (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))) (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unsub-neg_binary32 (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u)) (neg.f32 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u)) (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> +-commutative_binary32 (+.f32 (*.f32 -1 (*.f32 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i)) (pow.f32 u 3))) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (Rewrite=> associate-*r*_binary32 (*.f32 (*.f32 -1 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i))) (pow.f32 u 3))) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (*.f32 -1 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i))) (Rewrite=> sqr-pow_binary32 (*.f32 (pow.f32 u (/.f32 3 2)) (pow.f32 u (/.f32 3 2))))) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))): 1 points increase in error, 2 points decrease in error
   (+.f32 (Rewrite=> associate-*r*_binary32 (*.f32 (*.f32 (*.f32 -1 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i))) (pow.f32 u (/.f32 3 2))) (pow.f32 u (/.f32 3 2)))) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))): 2 points increase in error, 3 points decrease in error
   (Rewrite<= fma-udef_binary32 (fma.f32 (*.f32 (*.f32 -1 (+.f32 (*.f32 -1/6 n0_i) (*.f32 1/6 n1_i))) (pow.f32 u (/.f32 3 2))) (pow.f32 u (/.f32 3 2)) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u)))): 3 points increase in error, 3 points decrease in error
   (fma.f32 (*.f32 (Rewrite=> distribute-lft-in_binary32 (+.f32 (*.f32 -1 (*.f32 -1/6 n0_i)) (*.f32 -1 (*.f32 1/6 n1_i)))) (pow.f32 u (/.f32 3 2))) (pow.f32 u (/.f32 3 2)) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (*.f32 (+.f32 (Rewrite=> associate-*r*_binary32 (*.f32 (*.f32 -1 -1/6) n0_i)) (*.f32 -1 (*.f32 1/6 n1_i))) (pow.f32 u (/.f32 3 2))) (pow.f32 u (/.f32 3 2)) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (*.f32 (+.f32 (*.f32 (Rewrite=> metadata-eval 1/6) n0_i) (*.f32 -1 (*.f32 1/6 n1_i))) (pow.f32 u (/.f32 3 2))) (pow.f32 u (/.f32 3 2)) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (*.f32 (+.f32 (*.f32 1/6 n0_i) (Rewrite=> associate-*r*_binary32 (*.f32 (*.f32 -1 1/6) n1_i))) (pow.f32 u (/.f32 3 2))) (pow.f32 u (/.f32 3 2)) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (*.f32 (+.f32 (*.f32 1/6 n0_i) (*.f32 (Rewrite=> metadata-eval -1/6) n1_i)) (pow.f32 u (/.f32 3 2))) (pow.f32 u (/.f32 3 2)) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary32 (+.f32 (*.f32 (*.f32 (+.f32 (*.f32 1/6 n0_i) (*.f32 -1/6 n1_i)) (pow.f32 u (/.f32 3 2))) (pow.f32 u (/.f32 3 2))) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u)))): 3 points increase in error, 3 points decrease in error
   (+.f32 (Rewrite<= associate-*r*_binary32 (*.f32 (+.f32 (*.f32 1/6 n0_i) (*.f32 -1/6 n1_i)) (*.f32 (pow.f32 u (/.f32 3 2)) (pow.f32 u (/.f32 3 2))))) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))): 3 points increase in error, 2 points decrease in error
   (+.f32 (*.f32 (+.f32 (*.f32 1/6 n0_i) (*.f32 -1/6 n1_i)) (Rewrite<= sqr-pow_binary32 (pow.f32 u 3))) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))): 2 points increase in error, 1 points decrease in error
   (+.f32 (Rewrite<= *-commutative_binary32 (*.f32 (pow.f32 u 3) (+.f32 (*.f32 1/6 n0_i) (*.f32 -1/6 n1_i)))) (+.f32 (*.f32 -1/2 (*.f32 (pow.f32 u 2) n0_i)) (*.f32 (-.f32 (*.f32 1/2 n0_i) (*.f32 -1/6 (+.f32 n1_i (*.f32 -1 n0_i)))) u))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.4

   \[\leadsto \left(n0_i \cdot \left(u \cdot \left(0.5 + u \cdot -0.5\right)\right) + \left(n1_i - n0_i\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot {normAngle}^{2} + \left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) \]

Alternatives

Alternative 1
Error	0.4
Cost	4096

\[\left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.5 \cdot \left(n0_i \cdot u\right) + \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right)\right)\right) \]

Alternative 2
Error	0.5
Cost	3904

\[\left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right)\right) \]

Alternative 3
Error	0.5
Cost	3776

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

Alternative 4
Error	0.6
Cost	3360

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

Alternative 5
Error	9.3
Cost	296

\[\begin{array}{l} t_0 := n0_i - n0_i \cdot u\\ \mathbf{if}\;n0_i \leq -9.700000140248563 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n0_i \leq 2.199999944383646 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	4.4
Cost	296

\[\begin{array}{l} t_0 := n0_i + u \cdot n1_i\\ \mathbf{if}\;n1_i \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	12.6
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -9.700000140248563 \cdot 10^{-24}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 8
Error	0.7
Cost	224

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

Alternative 9
Error	16.9
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2022310 
(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)))