Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.4

Time: 17.4s

Precision: binary32

Cost: 4480

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

Math

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

↓

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

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 (* n0_i (- 1.0 u))))
   (+
    (* n1_i u)
    (-
     t_0
     (*
      (pow normAngle 2.0)
      (+
       (* -0.16666666666666666 (+ (* n1_i u) t_0))
       (*
        -0.16666666666666666
        (* n0_i (+ -1.0 (* u (- 3.0 (* u (- 3.0 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) {
	float t_0 = n0_i * (1.0f - u);
	return (n1_i * u) + (t_0 - (powf(normAngle, 2.0f) * ((-0.16666666666666666f * ((n1_i * u) + t_0)) + (-0.16666666666666666f * (n0_i * (-1.0f + (u * (3.0f - (u * (3.0f - 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
    real(4) :: t_0
    t_0 = n0_i * (1.0e0 - u)
    code = (n1_i * u) + (t_0 - ((normangle ** 2.0e0) * (((-0.16666666666666666e0) * ((n1_i * u) + t_0)) + ((-0.16666666666666666e0) * (n0_i * ((-1.0e0) + (u * (3.0e0 - (u * (3.0e0 - 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)
	t_0 = Float32(n0_i * Float32(Float32(1.0) - u))
	return Float32(Float32(n1_i * u) + Float32(t_0 - Float32((normAngle ^ Float32(2.0)) * Float32(Float32(Float32(-0.16666666666666666) * Float32(Float32(n1_i * u) + t_0)) + Float32(Float32(-0.16666666666666666) * Float32(n0_i * Float32(Float32(-1.0) + Float32(u * Float32(Float32(3.0) - Float32(u * Float32(Float32(3.0) - 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)
	t_0 = n0_i * (single(1.0) - u);
	tmp = (n1_i * u) + (t_0 - ((normAngle ^ single(2.0)) * ((single(-0.16666666666666666) * ((n1_i * u) + t_0)) + (single(-0.16666666666666666) * (n0_i * (single(-1.0) + (u * (single(3.0) - (u * (single(3.0) - 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 8.1

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(normAngle - u \cdot normAngle\right), n0_i, \sin \left(u \cdot normAngle\right) \cdot n1_i\right)}{\sin normAngle}} \]
   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, 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)): 5 points increase in error, 1 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)))): 8 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))): 24 points increase in error, 86 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))): 34 points increase in error, 9 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)))): 3 points increase in error, 72 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))): 16 points increase in error, 3 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 8.2

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

4. Taylor expanded in normAngle around 0 0.4

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

5. Taylor expanded in u around 0 0.4

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

6. Simplified 0.4

   \[\leadsto n1_i \cdot u + \left(\left(-0.16666666666666666 \cdot \left(\color{blue}{\left(1 + u \cdot \left(-3 + u \cdot \left(3 - u\right)\right)\right)} \cdot n0_i\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \cdot {normAngle}^{2} + \left(1 - u\right) \cdot n0_i\right) \]
   Proof
   (+.f32 1 (*.f32 u (+.f32 -3 (*.f32 u (-.f32 3 u))))): 0 points increase in error, 0 points decrease in error
   (+.f32 1 (*.f32 u (+.f32 -3 (Rewrite<= distribute-lft-out--_binary32 (-.f32 (*.f32 u 3) (*.f32 u u)))))): 3 points increase in error, 1 points decrease in error
   (+.f32 1 (*.f32 u (+.f32 -3 (-.f32 (*.f32 u 3) (Rewrite<= unpow2_binary32 (pow.f32 u 2)))))): 0 points increase in error, 0 points decrease in error
   (+.f32 1 (*.f32 u (+.f32 -3 (Rewrite<= unsub-neg_binary32 (+.f32 (*.f32 u 3) (neg.f32 (pow.f32 u 2))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 1 (Rewrite<= distribute-lft-out_binary32 (+.f32 (*.f32 u -3) (*.f32 u (+.f32 (*.f32 u 3) (neg.f32 (pow.f32 u 2))))))): 6 points increase in error, 5 points decrease in error
   (+.f32 1 (+.f32 (Rewrite<= *-commutative_binary32 (*.f32 -3 u)) (*.f32 u (+.f32 (*.f32 u 3) (neg.f32 (pow.f32 u 2)))))): 0 points increase in error, 0 points decrease in error
   (+.f32 1 (+.f32 (*.f32 -3 u) (Rewrite<= distribute-lft-out_binary32 (+.f32 (*.f32 u (*.f32 u 3)) (*.f32 u (neg.f32 (pow.f32 u 2))))))): 6 points increase in error, 3 points decrease in error
   (+.f32 1 (+.f32 (*.f32 -3 u) (+.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 u u) 3)) (*.f32 u (neg.f32 (pow.f32 u 2)))))): 2 points increase in error, 5 points decrease in error
   (+.f32 1 (+.f32 (*.f32 -3 u) (+.f32 (*.f32 (Rewrite<= unpow2_binary32 (pow.f32 u 2)) 3) (*.f32 u (neg.f32 (pow.f32 u 2)))))): 0 points increase in error, 0 points decrease in error
   (+.f32 1 (+.f32 (*.f32 -3 u) (+.f32 (Rewrite<= *-commutative_binary32 (*.f32 3 (pow.f32 u 2))) (*.f32 u (neg.f32 (pow.f32 u 2)))))): 0 points increase in error, 0 points decrease in error
   (+.f32 1 (+.f32 (*.f32 -3 u) (+.f32 (*.f32 3 (pow.f32 u 2)) (Rewrite<= distribute-rgt-neg-in_binary32 (neg.f32 (*.f32 u (pow.f32 u 2))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 1 (+.f32 (*.f32 -3 u) (+.f32 (*.f32 3 (pow.f32 u 2)) (neg.f32 (*.f32 u (Rewrite=> unpow2_binary32 (*.f32 u u))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 1 (+.f32 (*.f32 -3 u) (+.f32 (*.f32 3 (pow.f32 u 2)) (neg.f32 (Rewrite<= cube-mult_binary32 (pow.f32 u 3)))))): 0 points increase in error, 2 points decrease in error
   (+.f32 1 (+.f32 (*.f32 -3 u) (+.f32 (*.f32 3 (pow.f32 u 2)) (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (pow.f32 u 3)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-+r+_binary32 (+.f32 (+.f32 1 (*.f32 -3 u)) (+.f32 (*.f32 3 (pow.f32 u 2)) (*.f32 -1 (pow.f32 u 3))))): 21 points increase in error, 15 points decrease in error
   (+.f32 (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 -3 u) 1)) (+.f32 (*.f32 3 (pow.f32 u 2)) (*.f32 -1 (pow.f32 u 3)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+r+_binary32 (+.f32 (*.f32 -3 u) (+.f32 1 (+.f32 (*.f32 3 (pow.f32 u 2)) (*.f32 -1 (pow.f32 u 3)))))): 23 points increase in error, 35 points decrease in error

7. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	4352

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

Alternative 2
Error	0.4
Cost	4096

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

Alternative 3
Error	0.4
Cost	3904

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

Alternative 4
Error	0.5
Cost	3776

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

Alternative 5
Error	0.6
Cost	3360

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

Alternative 6
Error	9.4
Cost	296

\[\begin{array}{l} t_0 := n0_i - u \cdot n0_i\\ \mathbf{if}\;n0_i \leq -2.0000000063421537 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n0_i \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	4.4
Cost	296

\[\begin{array}{l} t_0 := n1_i \cdot u + n0_i\\ \mathbf{if}\;n1_i \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	12.8
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -2.0000000063421537 \cdot 10^{-28}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 9
Error	0.7
Cost	224

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

Alternative 10
Error	16.8
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2022295 
(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)))