Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.4

Time: 17.4s

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

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

↓

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

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

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

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 = (n1_i * u) + (((1.0e0 - u) * n0_i) + ((((-0.16666666666666666e0) * (((1.0e0 - u) ** 3.0e0) * n0_i)) + ((-0.16666666666666666e0) * ((n0_i * (u + (-1.0e0))) - (n1_i * u)))) * (normangle ** 2.0e0)))
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(n1_i * u) + Float32(Float32(Float32(Float32(1.0) - u) * n0_i) + Float32(Float32(Float32(Float32(-0.16666666666666666) * Float32((Float32(Float32(1.0) - u) ^ Float32(3.0)) * n0_i)) + Float32(Float32(-0.16666666666666666) * Float32(Float32(n0_i * Float32(u + Float32(-1.0))) - Float32(n1_i * u)))) * (normAngle ^ Float32(2.0)))))
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 = (n1_i * u) + (((single(1.0) - u) * n0_i) + (((single(-0.16666666666666666) * (((single(1.0) - u) ^ single(3.0)) * n0_i)) + (single(-0.16666666666666666) * ((n0_i * (u + single(-1.0))) - (n1_i * u)))) * (normAngle ^ single(2.0))));
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 8.5

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right), n0_i, \sin \left(u \cdot normAngle\right) \cdot n1_i\right)}{\sin normAngle}} \]
   Proof
   (/.f32 (fma.f32 (sin.f32 (*.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)): 8 points increase in error, 3 points decrease in error
   (/.f32 (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i) (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i))) (sin.f32 normAngle)): 0 points increase in error, 0 points decrease in error
   (/.f32 (+.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i) (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i)) (Rewrite<= /-rgt-identity_binary32 (/.f32 (sin.f32 normAngle) 1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 (+.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i) (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i)) 1) (sin.f32 normAngle))): 0 points increase in error, 0 points decrease in error
   (/.f32 (Rewrite<= *-commutative_binary32 (*.f32 1 (+.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i) (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i)))) (sin.f32 normAngle)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l/_binary32 (*.f32 (/.f32 1 (sin.f32 normAngle)) (+.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i) (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i)))): 29 points increase in error, 23 points decrease in error
   (Rewrite<= distribute-lft-out_binary32 (+.f32 (*.f32 (/.f32 1 (sin.f32 normAngle)) (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (*.f32 (/.f32 1 (sin.f32 normAngle)) (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i)))): 10 points increase in error, 6 points decrease in error
   (+.f32 (*.f32 (/.f32 1 (sin.f32 normAngle)) (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (/.f32 1 (sin.f32 normAngle)) (sin.f32 (*.f32 (-.f32 1 u) normAngle))) n0_i))): 34 points increase in error, 93 points decrease in error
   (+.f32 (*.f32 (/.f32 1 (sin.f32 normAngle)) (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i)) (*.f32 (Rewrite<= *-commutative_binary32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle)))) n0_i)): 0 points increase in error, 0 points decrease in error
   (+.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (/.f32 1 (sin.f32 normAngle)) (sin.f32 (*.f32 u normAngle))) n1_i)) (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i)): 6 points increase in error, 75 points decrease in error
   (+.f32 (*.f32 (Rewrite<= *-commutative_binary32 (*.f32 (sin.f32 (*.f32 u normAngle)) (/.f32 1 (sin.f32 normAngle)))) n1_i) (*.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_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 8.6

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

4. Simplified 8.6

   \[\leadsto \frac{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right), n0_i, \color{blue}{u \cdot \left(n1_i \cdot normAngle\right)}\right)}{\sin normAngle} \]
   Proof
   (*.f32 u (*.f32 n1_i normAngle)): 0 points increase in error, 0 points decrease in error
   (*.f32 u (Rewrite=> *-commutative_binary32 (*.f32 normAngle n1_i))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 u normAngle) n1_i)): 14 points increase in error, 18 points decrease in error
   (Rewrite<= *-commutative_binary32 (*.f32 n1_i (*.f32 u normAngle))): 0 points increase in error, 0 points decrease in error

5. Taylor expanded in normAngle around 0 0.4

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

6. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	3968

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

Alternative 2
Error	0.4
Cost	3904

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

Alternative 3
Error	0.6
Cost	3360

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

Alternative 4
Error	9.6
Cost	296

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 1.5000000583807998 \cdot 10^{-16}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 5
Error	9.5
Cost	296

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 1.5000000583807998 \cdot 10^{-16}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 6
Error	4.6
Cost	296

\[\begin{array}{l} t_0 := n1_i \cdot u + n0_i\\ \mathbf{if}\;n1_i \leq -2.199999944383646 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 2.500000007927692 \cdot 10^{-31}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	12.6
Cost	232

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 1.5000000583807998 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 8
Error	0.6
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 2022296 
(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)))