Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.4

Time: 14.1s

Precision: binary32

Cost: 3616

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

Math

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

↓

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

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
 (+ (* (- 1.0 u) n0_i) (* (* normAngle (/ u (sin normAngle))) n1_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 ((1.0f - u) * n0_i) + ((normAngle * (u / sinf(normAngle))) * n1_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 = ((1.0e0 - u) * n0_i) + ((normangle * (u / sin(normangle))) * n1_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(Float32(Float32(1.0) - u) * n0_i) + Float32(Float32(normAngle * Float32(u / sin(normAngle))) * n1_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 = ((single(1.0) - u) * n0_i) + ((normAngle * (u / sin(normAngle))) * n1_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. Taylor expanded in normAngle around 0 0.9

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

3. Taylor expanded in u around 0 0.9

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

4. Simplified 0.4

   \[\leadsto \left(1 - u\right) \cdot n0_i + \color{blue}{\left(\frac{u}{\sin normAngle} \cdot normAngle\right)} \cdot n1_i \]
   Proof
   (*.f32 (/.f32 u (sin.f32 normAngle)) normAngle): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r/_binary32 (/.f32 u (/.f32 (sin.f32 normAngle) normAngle))): 12 points increase in error, 17 points decrease in error
   (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 u normAngle) (sin.f32 normAngle))): 32 points increase in error, 37 points decrease in error

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.5
Cost	544

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

Alternative 2
Error	9.2
Cost	296

\[\begin{array}{l} t_0 := \left(1 - u\right) \cdot n0_i\\ \mathbf{if}\;n0_i \leq -1.2000000234497777 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n0_i \leq 1.0000000195414814 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	4.7
Cost	296

\[\begin{array}{l} t_0 := n0_i + u \cdot n1_i\\ \mathbf{if}\;n1_i \leq -4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 7.99999974612418 \cdot 10^{-21}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	4.7
Cost	296

\[\begin{array}{l} t_0 := n0_i + u \cdot n1_i\\ \mathbf{if}\;n1_i \leq -4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 7.99999974612418 \cdot 10^{-21}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	12.5
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -1.2000000234497777 \cdot 10^{-24}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 6
Error	0.6
Cost	224

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

Alternative 7
Error	17.2
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2022329 
(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)))