Curve intersection, scale width based on ribbon orientation

Average Error: 2.83% → 0.9%

Time: 20.3s

Precision: binary32

Cost: 3552

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

Math

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

↓

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

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 (* (- (* n1_i (/ normAngle (sin normAngle))) n0_i) 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 + (((n1_i * (normAngle / sinf(normAngle))) - n0_i) * 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 + (((n1_i * (normangle / sin(normangle))) - n0_i) * 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(n0_i + Float32(Float32(Float32(n1_i * Float32(normAngle / sin(normAngle))) - n0_i) * 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 + (((n1_i * (normAngle / sin(normAngle))) - n0_i) * u);
end

Derivation

1. Initial program 2.83

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

2. Taylor expanded in normAngle around 0 2.85

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

3. Taylor expanded in u around 0 10.09

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

4. Applied egg-rr 0.9

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

5. Final simplification 0.9

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

Alternatives

Alternative 1
Error	1.26%
Cost	480

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

Alternative 2
Error	14.39%
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.0000000063421537 \cdot 10^{-29} \lor \neg \left(n1_i \leq 2.0000000390829628 \cdot 10^{-24}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 3
Error	15.56%
Cost	297

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -2.00000009162741 \cdot 10^{-18} \lor \neg \left(n0_i \leq 1.999999936531045 \cdot 10^{-19}\right):\\ \;\;\;\;n0_i - n0_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i + n1_i \cdot u\\ \end{array} \]

Alternative 4
Error	29.9%
Cost	296

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.999999970665357 \cdot 10^{-10}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 6.800000016196628 \cdot 10^{-13}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 5
Error	39.87%
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 1.0000000195414814 \cdot 10^{-24}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 6
Error	1.95%
Cost	224

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

Alternative 7
Error	52.87%
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2023115 
(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)))