Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.4

Time: 14.5s

Precision: binary32

Cost: 480

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

Math

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

↓

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

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
   (+
    n1_i
    (- (* (* n1_i 0.16666666666666666) (* normAngle normAngle)) n0_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 n0_i + (u * (n1_i + (((n1_i * 0.16666666666666666f) * (normAngle * normAngle)) - n0_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 = n0_i + (u * (n1_i + (((n1_i * 0.16666666666666666e0) * (normangle * normangle)) - n0_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(n0_i + Float32(u * Float32(n1_i + Float32(Float32(Float32(n1_i * Float32(0.16666666666666666)) * Float32(normAngle * normAngle)) - n0_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 = n0_i + (u * (n1_i + (((n1_i * single(0.16666666666666666)) * (normAngle * normAngle)) - n0_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. Simplified 8.2

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)}{\sin normAngle}} \]
   Proof
   (/.f32 (fma.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 (Rewrite<= fma-def_binary32 (+.f32 (*.f32 (sin.f32 (*.f32 u normAngle)) n1_i) (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) n0_i))) (sin.f32 normAngle)): 1 points increase in error, 1 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)))): 39 points increase in error, 15 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)))): 8 points increase in error, 9 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))): 33 points increase in error, 85 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)): 8 points increase in error, 86 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 3.8

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

4. Simplified 3.8

   \[\leadsto \color{blue}{n0_i + u \cdot \left(\frac{n1_i \cdot normAngle}{\sin normAngle} + \frac{-\left(\cos normAngle \cdot n0_i\right) \cdot normAngle}{\sin normAngle}\right)} \]
   Proof
   (+.f32 n0_i (*.f32 u (+.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (/.f32 (neg.f32 (*.f32 (*.f32 (cos.f32 normAngle) n0_i) normAngle)) (sin.f32 normAngle))))): 0 points increase in error, 0 points decrease in error
   (+.f32 n0_i (*.f32 u (+.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (/.f32 (neg.f32 (Rewrite<= associate-*r*_binary32 (*.f32 (cos.f32 normAngle) (*.f32 n0_i normAngle)))) (sin.f32 normAngle))))): 0 points increase in error, 0 points decrease in error
   (+.f32 n0_i (*.f32 u (+.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (Rewrite<= distribute-neg-frac_binary32 (neg.f32 (/.f32 (*.f32 (cos.f32 normAngle) (*.f32 n0_i normAngle)) (sin.f32 normAngle))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 n0_i (*.f32 u (+.f32 (/.f32 (*.f32 n1_i normAngle) (sin.f32 normAngle)) (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (/.f32 (*.f32 (cos.f32 normAngle) (*.f32 n0_i normAngle)) (sin.f32 normAngle))))))): 0 points increase in error, 0 points decrease in error

5. Taylor expanded in normAngle around 0 3.1

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

6. Simplified 3.1

   \[\leadsto n0_i + u \cdot \left(\frac{n1_i \cdot normAngle}{\sin normAngle} + \color{blue}{\left(-n0_i\right)}\right) \]
   Proof
   (neg.f32 n0_i): 0 points increase in error, 0 points decrease in error
   (Rewrite=> neg-mul-1_binary32 (*.f32 -1 n0_i)): 0 points increase in error, 0 points decrease in error

7. Taylor expanded in normAngle around 0 0.4

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

8. Simplified 0.4

   \[\leadsto n0_i + u \cdot \color{blue}{\left(n1_i + \left(\left(0.16666666666666666 \cdot n1_i\right) \cdot \left(normAngle \cdot normAngle\right) - n0_i\right)\right)} \]
   Proof
   (+.f32 n1_i (-.f32 (*.f32 (*.f32 1/6 n1_i) (*.f32 normAngle normAngle)) n0_i)): 0 points increase in error, 0 points decrease in error
   (+.f32 n1_i (-.f32 (*.f32 (*.f32 1/6 n1_i) (Rewrite<= unpow2_binary32 (pow.f32 normAngle 2))) n0_i)): 0 points increase in error, 0 points decrease in error
   (+.f32 n1_i (-.f32 (Rewrite<= associate-*r*_binary32 (*.f32 1/6 (*.f32 n1_i (pow.f32 normAngle 2)))) n0_i)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 n1_i (*.f32 1/6 (*.f32 n1_i (pow.f32 normAngle 2)))) n0_i)): 1 points increase in error, 0 points decrease in error

9. Final simplification 0.4

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

Alternatives

Alternative 1
Error	9.5
Cost	296

\[\begin{array}{l} t_0 := n0_i \cdot \left(1 - u\right)\\ \mathbf{if}\;n0_i \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n0_i \leq 9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	4.4
Cost	296

\[\begin{array}{l} t_0 := n0_i + u \cdot n1_i\\ \mathbf{if}\;n1_i \leq -5.60000001775803 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 9.999999887266023 \cdot 10^{-27}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	4.3
Cost	296

\[\begin{array}{l} t_0 := n0_i + u \cdot n1_i\\ \mathbf{if}\;n1_i \leq -5.60000001775803 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 9.999999887266023 \cdot 10^{-27}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	12.6
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 5
Error	0.6
Cost	224

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

Alternative 6
Error	16.8
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2022339 
(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)))