Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.6

Time: 12.7s

Precision: binary32

Cost: 224

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

Math

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

↓

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

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 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 - 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 - 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 - 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 - 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 0.7

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]
   Proof
   (fma.f32 (/.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (sin.f32 normAngle)) n0_i (*.f32 (/.f32 (sin.f32 (*.f32 u normAngle)) (sin.f32 normAngle)) n1_i)): 0 points increase in error, 0 points decrease in error
   (fma.f32 (/.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) 1)) (sin.f32 normAngle)) n0_i (*.f32 (/.f32 (sin.f32 (*.f32 u normAngle)) (sin.f32 normAngle)) n1_i)): 0 points increase in error, 0 points decrease in error
   (fma.f32 (Rewrite<= associate-*r/_binary32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle)))) n0_i (*.f32 (/.f32 (sin.f32 (*.f32 u normAngle)) (sin.f32 normAngle)) n1_i)): 26 points increase in error, 7 points decrease in error
   (fma.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i (*.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
   (fma.f32 (*.f32 (sin.f32 (*.f32 (-.f32 1 u) normAngle)) (/.f32 1 (sin.f32 normAngle))) n0_i (*.f32 (Rewrite<= associate-*r/_binary32 (*.f32 (sin.f32 (*.f32 u normAngle)) (/.f32 1 (sin.f32 normAngle)))) n1_i)): 20 points increase in error, 8 points decrease in error
   (Rewrite<= fma-def_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))): 5 points increase in error, 2 points decrease in error

3. Taylor expanded in normAngle around 0 0.7

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

4. Simplified 0.7

   \[\leadsto \color{blue}{\mathsf{fma}\left(n1_i, u, \left(1 - u\right) \cdot n0_i\right)} \]
   Proof
   (fma.f32 n1_i u (*.f32 (-.f32 1 u) n0_i)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary32 (+.f32 (*.f32 n1_i u) (*.f32 (-.f32 1 u) n0_i))): 14 points increase in error, 2 points decrease in error

5. Taylor expanded in u around 0 0.6

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

6. Taylor expanded in n1_i around 0 0.7

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

7. Simplified 0.6

   \[\leadsto \color{blue}{n0_i + u \cdot \left(n1_i - n0_i\right)} \]
   Proof
   (+.f32 n0_i (*.f32 u (-.f32 n1_i n0_i))): 0 points increase in error, 0 points decrease in error
   (+.f32 n0_i (Rewrite<= distribute-lft-out--_binary32 (-.f32 (*.f32 u n1_i) (*.f32 u n0_i)))): 6 points increase in error, 2 points decrease in error
   (+.f32 n0_i (-.f32 (Rewrite<= *-commutative_binary32 (*.f32 n1_i u)) (*.f32 u n0_i))): 0 points increase in error, 0 points decrease in error
   (+.f32 n0_i (Rewrite<= unsub-neg_binary32 (+.f32 (*.f32 n1_i u) (neg.f32 (*.f32 u n0_i))))): 0 points increase in error, 0 points decrease in error
   (+.f32 n0_i (+.f32 (*.f32 n1_i u) (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (*.f32 u n0_i))))): 0 points increase in error, 0 points decrease in error
   (+.f32 n0_i (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 -1 (*.f32 u n0_i)) (*.f32 n1_i u)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary32 (+.f32 (+.f32 (*.f32 -1 (*.f32 u n0_i)) (*.f32 n1_i u)) n0_i)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+r+_binary32 (+.f32 (*.f32 -1 (*.f32 u n0_i)) (+.f32 (*.f32 n1_i u) n0_i))): 12 points increase in error, 2 points decrease in error

8. Final simplification 0.6

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

Alternatives

Alternative 1
Error	9.7
Cost	296

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 2
Error	4.6
Cost	296

\[\begin{array}{l} t_0 := n0_i + u \cdot n1_i\\ \mathbf{if}\;n1_i \leq -4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	4.5
Cost	296

\[\begin{array}{l} t_0 := n0_i + u \cdot n1_i\\ \mathbf{if}\;n1_i \leq -4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	12.6
Cost	232

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 5
Error	16.9
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2022338 
(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)))