Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.7

Time: 17.0s

Precision: binary32

Cost: 16480

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

Math

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

↓

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

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
 (fma
  (/ (sin (- normAngle (* u normAngle))) (sin normAngle))
  n0_i
  (* (/ (sin (* u normAngle)) (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 fmaf((sinf((normAngle - (u * normAngle))) / sinf(normAngle)), n0_i, ((sinf((u * normAngle)) / sinf(normAngle)) * n1_i));
}

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 fma(Float32(sin(Float32(normAngle - Float32(u * normAngle))) / sin(normAngle)), n0_i, Float32(Float32(sin(Float32(u * normAngle)) / 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. 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, 4 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)): 2 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 (/.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 (sin.f32 (*.f32 u normAngle)) 1)) (sin.f32 normAngle)) n1_i)): 0 points increase in error, 6 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)): 0 points increase in error, 0 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))): 6 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.7

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

4. Final simplification 0.7

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

Alternatives

Alternative 1
Error	0.6
Cost	3360

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

Alternative 2
Error	4.4
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -3.6999998504363515 \cdot 10^{-26} \lor \neg \left(n1_i \leq 5.000000156871975 \cdot 10^{-23}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 3
Error	4.5
Cost	297

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -8.00000025099516 \cdot 10^{-22} \lor \neg \left(n1_i \leq 5.000000156871975 \cdot 10^{-23}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \end{array} \]

Alternative 4
Error	9.5
Cost	296

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 5
Error	0.6
Cost	288

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

Alternative 6
Error	12.7
Cost	232

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.6600000119442528 \cdot 10^{-18}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 1.199999961918627 \cdot 10^{-21}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 7
Error	0.6
Cost	224

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

Alternative 8
Error	16.7
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2022340 
(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)))