Curve intersection, scale width based on ribbon orientation

Average Error: 0.9 → 0.4

Time: 15.6s

Precision: binary32

Cost: 10880

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

Math

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

↓

\[\begin{array}{l} t_0 := n1_i \cdot u + \left(1 - u\right) \cdot n0_i\\ t_0 + \left(t_0 \cdot 0.16666666666666666 - \left(\left({\left(1 - u\right)}^{3} \cdot n0_i\right) \cdot 0.16666666666666666 + \left(n1_i \cdot {u}^{3}\right) \cdot 0.16666666666666666\right)\right) \cdot {normAngle}^{2} \end{array} \]

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
 (let* ((t_0 (+ (* n1_i u) (* (- 1.0 u) n0_i))))
   (+
    t_0
    (*
     (-
      (* t_0 0.16666666666666666)
      (+
       (* (* (pow (- 1.0 u) 3.0) n0_i) 0.16666666666666666)
       (* (* n1_i (pow u 3.0)) 0.16666666666666666)))
     (pow normAngle 2.0)))))

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) {
	float t_0 = (n1_i * u) + ((1.0f - u) * n0_i);
	return t_0 + (((t_0 * 0.16666666666666666f) - (((powf((1.0f - u), 3.0f) * n0_i) * 0.16666666666666666f) + ((n1_i * powf(u, 3.0f)) * 0.16666666666666666f))) * powf(normAngle, 2.0f));
}

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
    real(4) :: t_0
    t_0 = (n1_i * u) + ((1.0e0 - u) * n0_i)
    code = t_0 + (((t_0 * 0.16666666666666666e0) - (((((1.0e0 - u) ** 3.0e0) * n0_i) * 0.16666666666666666e0) + ((n1_i * (u ** 3.0e0)) * 0.16666666666666666e0))) * (normangle ** 2.0e0))
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)
	t_0 = Float32(Float32(n1_i * u) + Float32(Float32(Float32(1.0) - u) * n0_i))
	return Float32(t_0 + Float32(Float32(Float32(t_0 * Float32(0.16666666666666666)) - Float32(Float32(Float32((Float32(Float32(1.0) - u) ^ Float32(3.0)) * n0_i) * Float32(0.16666666666666666)) + Float32(Float32(n1_i * (u ^ Float32(3.0))) * Float32(0.16666666666666666)))) * (normAngle ^ Float32(2.0))))
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)
	t_0 = (n1_i * u) + ((single(1.0) - u) * n0_i);
	tmp = t_0 + (((t_0 * single(0.16666666666666666)) - (((((single(1.0) - u) ^ single(3.0)) * n0_i) * single(0.16666666666666666)) + ((n1_i * (u ^ single(3.0))) * single(0.16666666666666666)))) * (normAngle ^ single(2.0)));
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 8.4

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)}{\sin normAngle}} \]
   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)): 3 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)))): 33 points increase in error, 17 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)))): 6 points increase in error, 5 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))): 36 points increase in error, 82 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)): 9 points increase in error, 75 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 normAngle around 0 0.4

   \[\leadsto \color{blue}{\left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)} \]

4. Final simplification 0.4

   \[\leadsto \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) + \left(\left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \cdot 0.16666666666666666 - \left(\left({\left(1 - u\right)}^{3} \cdot n0_i\right) \cdot 0.16666666666666666 + \left(n1_i \cdot {u}^{3}\right) \cdot 0.16666666666666666\right)\right) \cdot {normAngle}^{2} \]

Alternatives

Alternative 1
Error	0.4
Cost	3968

\[\left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.5 + -0.16666666666666666 \cdot \left(n0_i - n1_i\right)\right)\right) \]

Alternative 2
Error	0.5
Cost	608

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

Alternative 3
Error	9.5
Cost	296

\[\begin{array}{l} t_0 := \left(1 - u\right) \cdot n0_i\\ \mathbf{if}\;n0_i \leq -9.999999887266023 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n0_i \leq 3.499999888929329 \cdot 10^{-20}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	4.3
Cost	296

\[\begin{array}{l} t_0 := n0_i + n1_i \cdot u\\ \mathbf{if}\;n1_i \leq -1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 8.000000217961609 \cdot 10^{-27}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	4.3
Cost	296

\[\begin{array}{l} t_0 := n0_i + n1_i \cdot u\\ \mathbf{if}\;n1_i \leq -1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 8.000000217961609 \cdot 10^{-27}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	12.5
Cost	232

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 3.499999888929329 \cdot 10^{-20}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 7
Error	0.6
Cost	224

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

Alternative 8
Error	17.2
Cost	32

\[n0_i \]

Reproduce

herbie shell --seed 2022325 
(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)))