Average Error: 0.9 → 0.5
Time: 9.0s
Precision: binary32
\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]
\[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
↓
\[\begin{array}{l} t_0 := \left(normAngle \cdot normAngle\right) \cdot n0_i\\ \mathsf{fma}\left(1, \left(u \cdot t_0\right) \cdot 0.3333333333333333 - n0_i \cdot \mathsf{fma}\left(0.5, {\left(u \cdot normAngle\right)}^{2}, u\right), \mathsf{fma}\left(n1_i, u, \mathsf{fma}\left(0.16666666666666666, t_0 \cdot {u}^{3}, n0_i\right)\right)\right) \end{array} \]
(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 (* (* normAngle normAngle) n0_i)))
   (fma
    1.0
    (-
     (* (* u t_0) 0.3333333333333333)
     (* n0_i (fma 0.5 (pow (* u normAngle) 2.0) u)))
    (fma n1_i u (fma 0.16666666666666666 (* t_0 (pow u 3.0)) n0_i)))))
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 = (normAngle * normAngle) * n0_i;
	return fmaf(1.0f, (((u * t_0) * 0.3333333333333333f) - (n0_i * fmaf(0.5f, powf((u * normAngle), 2.0f), u))), fmaf(n1_i, u, fmaf(0.16666666666666666f, (t_0 * powf(u, 3.0f)), n0_i)));
}

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(normAngle * normAngle) * n0_i)
	return fma(Float32(1.0), Float32(Float32(Float32(u * t_0) * Float32(0.3333333333333333)) - Float32(n0_i * fma(Float32(0.5), (Float32(u * normAngle) ^ Float32(2.0)), u))), fma(n1_i, u, fma(Float32(0.16666666666666666), Float32(t_0 * (u ^ Float32(3.0))), n0_i)))
end

\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 := \left(normAngle \cdot normAngle\right) \cdot n0_i\\
\mathsf{fma}\left(1, \left(u \cdot t_0\right) \cdot 0.3333333333333333 - n0_i \cdot \mathsf{fma}\left(0.5, {\left(u \cdot normAngle\right)}^{2}, u\right), \mathsf{fma}\left(n1_i, u, \mathsf{fma}\left(0.16666666666666666, t_0 \cdot {u}^{3}, n0_i\right)\right)\right)
\end{array}

Error

Bits error versus normAngle

Bits error versus u

Bits error versus n0_i

Bits error versus n1_i

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. Taylor expanded in normAngle around 0 0.8

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

  3. Taylor expanded in normAngle around 0 0.6

    \[\leadsto \color{blue}{\left(0.5 \cdot \left(u \cdot \left(n0_i \cdot {normAngle}^{2}\right)\right) + \left(n1_i \cdot u + \left(0.16666666666666666 \cdot \left({u}^{3} \cdot \left(n0_i \cdot {normAngle}^{2}\right)\right) + n0_i\right)\right)\right) - \left(0.16666666666666666 \cdot \left(n0_i \cdot \left(u \cdot {normAngle}^{2}\right)\right) + \left(0.5 \cdot \left({u}^{2} \cdot \left(n0_i \cdot {normAngle}^{2}\right)\right) + u \cdot n0_i\right)\right)} \]

  4. Simplified0.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(n1_i, u, \mathsf{fma}\left(0.16666666666666666, \left(n0_i \cdot \left(normAngle \cdot normAngle\right)\right) \cdot {u}^{3}, n0_i\right)\right) + \left(\left(u \cdot \left(n0_i \cdot \left(normAngle \cdot normAngle\right)\right)\right) \cdot 0.3333333333333333 - n0_i \cdot \left(\left(0.5 \cdot \left(u \cdot u\right)\right) \cdot \left(normAngle \cdot normAngle\right) + u\right)\right)} \]

  5. Applied egg-rr0.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, \left(u \cdot \left(\left(normAngle \cdot normAngle\right) \cdot n0_i\right)\right) \cdot 0.3333333333333333 - n0_i \cdot \mathsf{fma}\left(0.5, {\left(u \cdot normAngle\right)}^{2}, u\right), \mathsf{fma}\left(n1_i, u, \mathsf{fma}\left(0.16666666666666666, \left(\left(normAngle \cdot normAngle\right) \cdot n0_i\right) \cdot {u}^{3}, n0_i\right)\right)\right)} \]

  6. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(1, \left(u \cdot \left(\left(normAngle \cdot normAngle\right) \cdot n0_i\right)\right) \cdot 0.3333333333333333 - n0_i \cdot \mathsf{fma}\left(0.5, {\left(u \cdot normAngle\right)}^{2}, u\right), \mathsf{fma}\left(n1_i, u, \mathsf{fma}\left(0.16666666666666666, \left(\left(normAngle \cdot normAngle\right) \cdot n0_i\right) \cdot {u}^{3}, n0_i\right)\right)\right) \]

Reproduce

herbie shell --seed 2022134 
(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)))