Average Error: 0.9 → 0.3
Time: 8.0s
Precision: binary32
\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]
\[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(n0_i, 1 - u, u \cdot n1_i\right)\\ \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(-0.16666666666666666, n0_i \cdot {\left(1 - u\right)}^{3}, 0.16666666666666666 \cdot t_0\right), t_0\right) \end{array} \]
(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 (fma n0_i (- 1.0 u) (* u n1_i))))
   (fma
    (* normAngle normAngle)
    (fma
     -0.16666666666666666
     (* n0_i (pow (- 1.0 u) 3.0))
     (* 0.16666666666666666 t_0))
    t_0)))
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 = fmaf(n0_i, (1.0f - u), (u * n1_i));
	return fmaf((normAngle * normAngle), fmaf(-0.16666666666666666f, (n0_i * powf((1.0f - u), 3.0f)), (0.16666666666666666f * t_0)), t_0);
}

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 = fma(n0_i, Float32(Float32(1.0) - u), Float32(u * n1_i))
	return fma(Float32(normAngle * normAngle), fma(Float32(-0.16666666666666666), Float32(n0_i * (Float32(Float32(1.0) - u) ^ Float32(3.0))), Float32(Float32(0.16666666666666666) * t_0)), t_0)
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 := \mathsf{fma}\left(n0_i, 1 - u, u \cdot n1_i\right)\\
\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(-0.16666666666666666, n0_i \cdot {\left(1 - u\right)}^{3}, 0.16666666666666666 \cdot t_0\right), t_0\right)
\end{array}

Error

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 u around 0 0.9

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

  3. Simplified0.4

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

  4. Taylor expanded in u around inf 8.2

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

  5. Simplified0.3

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

  6. Taylor expanded in normAngle around 0 0.4

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2022211 
(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)))