Average Error: 0.9 → 0.3
Time: 8.0s
Precision: binary32
\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]
\[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
↓
\[\mathsf{fma}\left(normAngle, normAngle \cdot \left(-0.16666666666666666 \cdot \mathsf{fma}\left(n0_i, u + \left({\left(1 - u\right)}^{3} + -1\right), -u \cdot n1_i\right)\right), \mathsf{fma}\left(u, n1_i, \mathsf{fma}\left(n0_i, -u, n0_i\right)\right)\right) \]
(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
  normAngle
  (*
   normAngle
   (*
    -0.16666666666666666
    (fma n0_i (+ u (+ (pow (- 1.0 u) 3.0) -1.0)) (- (* u n1_i)))))
  (fma u n1_i (fma n0_i (- u) 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) {
	return fmaf(normAngle, (normAngle * (-0.16666666666666666f * fmaf(n0_i, (u + (powf((1.0f - u), 3.0f) + -1.0f)), -(u * n1_i)))), fmaf(u, n1_i, fmaf(n0_i, -u, 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)
	return fma(normAngle, Float32(normAngle * Float32(Float32(-0.16666666666666666) * fma(n0_i, Float32(u + Float32((Float32(Float32(1.0) - u) ^ Float32(3.0)) + Float32(-1.0))), Float32(-Float32(u * n1_i))))), fma(u, n1_i, fma(n0_i, Float32(-u), 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
↓
\mathsf{fma}\left(normAngle, normAngle \cdot \left(-0.16666666666666666 \cdot \mathsf{fma}\left(n0_i, u + \left({\left(1 - u\right)}^{3} + -1\right), -u \cdot n1_i\right)\right), \mathsf{fma}\left(u, n1_i, \mathsf{fma}\left(n0_i, -u, n0_i\right)\right)\right)

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 normAngle around 0 0.4

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2022181 
(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)))