?

Average Error: 2.85% → 0.85%
Time: 18.1s
Precision: binary32
Cost: 6688

?

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]
\[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
↓
\[\mathsf{fma}\left(normAngle \cdot \frac{n1_i}{\sin normAngle} - n0_i, u, n0_i\right) \]
(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 (/ n1_i (sin normAngle))) 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 * (n1_i / sinf(normAngle))) - 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(Float32(Float32(normAngle * Float32(n1_i / sin(normAngle))) - n0_i), 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 \cdot \frac{n1_i}{\sin normAngle} - n0_i, u, n0_i\right)

Error?

Derivation?

  1. Initial program 2.85

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

  2. Taylor expanded in normAngle around 0 2.96

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

  3. Taylor expanded in u around 0 10.07

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

  4. Simplified0.85

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

    [Start]10.07

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

    fma-def [=>]9.98

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

    mul-1-neg [=>]9.98

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

    unsub-neg [=>]9.98

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

    associate-*l/ [<=]0.85

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

  5. Final simplification0.85

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

Alternatives

Alternative 1
Error2.75%
Cost3616
\[n0_i \cdot \left(1 - u\right) + n1_i \cdot \frac{normAngle \cdot u}{\sin normAngle} \]
Alternative 2
Error1.91%
Cost3360
\[\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \]
Alternative 3
Error31.62%
Cost297
\[\begin{array}{l} \mathbf{if}\;n0_i \leq -2.000000033724767 \cdot 10^{-16} \lor \neg \left(n0_i \leq 4.00000018325482 \cdot 10^{-18}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(n1_i + n0_i\right)\\ \end{array} \]
Alternative 4
Error13.91%
Cost297
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -3.999999954906409 \cdot 10^{-26} \lor \neg \left(n1_i \leq 3.0000001202542024 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]
Alternative 5
Error13.83%
Cost297
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -3.999999954906409 \cdot 10^{-26} \lor \neg \left(n1_i \leq 3.0000001202542024 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \]
Alternative 6
Error41.11%
Cost296
\[\begin{array}{l} \mathbf{if}\;n0_i \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 7.999999870201272 \cdot 10^{-17}:\\ \;\;\;\;u \cdot \left(n1_i + n0_i\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]
Alternative 7
Error42.32%
Cost232
\[\begin{array}{l} \mathbf{if}\;n0_i \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 7.999999870201272 \cdot 10^{-17}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]
Alternative 8
Error2.04%
Cost224
\[n0_i + u \cdot \left(n1_i - n0_i\right) \]
Alternative 9
Error53.04%
Cost32
\[n0_i \]

Error

Reproduce?

herbie shell --seed 2023089 
(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)))