Average Error: 0.9 → 0.6
Time: 16.9s
Precision: binary32
Cost: 3360
\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]
\[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
↓
\[\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \]
(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 u (- n1_i n0_i) 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(u, (n1_i - n0_i), 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(u, Float32(n1_i - n0_i), 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(u, n1_i - n0_i, n0_i\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. Simplified0.7

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

    [Start]0.9

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

    fma-def [=>]0.9

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

    associate-*r/ [=>]0.8

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

    *-rgt-identity [=>]0.8

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

    associate-*r/ [=>]0.7

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

    *-rgt-identity [=>]0.7

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

  3. Taylor expanded in normAngle around 0 0.7

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

  4. Simplified0.7

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

    [Start]0.7

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

    fma-def [=>]0.7

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

  5. Taylor expanded in n1_i around 0 0.7

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

  6. Simplified0.6

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

    [Start]0.7

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

    *-commutative [<=]0.7

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

    sub-neg [=>]0.7

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

    +-commutative [=>]0.7

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

    distribute-rgt1-in [<=]0.6

    \[ u \cdot n1_i + \color{blue}{\left(n0_i + \left(-u\right) \cdot n0_i\right)} \]

    *-commutative [<=]0.6

    \[ u \cdot n1_i + \left(n0_i + \color{blue}{n0_i \cdot \left(-u\right)}\right) \]

    +-commutative [<=]0.6

    \[ u \cdot n1_i + \color{blue}{\left(n0_i \cdot \left(-u\right) + n0_i\right)} \]

    associate-+r+ [=>]0.6

    \[ \color{blue}{\left(u \cdot n1_i + n0_i \cdot \left(-u\right)\right) + n0_i} \]

    cancel-sign-sub [<=]0.6

    \[ \color{blue}{\left(u \cdot n1_i - \left(-n0_i\right) \cdot \left(-u\right)\right)} + n0_i \]

    mul-1-neg [<=]0.6

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

    *-commutative [=>]0.6

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

    cancel-sign-sub [=>]0.6

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

    distribute-lft-in [<=]0.6

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

    fma-def [=>]0.6

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

    mul-1-neg [=>]0.6

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

    unsub-neg [=>]0.6

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

  7. Final simplification0.6

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

Alternatives

Alternative 1
Error0.6
Cost736
\[normAngle \cdot \left(n1_i \cdot \left(u \cdot \left(normAngle \cdot 0.16666666666666666\right)\right) + \left(\frac{u \cdot n1_i}{normAngle} + \frac{n0_i \cdot \left(1 - u\right)}{normAngle}\right)\right) \]
Alternative 2
Error9.0
Cost297
\[\begin{array}{l} \mathbf{if}\;n0_i \leq -2.0000000390829628 \cdot 10^{-25} \lor \neg \left(n0_i \leq 1.0000000195414814 \cdot 10^{-25}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]
Alternative 3
Error4.4
Cost297
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.500000007927692 \cdot 10^{-29} \lor \neg \left(n1_i \leq 1.0000000195414814 \cdot 10^{-24}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]
Alternative 4
Error0.7
Cost288
\[u \cdot n1_i + n0_i \cdot \left(1 - u\right) \]
Alternative 5
Error12.6
Cost232
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -8.999999714389703 \cdot 10^{-21}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 7.999999870201272 \cdot 10^{-17}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]
Alternative 6
Error0.6
Cost224
\[n0_i + u \cdot \left(n1_i - n0_i\right) \]
Alternative 7
Error17.0
Cost32
\[n0_i \]

Error

Reproduce

herbie shell --seed 2022356 
(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)))