Curve intersection, scale width based on ribbon orientation

?

Percentage Accurate: 97.3% → 98.1%
Time: 13.2s
Precision: binary32
Cost: 3680

?

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]
\[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
↓
\[\mathsf{fma}\left(n1_i - n0_i, u, n0_i - u \cdot \left(normAngle \cdot \left(normAngle \cdot \left(n0_i \cdot -0.3333333333333333\right)\right)\right)\right) \]
(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
  (- n1_i n0_i)
  u
  (- n0_i (* u (* normAngle (* normAngle (* n0_i -0.3333333333333333)))))))
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((n1_i - n0_i), u, (n0_i - (u * (normAngle * (normAngle * (n0_i * -0.3333333333333333f))))));
}

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(n1_i - n0_i), u, Float32(n0_i - Float32(u * Float32(normAngle * Float32(normAngle * Float32(n0_i * Float32(-0.3333333333333333)))))))
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(n1_i - n0_i, u, n0_i - u \cdot \left(normAngle \cdot \left(normAngle \cdot \left(n0_i \cdot -0.3333333333333333\right)\right)\right)\right)

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 8 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Derivation?

  1. Initial program 97.4%

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

  2. Simplified97.8%

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

    [Start]97.4%

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

    fma-def [=>]97.4%

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

    associate-*r/ [=>]97.6%

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

    *-rgt-identity [=>]97.6%

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

    associate-*r/ [=>]97.8%

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

    *-rgt-identity [=>]97.8%

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

  3. Taylor expanded in normAngle around 0 98.4%

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

  4. Taylor expanded in u around 0 92.4%

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

  5. Taylor expanded in normAngle around 0 98.8%

    \[\leadsto \color{blue}{\left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(-1 \cdot \left(\left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) \cdot \left(u \cdot {normAngle}^{2}\right)\right) + n0_i\right)} \]

  6. Simplified99.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(n1_i - n0_i, u, n0_i - u \cdot \left(normAngle \cdot \left(normAngle \cdot \left(n0_i \cdot -0.3333333333333333\right)\right)\right)\right)} \]
    Step-by-step derivation

    [Start]98.8%

    \[ \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(-1 \cdot \left(\left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) \cdot \left(u \cdot {normAngle}^{2}\right)\right) + n0_i\right) \]

    fma-def [=>]99.0%

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

    mul-1-neg [=>]99.0%

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

    sub-neg [<=]99.0%

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

    +-commutative [=>]99.0%

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

    mul-1-neg [=>]99.0%

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

    unsub-neg [=>]99.0%

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

    *-commutative [=>]99.0%

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

    associate-*l* [=>]99.0%

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

    unpow2 [=>]99.0%

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

    associate-*l* [=>]99.0%

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

    distribute-rgt-out-- [=>]99.0%

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

    metadata-eval [=>]99.0%

    \[ \mathsf{fma}\left(n1_i - n0_i, u, n0_i - u \cdot \left(normAngle \cdot \left(normAngle \cdot \left(n0_i \cdot \color{blue}{-0.3333333333333333}\right)\right)\right)\right) \]

  7. Final simplification99.0%

    \[\leadsto \mathsf{fma}\left(n1_i - n0_i, u, n0_i - u \cdot \left(normAngle \cdot \left(normAngle \cdot \left(n0_i \cdot -0.3333333333333333\right)\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy98.1%
Cost3680
\[\mathsf{fma}\left(n1_i - n0_i, u, n0_i - u \cdot \left(normAngle \cdot \left(normAngle \cdot \left(n0_i \cdot -0.3333333333333333\right)\right)\right)\right) \]
Alternative 2
Accuracy98.0%
Cost3360
\[\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \]
Alternative 3
Accuracy98.0%
Cost544
\[n0_i + \left(\left(n1_i - n0_i\right) \cdot u - u \cdot \left(normAngle \cdot \left(normAngle \cdot \left(n0_i \cdot -0.3333333333333333\right)\right)\right)\right) \]
Alternative 4
Accuracy70.5%
Cost296
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]
Alternative 5
Accuracy60.2%
Cost232
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 4.699999850847956 \cdot 10^{-20}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]
Alternative 6
Accuracy83.6%
Cost228
\[\begin{array}{l} \mathbf{if}\;n0_i \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]
Alternative 7
Accuracy97.9%
Cost224
\[n0_i + \left(n1_i - n0_i\right) \cdot u \]
Alternative 8
Accuracy47.2%
Cost32
\[n0_i \]

Reproduce?

herbie shell --seed 2023165 
(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)))