Curve intersection, scale width based on ribbon orientation

?

Percentage Accurate: 97.2% → 98.8%
Time: 27.5s
Precision: binary32
Cost: 10880

?

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]
\[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
↓
\[n1_i \cdot u + \left(\left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} + -0.16666666666666666 \cdot \left(u + -1\right)\right) \cdot n0_i\right) - \left(n1_i \cdot u\right) \cdot -0.16666666666666666\right) \cdot {normAngle}^{2} + \left(1 - u\right) \cdot n0_i\right) \]
(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
 (+
  (* n1_i u)
  (+
   (*
    (-
     (+
      (* -0.16666666666666666 (* n1_i (pow u 3.0)))
      (*
       (+
        (* -0.16666666666666666 (pow (- 1.0 u) 3.0))
        (* -0.16666666666666666 (+ u -1.0)))
       n0_i))
     (* (* n1_i u) -0.16666666666666666))
    (pow normAngle 2.0))
   (* (- 1.0 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 (n1_i * u) + (((((-0.16666666666666666f * (n1_i * powf(u, 3.0f))) + (((-0.16666666666666666f * powf((1.0f - u), 3.0f)) + (-0.16666666666666666f * (u + -1.0f))) * n0_i)) - ((n1_i * u) * -0.16666666666666666f)) * powf(normAngle, 2.0f)) + ((1.0f - u) * n0_i));
}

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = ((sin(((1.0e0 - u) * normangle)) * (1.0e0 / sin(normangle))) * n0_i) + ((sin((u * normangle)) * (1.0e0 / sin(normangle))) * n1_i)
end function
↓
real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = (n1_i * u) + ((((((-0.16666666666666666e0) * (n1_i * (u ** 3.0e0))) + ((((-0.16666666666666666e0) * ((1.0e0 - u) ** 3.0e0)) + ((-0.16666666666666666e0) * (u + (-1.0e0)))) * n0_i)) - ((n1_i * u) * (-0.16666666666666666e0))) * (normangle ** 2.0e0)) + ((1.0e0 - u) * n0_i))
end function

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 Float32(Float32(n1_i * u) + Float32(Float32(Float32(Float32(Float32(Float32(-0.16666666666666666) * Float32(n1_i * (u ^ Float32(3.0)))) + Float32(Float32(Float32(Float32(-0.16666666666666666) * (Float32(Float32(1.0) - u) ^ Float32(3.0))) + Float32(Float32(-0.16666666666666666) * Float32(u + Float32(-1.0)))) * n0_i)) - Float32(Float32(n1_i * u) * Float32(-0.16666666666666666))) * (normAngle ^ Float32(2.0))) + Float32(Float32(Float32(1.0) - u) * n0_i)))
end

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = ((sin(((single(1.0) - u) * normAngle)) * (single(1.0) / sin(normAngle))) * n0_i) + ((sin((u * normAngle)) * (single(1.0) / sin(normAngle))) * n1_i);
end
↓
function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n1_i * u) + (((((single(-0.16666666666666666) * (n1_i * (u ^ single(3.0)))) + (((single(-0.16666666666666666) * ((single(1.0) - u) ^ single(3.0))) + (single(-0.16666666666666666) * (u + single(-1.0)))) * n0_i)) - ((n1_i * u) * single(-0.16666666666666666))) * (normAngle ^ single(2.0))) + ((single(1.0) - 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
↓
n1_i \cdot u + \left(\left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} + -0.16666666666666666 \cdot \left(u + -1\right)\right) \cdot n0_i\right) - \left(n1_i \cdot u\right) \cdot -0.16666666666666666\right) \cdot {normAngle}^{2} + \left(1 - u\right) \cdot n0_i\right)

Local Percentage Accuracy?

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 9 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 97.5%

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

  2. Simplified97.9%

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

    [Start]97.5

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

    fma-def [=>]97.6

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

    associate-*r/ [=>]97.7

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

    *-rgt-identity [=>]97.7

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

    associate-*r/ [=>]97.9

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

    *-rgt-identity [=>]97.9

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

  3. Taylor expanded in normAngle around 0 99.0%

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

  4. Final simplification99.0%

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

Alternatives

Alternative 1
Accuracy98.2%
Cost3360
\[\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \]
Alternative 2
Accuracy61.0%
Cost297
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999967550318 \cdot 10^{-17} \lor \neg \left(n1_i \leq 9.9999998245167 \cdot 10^{-15}\right):\\ \;\;\;\;u \cdot \left(n1_i + n0_i\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]
Alternative 3
Accuracy71.1%
Cost297
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.0000000072549875 \cdot 10^{-15} \lor \neg \left(n1_i \leq 4.99999991225835 \cdot 10^{-14}\right):\\ \;\;\;\;u \cdot \left(n1_i + n0_i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \end{array} \]
Alternative 4
Accuracy85.7%
Cost297
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -3.000000157232057 \cdot 10^{-23} \lor \neg \left(n1_i \leq 4.999999999099794 \cdot 10^{-24}\right):\\ \;\;\;\;n1_i \cdot u + n0_i\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \end{array} \]
Alternative 5
Accuracy85.8%
Cost297
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -3.000000157232057 \cdot 10^{-23} \lor \neg \left(n1_i \leq 4.999999999099794 \cdot 10^{-24}\right):\\ \;\;\;\;n1_i \cdot u + n0_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \end{array} \]
Alternative 6
Accuracy60.3%
Cost232
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]
Alternative 7
Accuracy98.1%
Cost224
\[n0_i + u \cdot \left(n1_i - n0_i\right) \]
Alternative 8
Accuracy47.4%
Cost32
\[n0_i \]

Reproduce?

herbie shell --seed 2023161 
(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)))