Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.5%

Time: 14.7s

Alternatives: 9

Speedup: 45.9×

Specification

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0\_i \land n0\_i \leq 1\right)\right) \land \left(-1 \leq n1\_i \land n1\_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t\_0\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot t\_0\right) \cdot n1\_i \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sin normAngle))))
   (+
    (* (* (sin (* (- 1.0 u) normAngle)) t_0) n0_i)
    (* (* (sin (* u normAngle)) t_0) n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = 1.0f / sinf(normAngle);
	return ((sinf(((1.0f - u) * normAngle)) * t_0) * n0_i) + ((sinf((u * normAngle)) * t_0) * n1_i);
}

Fortran

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
    real(4) :: t_0
    t_0 = 1.0e0 / sin(normangle)
    code = ((sin(((1.0e0 - u) * normangle)) * t_0) * n0_i) + ((sin((u * normangle)) * t_0) * n1_i)
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(1.0) / sin(normAngle))
	return Float32(Float32(Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * t_0) * n0_i) + Float32(Float32(sin(Float32(u * normAngle)) * t_0) * n1_i))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	t_0 = single(1.0) / sin(normAngle);
	tmp = ((sin(((single(1.0) - u) * normAngle)) * t_0) * n0_i) + ((sin((u * normAngle)) * t_0) * n1_i);
end

Initial Program: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t\_0\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot t\_0\right) \cdot n1\_i \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sin normAngle))))
   (+
    (* (* (sin (* (- 1.0 u) normAngle)) t_0) n0_i)
    (* (* (sin (* u normAngle)) t_0) n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = 1.0f / sinf(normAngle);
	return ((sinf(((1.0f - u) * normAngle)) * t_0) * n0_i) + ((sinf((u * normAngle)) * t_0) * n1_i);
}

Fortran

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
    real(4) :: t_0
    t_0 = 1.0e0 / sin(normangle)
    code = ((sin(((1.0e0 - u) * normangle)) * t_0) * n0_i) + ((sin((u * normangle)) * t_0) * n1_i)
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(1.0) / sin(normAngle))
	return Float32(Float32(Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * t_0) * n0_i) + Float32(Float32(sin(Float32(u * normAngle)) * t_0) * n1_i))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	t_0 = single(1.0) / sin(normAngle);
	tmp = ((sin(((single(1.0) - u) * normAngle)) * t_0) * n0_i) + ((sin((u * normAngle)) * t_0) * n1_i);
end

Alternative 1: 99.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.022222222222222223, \mathsf{fma}\left(normAngle \cdot normAngle, n1\_i \cdot 0.00205026455026455, n1\_i \cdot 0.019444444444444445\right)\right), \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* normAngle normAngle)
   (fma
    (* normAngle normAngle)
    (fma
     n0_i
     0.022222222222222223
     (fma
      (* normAngle normAngle)
      (* n1_i 0.00205026455026455)
      (* n1_i 0.019444444444444445)))
    (fma n0_i (fma -0.5 u 0.3333333333333333) (* n1_i 0.16666666666666666)))
   (- n1_i n0_i))
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), fmaf(n0_i, 0.022222222222222223f, fmaf((normAngle * normAngle), (n1_i * 0.00205026455026455f), (n1_i * 0.019444444444444445f))), fmaf(n0_i, fmaf(-0.5f, u, 0.3333333333333333f), (n1_i * 0.16666666666666666f))), (n1_i - n0_i)), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), fma(n0_i, Float32(0.022222222222222223), fma(Float32(normAngle * normAngle), Float32(n1_i * Float32(0.00205026455026455)), Float32(n1_i * Float32(0.019444444444444445)))), fma(n0_i, fma(Float32(-0.5), u, Float32(0.3333333333333333)), Float32(n1_i * Float32(0.16666666666666666)))), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 98.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) + n0\_i} \]

   2. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, -1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right), n0\_i\right)} \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(u, -0.5 \cdot \left(normAngle \cdot normAngle\right), \frac{normAngle \cdot \cos normAngle}{-\sin normAngle}\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right)} \]

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{45} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{2}{945} \cdot n0\_i - \left(\frac{-1}{5040} \cdot n1\_i + \left(\frac{1}{720} \cdot n1\_i + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right)\right)\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)}, n0\_i\right) \]

8. Simplified 99.7%

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.022222222222222223, \mathsf{fma}\left(normAngle \cdot normAngle, n0\_i \cdot 0.0021164021164021165 - \mathsf{fma}\left(n1\_i, 0.0011904761904761906, n1\_i \cdot -0.0032407407407407406\right), n1\_i \cdot 0.019444444444444445\right)\right), \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right)}, n0\_i\right) \]

9. Taylor expanded in n0_i around 0

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \frac{1}{45}, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{-1 \cdot \left(\frac{-7}{2160} \cdot n1\_i + \frac{1}{840} \cdot n1\_i\right)}, n1\_i \cdot \frac{7}{360}\right)\right), \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), n1\_i \cdot \frac{1}{6}\right)\right), n1\_i - n0\_i\right), n0\_i\right) \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \frac{1}{45}, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{\mathsf{neg}\left(\left(\frac{-7}{2160} \cdot n1\_i + \frac{1}{840} \cdot n1\_i\right)\right)}, n1\_i \cdot \frac{7}{360}\right)\right), \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), n1\_i \cdot \frac{1}{6}\right)\right), n1\_i - n0\_i\right), n0\_i\right) \]

    2. distribute-rgt-out N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \frac{1}{45}, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{neg}\left(\color{blue}{n1\_i \cdot \left(\frac{-7}{2160} + \frac{1}{840}\right)}\right), n1\_i \cdot \frac{7}{360}\right)\right), \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), n1\_i \cdot \frac{1}{6}\right)\right), n1\_i - n0\_i\right), n0\_i\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \frac{1}{45}, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{neg}\left(n1\_i \cdot \color{blue}{\frac{-31}{15120}}\right), n1\_i \cdot \frac{7}{360}\right)\right), \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), n1\_i \cdot \frac{1}{6}\right)\right), n1\_i - n0\_i\right), n0\_i\right) \]

    4. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \frac{1}{45}, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{n1\_i \cdot \left(\mathsf{neg}\left(\frac{-31}{15120}\right)\right)}, n1\_i \cdot \frac{7}{360}\right)\right), \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), n1\_i \cdot \frac{1}{6}\right)\right), n1\_i - n0\_i\right), n0\_i\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \frac{1}{45}, \mathsf{fma}\left(normAngle \cdot normAngle, n1\_i \cdot \color{blue}{\frac{31}{15120}}, n1\_i \cdot \frac{7}{360}\right)\right), \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), n1\_i \cdot \frac{1}{6}\right)\right), n1\_i - n0\_i\right), n0\_i\right) \]

    6. lower-*.f32 99.7

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.022222222222222223, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{n1\_i \cdot 0.00205026455026455}, n1\_i \cdot 0.019444444444444445\right)\right), \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right), n0\_i\right) \]

11. Simplified 99.7%

    \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.022222222222222223, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{n1\_i \cdot 0.00205026455026455}, n1\_i \cdot 0.019444444444444445\right)\right), \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right), n0\_i\right) \]
12. Add Preprocessing

Alternative 2: 99.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.022222222222222223, n1\_i \cdot 0.019444444444444445\right), n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* normAngle normAngle)
   (fma
    n0_i
    (fma -0.5 u 0.3333333333333333)
    (fma
     (* normAngle normAngle)
     (fma n0_i 0.022222222222222223 (* n1_i 0.019444444444444445))
     (* n1_i 0.16666666666666666)))
   (- n1_i n0_i))
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, fmaf((normAngle * normAngle), fmaf(n0_i, fmaf(-0.5f, u, 0.3333333333333333f), fmaf((normAngle * normAngle), fmaf(n0_i, 0.022222222222222223f, (n1_i * 0.019444444444444445f)), (n1_i * 0.16666666666666666f))), (n1_i - n0_i)), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, fma(Float32(normAngle * normAngle), fma(n0_i, fma(Float32(-0.5), u, Float32(0.3333333333333333)), fma(Float32(normAngle * normAngle), fma(n0_i, Float32(0.022222222222222223), Float32(n1_i * Float32(0.019444444444444445))), Float32(n1_i * Float32(0.16666666666666666)))), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 98.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) + n0\_i} \]

   2. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, -1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right), n0\_i\right)} \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(u, -0.5 \cdot \left(normAngle \cdot normAngle\right), \frac{normAngle \cdot \cos normAngle}{-\sin normAngle}\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right)} \]

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)}, n0\_i\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right) + n1\_i}, n0\_i\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left({normAngle}^{2} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right) + -1 \cdot n0\_i\right)} + n1\_i, n0\_i\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{{normAngle}^{2} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right) + \left(-1 \cdot n0\_i + n1\_i\right)}, n0\_i\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(u, {normAngle}^{2} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right) + \color{blue}{\left(n1\_i + -1 \cdot n0\_i\right)}, n0\_i\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left({normAngle}^{2}, \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i, n1\_i + -1 \cdot n0\_i\right)}, n0\_i\right) \]

8. Simplified 99.6%

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.022222222222222223, n1\_i \cdot 0.019444444444444445\right), n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right)}, n0\_i\right) \]
9. Add Preprocessing

Alternative 3: 99.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, n1\_i - n0\_i, \mathsf{fma}\left(normAngle \cdot \left(u \cdot normAngle\right), \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), n1\_i \cdot 0.16666666666666666\right), n0\_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (- n1_i n0_i)
  (fma
   (* normAngle (* u normAngle))
   (fma n0_i (fma -0.5 u 0.3333333333333333) (* n1_i 0.16666666666666666))
   n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, (n1_i - n0_i), fmaf((normAngle * (u * normAngle)), fmaf(n0_i, fmaf(-0.5f, u, 0.3333333333333333f), (n1_i * 0.16666666666666666f)), n0_i));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, Float32(n1_i - n0_i), fma(Float32(normAngle * Float32(u * normAngle)), fma(n0_i, fma(Float32(-0.5), u, Float32(0.3333333333333333)), Float32(n1_i * Float32(0.16666666666666666))), n0_i))
end

Derivation

1. Initial program 98.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) + n0\_i} \]

   2. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, -1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right), n0\_i\right)} \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(u, -0.5 \cdot \left(normAngle \cdot normAngle\right), \frac{normAngle \cdot \cos normAngle}{-\sin normAngle}\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right)} \]

6. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right) + n0\_i} \]

   2. associate-+l+ N/A

      \[\leadsto \color{blue}{u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + \left({normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right)\right) + n0\_i\right)} \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1\_i + -1 \cdot n0\_i, {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right)\right) + n0\_i\right)} \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(u, n1\_i + \color{blue}{\left(\mathsf{neg}\left(n0\_i\right)\right)}, {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right)\right) + n0\_i\right) \]

   5. unsub-neg N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i - n0\_i}, {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right)\right) + n0\_i\right) \]

   6. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i - n0\_i}, {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right)\right) + n0\_i\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(u, n1\_i - n0\_i, \color{blue}{\left({normAngle}^{2} \cdot u\right) \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right)} + n0\_i\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(u, n1\_i - n0\_i, \color{blue}{\mathsf{fma}\left({normAngle}^{2} \cdot u, n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i, n0\_i\right)}\right) \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1\_i - n0\_i, \mathsf{fma}\left(normAngle \cdot \left(normAngle \cdot u\right), \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), n1\_i \cdot 0.16666666666666666\right), n0\_i\right)\right)} \]

9. Final simplification 99.4%

   \[\leadsto \mathsf{fma}\left(u, n1\_i - n0\_i, \mathsf{fma}\left(normAngle \cdot \left(u \cdot normAngle\right), \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), n1\_i \cdot 0.16666666666666666\right), n0\_i\right)\right) \]
10. Add Preprocessing

Alternative 4: 99.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), n1\_i \cdot 0.16666666666666666\right), n1\_i - n0\_i\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* normAngle normAngle)
   (fma n0_i (fma -0.5 u 0.3333333333333333) (* n1_i 0.16666666666666666))
   (- n1_i n0_i))
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, fmaf((normAngle * normAngle), fmaf(n0_i, fmaf(-0.5f, u, 0.3333333333333333f), (n1_i * 0.16666666666666666f)), (n1_i - n0_i)), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, fma(Float32(normAngle * normAngle), fma(n0_i, fma(Float32(-0.5), u, Float32(0.3333333333333333)), Float32(n1_i * Float32(0.16666666666666666))), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 98.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) + n0\_i} \]

   2. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, -1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right), n0\_i\right)} \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(u, -0.5 \cdot \left(normAngle \cdot normAngle\right), \frac{normAngle \cdot \cos normAngle}{-\sin normAngle}\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right)} \]

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right)\right)}, n0\_i\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right)\right) + n1\_i}, n0\_i\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left({normAngle}^{2} \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right) + -1 \cdot n0\_i\right)} + n1\_i, n0\_i\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{{normAngle}^{2} \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right) + \left(-1 \cdot n0\_i + n1\_i\right)}, n0\_i\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(u, {normAngle}^{2} \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right) + \color{blue}{\left(n1\_i + -1 \cdot n0\_i\right)}, n0\_i\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left({normAngle}^{2}, n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i, n1\_i + -1 \cdot n0\_i\right)}, n0\_i\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i, n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i, n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right)}, n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   9. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{\mathsf{fma}\left(n0\_i, \frac{1}{3} + \frac{-1}{2} \cdot u, \mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right)}, n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \color{blue}{\frac{-1}{2} \cdot u + \frac{1}{3}}, \mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right), n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   11. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right)}, \mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right), n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), \mathsf{neg}\left(\color{blue}{n1\_i \cdot \frac{-1}{6}}\right)\right), n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   13. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), \color{blue}{n1\_i \cdot \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right)}\right), n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), n1\_i \cdot \color{blue}{\frac{1}{6}}\right), n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   15. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), \color{blue}{n1\_i \cdot \frac{1}{6}}\right), n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   16. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), n1\_i \cdot \frac{1}{6}\right), n1\_i + \color{blue}{\left(\mathsf{neg}\left(n0\_i\right)\right)}\right), n0\_i\right) \]

   17. unsub-neg N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), n1\_i \cdot \frac{1}{6}\right), \color{blue}{n1\_i - n0\_i}\right), n0\_i\right) \]

   18. lower--.f32 99.4

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), n1\_i \cdot 0.16666666666666666\right), \color{blue}{n1\_i - n0\_i}\right), n0\_i\right) \]

8. Simplified 99.4%

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), n1\_i \cdot 0.16666666666666666\right), n1\_i - n0\_i\right)}, n0\_i\right) \]
9. Add Preprocessing

Alternative 5: 98.8% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, n1\_i \cdot 0.16666666666666666, n1\_i - n0\_i\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma (* normAngle normAngle) (* n1_i 0.16666666666666666) (- n1_i n0_i))
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, fmaf((normAngle * normAngle), (n1_i * 0.16666666666666666f), (n1_i - n0_i)), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, fma(Float32(normAngle * normAngle), Float32(n1_i * Float32(0.16666666666666666)), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 98.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) + n0\_i} \]

   2. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, -1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right), n0\_i\right)} \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(u, -0.5 \cdot \left(normAngle \cdot normAngle\right), \frac{normAngle \cdot \cos normAngle}{-\sin normAngle}\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right)} \]

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right)\right)}, n0\_i\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right)\right) + n1\_i}, n0\_i\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left({normAngle}^{2} \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right) + -1 \cdot n0\_i\right)} + n1\_i, n0\_i\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{{normAngle}^{2} \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right) + \left(-1 \cdot n0\_i + n1\_i\right)}, n0\_i\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(u, {normAngle}^{2} \cdot \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i\right) + \color{blue}{\left(n1\_i + -1 \cdot n0\_i\right)}, n0\_i\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left({normAngle}^{2}, n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i, n1\_i + -1 \cdot n0\_i\right)}, n0\_i\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i, n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) - \frac{-1}{6} \cdot n1\_i, n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right)}, n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   9. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{\mathsf{fma}\left(n0\_i, \frac{1}{3} + \frac{-1}{2} \cdot u, \mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right)}, n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \color{blue}{\frac{-1}{2} \cdot u + \frac{1}{3}}, \mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right), n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   11. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right)}, \mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right), n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), \mathsf{neg}\left(\color{blue}{n1\_i \cdot \frac{-1}{6}}\right)\right), n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   13. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), \color{blue}{n1\_i \cdot \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right)}\right), n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), n1\_i \cdot \color{blue}{\frac{1}{6}}\right), n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   15. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), \color{blue}{n1\_i \cdot \frac{1}{6}}\right), n1\_i + -1 \cdot n0\_i\right), n0\_i\right) \]

   16. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), n1\_i \cdot \frac{1}{6}\right), n1\_i + \color{blue}{\left(\mathsf{neg}\left(n0\_i\right)\right)}\right), n0\_i\right) \]

   17. unsub-neg N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{-1}{2}, u, \frac{1}{3}\right), n1\_i \cdot \frac{1}{6}\right), \color{blue}{n1\_i - n0\_i}\right), n0\_i\right) \]

   18. lower--.f32 99.4

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), n1\_i \cdot 0.16666666666666666\right), \color{blue}{n1\_i - n0\_i}\right), n0\_i\right) \]

8. Simplified 99.4%

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(-0.5, u, 0.3333333333333333\right), n1\_i \cdot 0.16666666666666666\right), n1\_i - n0\_i\right)}, n0\_i\right) \]

9. Taylor expanded in n0_i around 0

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{\frac{1}{6} \cdot n1\_i}, n1\_i - n0\_i\right), n0\_i\right) \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{n1\_i \cdot \frac{1}{6}}, n1\_i - n0\_i\right), n0\_i\right) \]

    2. lower-*.f32 99.2

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{n1\_i \cdot 0.16666666666666666}, n1\_i - n0\_i\right), n0\_i\right) \]

11. Simplified 99.2%

    \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{n1\_i \cdot 0.16666666666666666}, n1\_i - n0\_i\right), n0\_i\right) \]
12. Add Preprocessing

Alternative 6: 98.1% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(u, -\frac{n0\_i}{n1\_i}, u\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma n1_i (fma u (- (/ n0_i n1_i)) u) n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(n1_i, fmaf(u, -(n0_i / n1_i), u), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(n1_i, fma(u, Float32(-Float32(n0_i / n1_i)), u), n0_i)
end

Derivation

1. Initial program 98.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
4. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(n0\_i, 1 - u, n1\_i \cdot u\right)} \]

   2. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(n0\_i, \color{blue}{1 - u}, n1\_i \cdot u\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \color{blue}{u \cdot n1\_i}\right) \]

   4. lower-*.f32 98.3

      \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \color{blue}{u \cdot n1\_i}\right) \]

5. Simplified 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right)} \]

6. Taylor expanded in n1_i around inf

   \[\leadsto \color{blue}{n1\_i \cdot \left(u + \frac{n0\_i \cdot \left(1 - u\right)}{n1\_i}\right)} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{n1\_i \cdot \left(u + \frac{n0\_i \cdot \left(1 - u\right)}{n1\_i}\right)} \]

   2. +-commutative N/A

      \[\leadsto n1\_i \cdot \color{blue}{\left(\frac{n0\_i \cdot \left(1 - u\right)}{n1\_i} + u\right)} \]

   3. associate-/l* N/A

      \[\leadsto n1\_i \cdot \left(\color{blue}{n0\_i \cdot \frac{1 - u}{n1\_i}} + u\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto n1\_i \cdot \color{blue}{\mathsf{fma}\left(n0\_i, \frac{1 - u}{n1\_i}, u\right)} \]

   5. lower-/.f32 N/A

      \[\leadsto n1\_i \cdot \mathsf{fma}\left(n0\_i, \color{blue}{\frac{1 - u}{n1\_i}}, u\right) \]

   6. lower--.f32 97.6

      \[\leadsto n1\_i \cdot \mathsf{fma}\left(n0\_i, \frac{\color{blue}{1 - u}}{n1\_i}, u\right) \]

8. Simplified 97.6%

   \[\leadsto \color{blue}{n1\_i \cdot \mathsf{fma}\left(n0\_i, \frac{1 - u}{n1\_i}, u\right)} \]

9. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + n1\_i \cdot \left(u \cdot \left(1 + -1 \cdot \frac{n0\_i}{n1\_i}\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{n1\_i \cdot \left(u \cdot \left(1 + -1 \cdot \frac{n0\_i}{n1\_i}\right)\right) + n0\_i} \]

    2. lower-fma.f32 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(n1\_i, u \cdot \left(1 + -1 \cdot \frac{n0\_i}{n1\_i}\right), n0\_i\right)} \]

    3. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(n1\_i, u \cdot \color{blue}{\left(-1 \cdot \frac{n0\_i}{n1\_i} + 1\right)}, n0\_i\right) \]

    4. distribute-lft-in N/A

       \[\leadsto \mathsf{fma}\left(n1\_i, \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i}{n1\_i}\right) + u \cdot 1}, n0\_i\right) \]

    5. *-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(n1\_i, u \cdot \left(-1 \cdot \frac{n0\_i}{n1\_i}\right) + \color{blue}{u}, n0\_i\right) \]

    6. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(n1\_i, \color{blue}{\mathsf{fma}\left(u, -1 \cdot \frac{n0\_i}{n1\_i}, u\right)}, n0\_i\right) \]

    7. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(u, \color{blue}{\mathsf{neg}\left(\frac{n0\_i}{n1\_i}\right)}, u\right), n0\_i\right) \]

    8. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(u, \color{blue}{\frac{n0\_i}{\mathsf{neg}\left(n1\_i\right)}}, u\right), n0\_i\right) \]

    9. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(u, \frac{n0\_i}{\color{blue}{-1 \cdot n1\_i}}, u\right), n0\_i\right) \]

    10. lower-/.f32 N/A

        \[\leadsto \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(u, \color{blue}{\frac{n0\_i}{-1 \cdot n1\_i}}, u\right), n0\_i\right) \]

    11. mul-1-neg N/A

        \[\leadsto \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(u, \frac{n0\_i}{\color{blue}{\mathsf{neg}\left(n1\_i\right)}}, u\right), n0\_i\right) \]

    12. lower-neg.f32 98.6

        \[\leadsto \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(u, \frac{n0\_i}{\color{blue}{-n1\_i}}, u\right), n0\_i\right) \]

11. Simplified 98.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(n1\_i, \mathsf{fma}\left(u, \frac{n0\_i}{-n1\_i}, u\right), n0\_i\right)} \]

12. Final simplification 98.6%

    \[\leadsto \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(u, -\frac{n0\_i}{n1\_i}, u\right), n0\_i\right) \]
13. Add Preprocessing

Alternative 7: 98.1% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, n1\_i - n0\_i, n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i) :precision binary32 (fma u (- n1_i n0_i) n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, (n1_i - n0_i), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, Float32(n1_i - n0_i), n0_i)
end

Derivation

1. Initial program 98.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) + n0\_i} \]

   2. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, -1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right), n0\_i\right)} \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(u, -0.5 \cdot \left(normAngle \cdot normAngle\right), \frac{normAngle \cdot \cos normAngle}{-\sin normAngle}\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right)} \]

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i + -1 \cdot n0\_i}, n0\_i\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(u, n1\_i + \color{blue}{\left(\mathsf{neg}\left(n0\_i\right)\right)}, n0\_i\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i - n0\_i}, n0\_i\right) \]

   3. lower--.f32 98.6

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i - n0\_i}, n0\_i\right) \]

8. Simplified 98.6%

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i - n0\_i}, n0\_i\right) \]
9. Add Preprocessing

Alternative 8: 81.4% accurate, 65.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(n1\_i, u, n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i) :precision binary32 (fma n1_i u n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(n1_i, u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(n1_i, u, n0_i)
end

Derivation

1. Initial program 98.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{1} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
4. Step-by-step derivation

5. Simplified 82.5%

   \[\leadsto \color{blue}{1} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

6. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i + n1\_i \cdot u} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{n1\_i \cdot u + n0\_i} \]

   2. lower-fma.f32 82.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(n1\_i, u, n0\_i\right)} \]

8. Simplified 82.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(n1\_i, u, n0\_i\right)} \]
9. Add Preprocessing

Alternative 9: 38.1% accurate, 76.5× speedup

Math

\[\begin{array}{l} \\ u \cdot n1\_i \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i) :precision binary32 (* u n1_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return u * n1_i;
}

Fortran

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 = u * n1_i
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(u * n1_i)
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = u * n1_i;
end

Derivation

1. Initial program 98.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
4. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(n0\_i, 1 - u, n1\_i \cdot u\right)} \]

   2. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(n0\_i, \color{blue}{1 - u}, n1\_i \cdot u\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \color{blue}{u \cdot n1\_i}\right) \]

   4. lower-*.f32 98.3

      \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \color{blue}{u \cdot n1\_i}\right) \]

5. Simplified 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right)} \]

6. Taylor expanded in n0_i around 0

   \[\leadsto \color{blue}{n1\_i \cdot u} \]
7. Step-by-step derivation

   1. lower-*.f32 37.0

      \[\leadsto \color{blue}{n1\_i \cdot u} \]

8. Simplified 37.0%

   \[\leadsto \color{blue}{n1\_i \cdot u} \]

9. Final simplification 37.0%

   \[\leadsto u \cdot n1\_i \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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)))