Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.4%

Time: 16.3s

Alternatives: 13

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: 99.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\sin normAngle}\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   n0_i
   (fma 0.3333333333333333 (* normAngle normAngle) -1.0)
   (* n1_i (/ normAngle (sin normAngle))))
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, fmaf(n0_i, fmaf(0.3333333333333333f, (normAngle * normAngle), -1.0f), (n1_i * (normAngle / sinf(normAngle)))), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, fma(n0_i, fma(Float32(0.3333333333333333), Float32(normAngle * normAngle), Float32(-1.0)), Float32(n1_i * Float32(normAngle / sin(normAngle)))), n0_i)
end

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. Add Preprocessing

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 97.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle, normAngle \cdot \left(\mathsf{fma}\left(u, 0.16666666666666666, -0.16666666666666666\right) \cdot \mathsf{fma}\left(1 - u, 1 - u, -1\right)\right), 1 - u\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-/l* N/A

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

   10. *-lowering-*.f32 N/A

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

   11. /-lowering-/.f32 N/A

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

   12. sin-lowering-sin.f32 99.4

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\color{blue}{\sin normAngle}}\right), n0\_i\right) \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\sin normAngle}\right), n0\_i\right)} \]
9. Add Preprocessing

Alternative 2: 99.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\left(normAngle \cdot normAngle\right) \cdot \left(0 - \mathsf{fma}\left(n1\_i, -0.0032407407407407406, n1\_i \cdot 0.0011904761904761906\right)\right), u, \left(u \cdot n1\_i\right) \cdot 0.019444444444444445\right), u \cdot \mathsf{fma}\left(n1\_i, 0.16666666666666666, n0\_i \cdot 0.3333333333333333\right)\right), \mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (* normAngle normAngle)
  (fma
   (* normAngle normAngle)
   (fma
    (*
     (* normAngle normAngle)
     (- 0.0 (fma n1_i -0.0032407407407407406 (* n1_i 0.0011904761904761906))))
    u
    (* (* u n1_i) 0.019444444444444445))
   (* u (fma n1_i 0.16666666666666666 (* n0_i 0.3333333333333333))))
  (fma (- n1_i n0_i) u n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), fmaf(((normAngle * normAngle) * (0.0f - fmaf(n1_i, -0.0032407407407407406f, (n1_i * 0.0011904761904761906f)))), u, ((u * n1_i) * 0.019444444444444445f)), (u * fmaf(n1_i, 0.16666666666666666f, (n0_i * 0.3333333333333333f)))), fmaf((n1_i - n0_i), u, n0_i));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), fma(Float32(Float32(normAngle * normAngle) * Float32(Float32(0.0) - fma(n1_i, Float32(-0.0032407407407407406), Float32(n1_i * Float32(0.0011904761904761906))))), u, Float32(Float32(u * n1_i) * Float32(0.019444444444444445))), Float32(u * fma(n1_i, Float32(0.16666666666666666), Float32(n0_i * Float32(0.3333333333333333))))), fma(Float32(n1_i - n0_i), u, n0_i))
end

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. Add Preprocessing

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 97.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle, normAngle \cdot \left(\mathsf{fma}\left(u, 0.16666666666666666, -0.16666666666666666\right) \cdot \mathsf{fma}\left(1 - u, 1 - u, -1\right)\right), 1 - u\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-/l* N/A

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

   10. *-lowering-*.f32 N/A

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

   11. /-lowering-/.f32 N/A

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

   12. sin-lowering-sin.f32 99.4

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\color{blue}{\sin normAngle}}\right), n0\_i\right) \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\sin normAngle}\right), n0\_i\right)} \]

9. Taylor expanded in normAngle around 0

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

11. Simplified 98.8%

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

12. Final simplification 98.8%

    \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\left(normAngle \cdot normAngle\right) \cdot \left(0 - \mathsf{fma}\left(n1\_i, -0.0032407407407407406, n1\_i \cdot 0.0011904761904761906\right)\right), u, \left(u \cdot n1\_i\right) \cdot 0.019444444444444445\right), u \cdot \mathsf{fma}\left(n1\_i, 0.16666666666666666, n0\_i \cdot 0.3333333333333333\right)\right), \mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right)\right) \]
13. Add Preprocessing

Alternative 3: 99.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle, normAngle \cdot 0.00205026455026455, 0.019444444444444445\right), 0.16666666666666666\right), 1\right)\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   n0_i
   (fma 0.3333333333333333 (* normAngle normAngle) -1.0)
   (*
    n1_i
    (fma
     (* normAngle normAngle)
     (fma
      (* normAngle normAngle)
      (fma normAngle (* normAngle 0.00205026455026455) 0.019444444444444445)
      0.16666666666666666)
     1.0)))
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, fmaf(n0_i, fmaf(0.3333333333333333f, (normAngle * normAngle), -1.0f), (n1_i * fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), fmaf(normAngle, (normAngle * 0.00205026455026455f), 0.019444444444444445f), 0.16666666666666666f), 1.0f))), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, fma(n0_i, fma(Float32(0.3333333333333333), Float32(normAngle * normAngle), Float32(-1.0)), Float32(n1_i * fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), fma(normAngle, Float32(normAngle * Float32(0.00205026455026455)), Float32(0.019444444444444445)), Float32(0.16666666666666666)), Float32(1.0)))), n0_i)
end

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. Add Preprocessing

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 97.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle, normAngle \cdot \left(\mathsf{fma}\left(u, 0.16666666666666666, -0.16666666666666666\right) \cdot \mathsf{fma}\left(1 - u, 1 - u, -1\right)\right), 1 - u\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-/l* N/A

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

   10. *-lowering-*.f32 N/A

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

   11. /-lowering-/.f32 N/A

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

   12. sin-lowering-sin.f32 99.4

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\color{blue}{\sin normAngle}}\right), n0\_i\right) \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\sin normAngle}\right), n0\_i\right)} \]

9. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(\frac{1}{3}, normAngle \cdot normAngle, -1\right), n1\_i \cdot \color{blue}{\left(1 + {normAngle}^{2} \cdot \left(\frac{1}{6} + {normAngle}^{2} \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {normAngle}^{2}\right)\right)\right)}\right), n0\_i\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

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

    2. accelerator-lowering-fma.f32 N/A

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

    3. unpow2 N/A

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

    4. *-lowering-*.f32 N/A

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

    5. +-commutative N/A

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

    6. accelerator-lowering-fma.f32 N/A

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

    7. unpow2 N/A

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

    8. *-lowering-*.f32 N/A

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

    9. +-commutative N/A

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

    10. *-commutative N/A

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

    11. unpow2 N/A

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

    12. associate-*l* N/A

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

    13. accelerator-lowering-fma.f32 N/A

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

    14. *-lowering-*.f32 98.8

        \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle, \color{blue}{normAngle \cdot 0.00205026455026455}, 0.019444444444444445\right), 0.16666666666666666\right), 1\right)\right), n0\_i\right) \]

11. Simplified 98.8%

    \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle, normAngle \cdot 0.00205026455026455, 0.019444444444444445\right), 0.16666666666666666\right), 1\right)}\right), n0\_i\right) \]
12. Add Preprocessing

Alternative 4: 99.2% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

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. Add Preprocessing

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 97.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle, normAngle \cdot \left(\mathsf{fma}\left(u, 0.16666666666666666, -0.16666666666666666\right) \cdot \mathsf{fma}\left(1 - u, 1 - u, -1\right)\right), 1 - u\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-/l* N/A

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

   10. *-lowering-*.f32 N/A

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

   11. /-lowering-/.f32 N/A

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

   12. sin-lowering-sin.f32 99.4

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\color{blue}{\sin normAngle}}\right), n0\_i\right) \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\sin normAngle}\right), n0\_i\right)} \]

9. Taylor expanded in normAngle around 0

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

11. Simplified 98.8%

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

12. Taylor expanded in n1_i around inf

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

14. Simplified 98.6%

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

Alternative 5: 99.3% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* normAngle normAngle)
   (fma
    (* normAngle normAngle)
    (* n1_i 0.019444444444444445)
    (fma n1_i 0.16666666666666666 (* n0_i 0.3333333333333333)))
   (- 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), (n1_i * 0.019444444444444445f), fmaf(n1_i, 0.16666666666666666f, (n0_i * 0.3333333333333333f))), (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), Float32(n1_i * Float32(0.019444444444444445)), fma(n1_i, Float32(0.16666666666666666), Float32(n0_i * Float32(0.3333333333333333)))), Float32(n1_i - n0_i)), n0_i)
end

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. Add Preprocessing

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 97.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle, normAngle \cdot \left(\mathsf{fma}\left(u, 0.16666666666666666, -0.16666666666666666\right) \cdot \mathsf{fma}\left(1 - u, 1 - u, -1\right)\right), 1 - u\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-/l* N/A

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

   10. *-lowering-*.f32 N/A

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

   11. /-lowering-/.f32 N/A

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

   12. sin-lowering-sin.f32 99.4

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\color{blue}{\sin normAngle}}\right), n0\_i\right) \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\sin normAngle}\right), n0\_i\right)} \]

9. Taylor expanded in normAngle around 0

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

    1. associate-+r+ N/A

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

    2. +-commutative N/A

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

    3. accelerator-lowering-fma.f32 N/A

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

11. Simplified 98.6%

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

Alternative 6: 99.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

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. Add Preprocessing

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 97.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle, normAngle \cdot \left(\mathsf{fma}\left(u, 0.16666666666666666, -0.16666666666666666\right) \cdot \mathsf{fma}\left(1 - u, 1 - u, -1\right)\right), 1 - u\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-/l* N/A

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

   10. *-lowering-*.f32 N/A

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

   11. /-lowering-/.f32 N/A

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

   12. sin-lowering-sin.f32 99.4

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\color{blue}{\sin normAngle}}\right), n0\_i\right) \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\sin normAngle}\right), n0\_i\right)} \]

9. Taylor expanded in normAngle around 0

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

11. Simplified 98.8%

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

12. Taylor expanded in normAngle around 0

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

    1. *-lowering-*.f32 N/A

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

    2. *-commutative N/A

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

    3. accelerator-lowering-fma.f32 N/A

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

    4. *-lowering-*.f32 98.2

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

14. Simplified 98.2%

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

15. Final simplification 98.2%

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

Alternative 7: 99.1% accurate, 14.3× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* normAngle normAngle)
   (fma n1_i 0.16666666666666666 (* n0_i 0.3333333333333333))
   (- 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(n1_i, 0.16666666666666666f, (n0_i * 0.3333333333333333f)), (n1_i - n0_i)), n0_i);
}

Julia

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

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. Add Preprocessing

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 97.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle, normAngle \cdot \left(\mathsf{fma}\left(u, 0.16666666666666666, -0.16666666666666666\right) \cdot \mathsf{fma}\left(1 - u, 1 - u, -1\right)\right), 1 - u\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-/l* N/A

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

   10. *-lowering-*.f32 N/A

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

   11. /-lowering-/.f32 N/A

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

   12. sin-lowering-sin.f32 99.4

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\color{blue}{\sin normAngle}}\right), n0\_i\right) \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\sin normAngle}\right), n0\_i\right)} \]

9. Taylor expanded in normAngle around 0

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

    1. associate-+r+ N/A

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

    2. cancel-sign-sub-inv N/A

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

    3. metadata-eval N/A

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

    4. +-commutative N/A

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

    5. +-commutative N/A

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

    6. accelerator-lowering-fma.f32 N/A

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

    7. unpow2 N/A

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

    8. *-lowering-*.f32 N/A

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

    9. *-commutative N/A

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

    10. accelerator-lowering-fma.f32 N/A

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

    11. *-commutative N/A

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

    12. *-lowering-*.f32 N/A

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

    13. mul-1-neg N/A

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

    14. unsub-neg N/A

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

    15. --lowering--.f32 98.2

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

11. Simplified 98.2%

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

Alternative 8: 98.3% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Julia

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

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. Add Preprocessing

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 97.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle, normAngle \cdot \left(\mathsf{fma}\left(u, 0.16666666666666666, -0.16666666666666666\right) \cdot \mathsf{fma}\left(1 - u, 1 - u, -1\right)\right), 1 - u\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-/l* N/A

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

   10. *-lowering-*.f32 N/A

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

   11. /-lowering-/.f32 N/A

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

   12. sin-lowering-sin.f32 99.4

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\color{blue}{\sin normAngle}}\right), n0\_i\right) \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\sin normAngle}\right), n0\_i\right)} \]

9. Taylor expanded in normAngle around 0

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

11. Simplified 98.8%

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

12. Taylor expanded in n1_i around 0

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. *-lowering-*.f32 N/A

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

    4. *-lowering-*.f32 97.6

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

14. Simplified 97.6%

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

15. Final simplification 97.6%

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

Alternative 9: 98.3% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma n0_i (fma 0.3333333333333333 (* normAngle normAngle) -1.0) n1_i)
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, fmaf(n0_i, fmaf(0.3333333333333333f, (normAngle * normAngle), -1.0f), n1_i), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, fma(n0_i, fma(Float32(0.3333333333333333), Float32(normAngle * normAngle), Float32(-1.0)), n1_i), n0_i)
end

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. Add Preprocessing

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 97.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle, normAngle \cdot \left(\mathsf{fma}\left(u, 0.16666666666666666, -0.16666666666666666\right) \cdot \mathsf{fma}\left(1 - u, 1 - u, -1\right)\right), 1 - u\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-/l* N/A

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

   10. *-lowering-*.f32 N/A

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

   11. /-lowering-/.f32 N/A

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

   12. sin-lowering-sin.f32 99.4

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\color{blue}{\sin normAngle}}\right), n0\_i\right) \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\sin normAngle}\right), n0\_i\right)} \]

9. Taylor expanded in normAngle around 0

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

11. Simplified 97.6%

    \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), \color{blue}{n1\_i}\right), n0\_i\right) \]
12. Add Preprocessing

Alternative 10: 60.1% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -3.99999987306209 \cdot 10^{-20}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -3.99999987306209e-20)
   n0_i
   (if (<= n0_i 3.999999935100636e-17) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -3.99999987306209e-20f) {
		tmp = n0_i;
	} else if (n0_i <= 3.999999935100636e-17f) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

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) :: tmp
    if (n0_i <= (-3.99999987306209e-20)) then
        tmp = n0_i
    else if (n0_i <= 3.999999935100636e-17) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-3.99999987306209e-20))
		tmp = n0_i;
	elseif (n0_i <= Float32(3.999999935100636e-17))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-3.99999987306209e-20))
		tmp = n0_i;
	elseif (n0_i <= single(3.999999935100636e-17))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -3.99999987e-20 or 3.99999994e-17 < n0_i

   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

      \[\leadsto \color{blue}{n0\_i} \]
   4. Step-by-step derivation

   5. Simplified 62.6%

      \[\leadsto \color{blue}{n0\_i} \]

   if -3.99999987e-20 < n0_i < 3.99999994e-17

   1. Initial program 96.7%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 86.5%

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

   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f32 85.8

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

   8. Simplified 85.8%

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

   9. Taylor expanded in u around inf

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

       1. *-lowering-*.f32 61.1

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

   11. Simplified 61.1%

       \[\leadsto \color{blue}{n1\_i \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -3.99999987306209 \cdot 10^{-20}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.2% accurate, 45.9× speedup

Math

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

FPCore

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

C

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

Julia

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

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. Add Preprocessing

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

5. Simplified 97.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle, normAngle \cdot \left(\mathsf{fma}\left(u, 0.16666666666666666, -0.16666666666666666\right) \cdot \mathsf{fma}\left(1 - u, 1 - u, -1\right)\right), 1 - u\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-/l* N/A

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

   10. *-lowering-*.f32 N/A

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

   11. /-lowering-/.f32 N/A

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

   12. sin-lowering-sin.f32 99.4

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\color{blue}{\sin normAngle}}\right), n0\_i\right) \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(0.3333333333333333, normAngle \cdot normAngle, -1\right), n1\_i \cdot \frac{normAngle}{\sin normAngle}\right), n0\_i\right)} \]

9. Taylor expanded in normAngle around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. accelerator-lowering-fma.f32 N/A

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

    4. mul-1-neg N/A

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

    5. unsub-neg N/A

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

    6. --lowering--.f32 97.4

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

11. Simplified 97.4%

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

Alternative 12: 82.1% accurate, 65.6× speedup

Math

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

FPCore

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

C

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

Julia

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

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. Add Preprocessing

3. Taylor expanded in u around 0

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

5. Simplified 82.1%

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

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. *-commutative N/A

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

   3. accelerator-lowering-fma.f32 82.0

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

8. Simplified 82.0%

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

Alternative 13: 47.4% accurate, 459.0× speedup

Math

\[\begin{array}{l} \\ n0\_i \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_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 = n0_i
end function

Julia

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

MATLAB

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

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. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i} \]
4. Step-by-step derivation

5. Simplified 43.8%

   \[\leadsto \color{blue}{n0\_i} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(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)))