Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.2%

Time: 15.6s

Alternatives: 12

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: 99.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, 0.00205026455026455, 0.019444444444444445\right), 0.16666666666666666\right)\\ t_1 := n1\_i \cdot \left(normAngle \cdot normAngle\right)\\ \mathsf{fma}\left(u, \mathsf{fma}\left(\left(t\_1 \cdot -0.16666666666666666\right) \cdot \left(u \cdot u\right), \mathsf{fma}\left(normAngle \cdot normAngle, t\_0, 1\right), \mathsf{fma}\left(t\_1, t\_0, n1\_i\right) - n0\_i\right), n0\_i\right) \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0
         (fma
          (* normAngle normAngle)
          (fma
           (* normAngle normAngle)
           0.00205026455026455
           0.019444444444444445)
          0.16666666666666666))
        (t_1 (* n1_i (* normAngle normAngle))))
   (fma
    u
    (fma
     (* (* t_1 -0.16666666666666666) (* u u))
     (fma (* normAngle normAngle) t_0 1.0)
     (- (fma t_1 t_0 n1_i) n0_i))
    n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), 0.00205026455026455f, 0.019444444444444445f), 0.16666666666666666f);
	float t_1 = n1_i * (normAngle * normAngle);
	return fmaf(u, fmaf(((t_1 * -0.16666666666666666f) * (u * u)), fmaf((normAngle * normAngle), t_0, 1.0f), (fmaf(t_1, t_0, n1_i) - n0_i)), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), Float32(0.00205026455026455), Float32(0.019444444444444445)), Float32(0.16666666666666666))
	t_1 = Float32(n1_i * Float32(normAngle * normAngle))
	return fma(u, fma(Float32(Float32(t_1 * Float32(-0.16666666666666666)) * Float32(u * u)), fma(Float32(normAngle * normAngle), t_0, Float32(1.0)), Float32(fma(t_1, t_0, n1_i) - n0_i)), n0_i)
end

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

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

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

   6. neg-lowering-neg.f32 97.4

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

5. Simplified 97.4%

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

6. Taylor expanded in normAngle around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

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

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 97.1

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

8. Simplified 97.1%

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

9. Taylor expanded in u around 0

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

11. Simplified 99.3%

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

Alternative 2: 99.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(-0.16666666666666666, u \cdot u, 0.16666666666666666\right), \left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot \mathsf{fma}\left(u \cdot u, -0.027777777777777776, 0.019444444444444445\right)\right)\right), n1\_i - n0\_i\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* normAngle normAngle)
   (fma
    n1_i
    (fma -0.16666666666666666 (* u u) 0.16666666666666666)
    (*
     (* normAngle normAngle)
     (* n1_i (fma (* u u) -0.027777777777777776 0.019444444444444445))))
   (- 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, fmaf(-0.16666666666666666f, (u * u), 0.16666666666666666f), ((normAngle * normAngle) * (n1_i * fmaf((u * u), -0.027777777777777776f, 0.019444444444444445f)))), (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, fma(Float32(-0.16666666666666666), Float32(u * u), Float32(0.16666666666666666)), Float32(Float32(normAngle * normAngle) * Float32(n1_i * fma(Float32(u * u), Float32(-0.027777777777777776), Float32(0.019444444444444445))))), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

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

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

   6. neg-lowering-neg.f32 97.4

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

5. Simplified 97.4%

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

6. Taylor expanded in normAngle around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

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

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 97.1

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

8. Simplified 97.1%

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

9. Taylor expanded in u around 0

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

11. Simplified 99.3%

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

12. Taylor expanded in normAngle around 0

    \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left(n1\_i + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{2}\right) + \left(\frac{1}{6} \cdot n1\_i + {normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot \left(n1\_i \cdot {u}^{2}\right) + \frac{7}{360} \cdot n1\_i\right)\right)\right)\right) - n0\_i}, n0\_i\right) \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left({normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{2}\right) + \left(\frac{1}{6} \cdot n1\_i + {normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot \left(n1\_i \cdot {u}^{2}\right) + \frac{7}{360} \cdot n1\_i\right)\right)\right) + n1\_i\right)} - n0\_i, n0\_i\right) \]

    2. associate--l+ N/A

       \[\leadsto \mathsf{fma}\left(u, \color{blue}{{normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{2}\right) + \left(\frac{1}{6} \cdot n1\_i + {normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot \left(n1\_i \cdot {u}^{2}\right) + \frac{7}{360} \cdot n1\_i\right)\right)\right) + \left(n1\_i - n0\_i\right)}, n0\_i\right) \]

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

       \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left({normAngle}^{2}, \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{2}\right) + \left(\frac{1}{6} \cdot n1\_i + {normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot \left(n1\_i \cdot {u}^{2}\right) + \frac{7}{360} \cdot n1\_i\right)\right), n1\_i - n0\_i\right)}, n0\_i\right) \]

14. Simplified 99.2%

    \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(-0.16666666666666666, u \cdot u, 0.16666666666666666\right), \left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot \mathsf{fma}\left(u \cdot u, -0.027777777777777776, 0.019444444444444445\right)\right)\right), n1\_i - n0\_i\right)}, n0\_i\right) \]
15. Add Preprocessing

Alternative 3: 99.1% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

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

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

   6. neg-lowering-neg.f32 97.4

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

5. Simplified 97.4%

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

6. Taylor expanded in normAngle around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

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

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 97.1

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

8. Simplified 97.1%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

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

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

11. Simplified 99.1%

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

Alternative 4: 98.9% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

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

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

   6. neg-lowering-neg.f32 97.4

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

5. Simplified 97.4%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-+l+ N/A

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

   7. *-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

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

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

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

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

   \[\leadsto \mathsf{fma}\left(n1\_i - n0\_i, u, \mathsf{fma}\left(normAngle \cdot normAngle, -0.16666666666666666 \cdot \left(\left(u \cdot n1\_i\right) \cdot \mathsf{fma}\left(u, u, -1\right)\right), n0\_i\right)\right) \]
10. Add Preprocessing

Alternative 5: 98.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

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

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

   6. neg-lowering-neg.f32 97.4

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

5. Simplified 97.4%

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

6. Taylor expanded in normAngle around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. +-commutative N/A

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

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

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 97.1

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

8. Simplified 97.1%

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

9. Taylor expanded in u around 0

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

11. Simplified 99.3%

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

12. Taylor expanded in normAngle around 0

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

    1. +-commutative N/A

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

    2. associate--l+ N/A

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

    3. unpow2 N/A

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

    4. associate-*l* N/A

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

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

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

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

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

    7. associate-*r* N/A

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

    8. *-commutative N/A

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

    9. associate-*l* N/A

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

    10. *-commutative N/A

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

    11. distribute-lft-out N/A

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

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

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

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

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

    14. unpow2 N/A

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

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

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

    16. --lowering--.f32 98.9

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

14. Simplified 98.9%

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

Alternative 6: 99.0% accurate, 14.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.3%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

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

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

8. Simplified 98.7%

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

Alternative 7: 98.8% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

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

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

   6. neg-lowering-neg.f32 97.4

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

5. Simplified 97.4%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-+l+ N/A

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

   7. *-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

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

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

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

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

8. Simplified 98.9%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

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

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

    3. +-commutative N/A

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

    4. associate--l+ N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

    7. associate-*l* N/A

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

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

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

    9. *-commutative N/A

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

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

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

    11. unpow2 N/A

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

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

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

    13. --lowering--.f32 98.7

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

11. Simplified 98.7%

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

Alternative 8: 61.3% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 2.5000000488537034 \cdot 10^{-26}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.999999936531045e-20)
   n0_i
   (if (<= n0_i 2.5000000488537034e-26) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.999999936531045e-20f) {
		tmp = n0_i;
	} else if (n0_i <= 2.5000000488537034e-26f) {
		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 <= (-1.999999936531045e-20)) then
        tmp = n0_i
    else if (n0_i <= 2.5000000488537034e-26) 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(-1.999999936531045e-20))
		tmp = n0_i;
	elseif (n0_i <= Float32(2.5000000488537034e-26))
		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(-1.999999936531045e-20))
		tmp = n0_i;
	elseif (n0_i <= single(2.5000000488537034e-26))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.99999994e-20 or 2.50000005e-26 < n0_i

   1. Initial program 97.8%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 59.0%

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

   if -1.99999994e-20 < n0_i < 2.50000005e-26

   1. Initial program 95.3%

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

   3. Taylor expanded in normAngle around 0

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

   5. Simplified 97.6%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 89.3%

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

   9. Taylor expanded in u around inf

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

       1. *-lowering-*.f32 70.2

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

   11. Simplified 70.2%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 2.5000000488537034 \cdot 10^{-26}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.3% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(n0\_i, -u, n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, n1\_i, n0\_i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -5.00000006675716e-11) (fma n0_i (- u) n0_i) (fma u n1_i n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -5.00000006675716e-11f) {
		tmp = fmaf(n0_i, -u, n0_i);
	} else {
		tmp = fmaf(u, n1_i, n0_i);
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-5.00000006675716e-11))
		tmp = fma(n0_i, Float32(-u), n0_i);
	else
		tmp = fma(u, n1_i, n0_i);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -5.00000007e-11

   1. Initial program 98.3%

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

   3. Taylor expanded in n0_i around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-lft-identity N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. neg-lowering-neg.f32 N/A

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

      15. /-lowering-/.f32 N/A

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

      16. sin-lowering-sin.f32 91.0

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

   5. Simplified 91.0%

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

   6. Taylor expanded in normAngle around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f32 91.6

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

   8. Simplified 91.6%

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

   if -5.00000007e-11 < n0_i

   1. Initial program 96.5%

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

   3. Taylor expanded in normAngle around 0

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

   5. Simplified 98.7%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 82.9%

      \[\leadsto \mathsf{fma}\left(u, n1\_i, \color{blue}{n0\_i}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 98.1% 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 96.8%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.3%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + -1 \cdot n0\_i\right)} \]
7. 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 97.6

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

8. Simplified 97.6%

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

Alternative 11: 82.0% 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 96.8%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.8%

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

6. Taylor expanded in u around 0

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

8. Simplified 79.8%

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

Alternative 12: 47.1% 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 96.8%

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

3. Taylor expanded in u around 0

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

5. Simplified 43.3%

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

Reproduce

herbie shell --seed 2024199 
(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)))