Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 99.3%

Time: 15.6s

Alternatives: 15

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: 97.1% 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: 99.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, 0.0021164021164021165, 0.022222222222222223\right), 0.3333333333333333\right), -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   n0_i
   (fma
    (* normAngle normAngle)
    (fma
     (* normAngle normAngle)
     (fma (* normAngle normAngle) 0.0021164021164021165 0.022222222222222223)
     0.3333333333333333)
    -1.0)
   (* normAngle (/ n1_i (sin normAngle))))
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, fmaf(n0_i, fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), 0.0021164021164021165f, 0.022222222222222223f), 0.3333333333333333f), -1.0f), (normAngle * (n1_i / sinf(normAngle)))), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, fma(n0_i, fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), Float32(0.0021164021164021165), Float32(0.022222222222222223)), Float32(0.3333333333333333)), Float32(-1.0)), Float32(normAngle * Float32(n1_i / sin(normAngle)))), n0_i)
end

Derivation

1. Initial program 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. 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. lower-fma.f32 N/A

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

5. Simplified 99.5%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{{normAngle}^{2} \cdot \left(\frac{1}{3} + {normAngle}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {normAngle}^{2}\right)\right) - 1}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{{normAngle}^{2} \cdot \left(\frac{1}{3} + {normAngle}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {normAngle}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

   2. metadata-eval N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, \frac{2}{945}, \frac{1}{45}\right), \frac{1}{3}\right), -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

   14. lower-*.f32 99.5

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, 0.0021164021164021165, 0.022222222222222223\right), 0.3333333333333333\right), -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

8. Simplified 99.5%

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, 0.0021164021164021165, 0.022222222222222223\right), 0.3333333333333333\right), -1\right)}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]
9. Add Preprocessing

Alternative 2: 99.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, n0\_i \cdot 0.0021164021164021165 - \mathsf{fma}\left(n1\_i, 0.0011904761904761906, n1\_i \cdot -0.0032407407407407406\right), \mathsf{fma}\left(n1\_i, 0.019444444444444445, n0\_i \cdot 0.022222222222222223\right)\right), \mathsf{fma}\left(n1\_i, 0.16666666666666666, n0\_i \cdot 0.3333333333333333\right)\right), n1\_i - n0\_i\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* normAngle normAngle)
   (fma
    (* normAngle normAngle)
    (fma
     (* normAngle normAngle)
     (-
      (* n0_i 0.0021164021164021165)
      (fma n1_i 0.0011904761904761906 (* n1_i -0.0032407407407407406)))
     (fma n1_i 0.019444444444444445 (* n0_i 0.022222222222222223)))
    (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), fmaf((normAngle * normAngle), ((n0_i * 0.0021164021164021165f) - fmaf(n1_i, 0.0011904761904761906f, (n1_i * -0.0032407407407407406f))), fmaf(n1_i, 0.019444444444444445f, (n0_i * 0.022222222222222223f))), 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), fma(Float32(normAngle * normAngle), Float32(Float32(n0_i * Float32(0.0021164021164021165)) - fma(n1_i, Float32(0.0011904761904761906), Float32(n1_i * Float32(-0.0032407407407407406)))), fma(n1_i, Float32(0.019444444444444445), Float32(n0_i * Float32(0.022222222222222223)))), fma(n1_i, Float32(0.16666666666666666), Float32(n0_i * Float32(0.3333333333333333)))), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in 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. lower-fma.f32 N/A

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

5. Simplified 99.5%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{{normAngle}^{2} \cdot \left(\frac{1}{3} + {normAngle}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {normAngle}^{2}\right)\right) - 1}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{{normAngle}^{2} \cdot \left(\frac{1}{3} + {normAngle}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {normAngle}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

   2. metadata-eval N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, \frac{2}{945}, \frac{1}{45}\right), \frac{1}{3}\right), -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

   14. lower-*.f32 99.5

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, 0.0021164021164021165, 0.022222222222222223\right), 0.3333333333333333\right), -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

8. Simplified 99.5%

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, 0.0021164021164021165, 0.022222222222222223\right), 0.3333333333333333\right), -1\right)}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

9. Taylor expanded in normAngle around 0

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

11. Simplified 99.2%

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

Alternative 3: 99.2% accurate, 8.5× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* normAngle normAngle)
   (fma
    n1_i
    0.16666666666666666
    (fma
     normAngle
     (*
      normAngle
      (fma n1_i 0.019444444444444445 (* n0_i 0.022222222222222223)))
     (* 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, fmaf(normAngle, (normAngle * fmaf(n1_i, 0.019444444444444445f, (n0_i * 0.022222222222222223f))), (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), fma(normAngle, Float32(normAngle * fma(n1_i, Float32(0.019444444444444445), Float32(n0_i * Float32(0.022222222222222223)))), Float32(n0_i * Float32(0.3333333333333333)))), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. 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. lower-fma.f32 N/A

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

5. Simplified 99.5%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{{normAngle}^{2} \cdot \left(\frac{1}{3} + {normAngle}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {normAngle}^{2}\right)\right) - 1}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{{normAngle}^{2} \cdot \left(\frac{1}{3} + {normAngle}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {normAngle}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

   2. metadata-eval N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, \frac{2}{945}, \frac{1}{45}\right), \frac{1}{3}\right), -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

   14. lower-*.f32 99.5

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, 0.0021164021164021165, 0.022222222222222223\right), 0.3333333333333333\right), -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

8. Simplified 99.5%

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, 0.0021164021164021165, 0.022222222222222223\right), 0.3333333333333333\right), -1\right)}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

9. Taylor expanded in normAngle around 0

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

    1. associate-+r+ N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f32 N/A

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

11. Simplified 98.9%

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

Alternative 4: 99.1% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

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

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

   12. metadata-eval N/A

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

   13. distribute-lft-out N/A

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

   14. lower-*.f32 N/A

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

   15. lower-fma.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 97.8

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

5. Simplified 97.8%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

8. Simplified 98.4%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f32 N/A

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

11. Simplified 98.5%

    \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, -0.5 \cdot \left(normAngle \cdot \left(normAngle \cdot u\right)\right), \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(normAngle \cdot normAngle, 0.16666666666666666, 1\right), 0.3333333333333333 \cdot \left(n0\_i \cdot \left(normAngle \cdot normAngle\right)\right)\right) - n0\_i\right), n0\_i\right)} \]

12. Taylor expanded in normAngle around 0

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

    1. associate-+r+ N/A

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

    2. +-commutative N/A

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

    3. associate-*r* N/A

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

    4. lower-fma.f32 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f32 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f32 N/A

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

    9. associate-+r+ N/A

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

    10. +-commutative N/A

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

    11. *-commutative N/A

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

    12. lower-fma.f32 N/A

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

    13. *-commutative N/A

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

    14. associate-*l* N/A

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

    15. lower-fma.f32 N/A

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

    16. lower-*.f32 N/A

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

    17. *-commutative N/A

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

    18. lower-*.f32 N/A

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

14. Simplified 98.5%

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

Alternative 5: 99.1% accurate, 10.7× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* normAngle normAngle)
   (fma
    n0_i
    0.3333333333333333
    (fma n0_i (* u -0.5) (* 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, fmaf(n0_i, (u * -0.5f), (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), fma(n0_i, Float32(u * Float32(-0.5)), Float32(n1_i * Float32(0.16666666666666666)))), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

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

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

   12. metadata-eval N/A

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

   13. distribute-lft-out N/A

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

   14. lower-*.f32 N/A

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

   15. lower-fma.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 97.8

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

5. Simplified 97.8%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

8. Simplified 98.4%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f32 N/A

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

11. Simplified 98.5%

    \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, -0.5 \cdot \left(normAngle \cdot \left(normAngle \cdot u\right)\right), \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(normAngle \cdot normAngle, 0.16666666666666666, 1\right), 0.3333333333333333 \cdot \left(n0\_i \cdot \left(normAngle \cdot normAngle\right)\right)\right) - n0\_i\right), n0\_i\right)} \]

12. Taylor expanded in normAngle around 0

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

    1. +-commutative N/A

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

    2. associate--l+ N/A

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

    3. lower-fma.f32 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f32 N/A

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

    6. associate-+r+ N/A

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

    7. +-commutative N/A

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

    8. *-commutative N/A

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

    9. lower-fma.f32 N/A

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

    10. *-commutative N/A

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

    11. associate-*l* N/A

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

    12. lower-fma.f32 N/A

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

    13. lower-*.f32 N/A

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

    14. *-commutative N/A

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

    15. lower-*.f32 N/A

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

    16. lower--.f32 98.5

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

14. Simplified 98.5%

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

Alternative 6: 99.0% accurate, 12.4× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. 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. lower-fma.f32 N/A

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

5. Simplified 99.5%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. lower--.f32 N/A

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

   8. lower-fma.f32 N/A

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

8. Simplified 98.5%

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

Alternative 7: 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 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. 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. lower-fma.f32 N/A

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

5. Simplified 99.5%

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

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. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

8. Simplified 98.5%

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

Alternative 8: 98.8% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. 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. lower-fma.f32 N/A

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

5. Simplified 99.5%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{{normAngle}^{2} \cdot \left(\frac{1}{3} + {normAngle}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {normAngle}^{2}\right)\right) - 1}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{{normAngle}^{2} \cdot \left(\frac{1}{3} + {normAngle}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {normAngle}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

   2. metadata-eval N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, \frac{2}{945}, \frac{1}{45}\right), \frac{1}{3}\right), -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

   14. lower-*.f32 99.5

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, 0.0021164021164021165, 0.022222222222222223\right), 0.3333333333333333\right), -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

8. Simplified 99.5%

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, 0.0021164021164021165, 0.022222222222222223\right), 0.3333333333333333\right), -1\right)}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

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. +-commutative N/A

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

    3. *-commutative N/A

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

    4. unpow2 N/A

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

    5. associate-*r* N/A

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

    6. lower-fma.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. sub-neg N/A

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

    9. +-commutative N/A

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

    10. *-commutative N/A

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

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

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

    12. metadata-eval N/A

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

    13. lower-fma.f32 N/A

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

    14. *-commutative N/A

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

    15. lower-*.f32 N/A

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

    16. mul-1-neg N/A

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

    17. unsub-neg N/A

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

    18. lower--.f32 98.5

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

11. Simplified 98.5%

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

12. Taylor expanded in n1_i around inf

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

    1. *-commutative N/A

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

    2. lower-*.f32 98.3

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

14. Simplified 98.3%

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

15. Final simplification 98.3%

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

Alternative 9: 98.3% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. 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. lower-fma.f32 N/A

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

5. Simplified 99.5%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{{normAngle}^{2} \cdot \left(\frac{1}{3} + {normAngle}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {normAngle}^{2}\right)\right) - 1}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{{normAngle}^{2} \cdot \left(\frac{1}{3} + {normAngle}^{2} \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {normAngle}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

   2. metadata-eval N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, \frac{2}{945}, \frac{1}{45}\right), \frac{1}{3}\right), -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

   14. lower-*.f32 99.5

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, 0.0021164021164021165, 0.022222222222222223\right), 0.3333333333333333\right), -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

8. Simplified 99.5%

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, 0.0021164021164021165, 0.022222222222222223\right), 0.3333333333333333\right), -1\right)}, normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

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. +-commutative N/A

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

    3. *-commutative N/A

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

    4. unpow2 N/A

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

    5. associate-*r* N/A

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

    6. lower-fma.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. sub-neg N/A

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

    9. +-commutative N/A

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

    10. *-commutative N/A

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

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

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

    12. metadata-eval N/A

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

    13. lower-fma.f32 N/A

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

    14. *-commutative N/A

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

    15. lower-*.f32 N/A

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

    16. mul-1-neg N/A

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

    17. unsub-neg N/A

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

    18. lower--.f32 98.5

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

11. Simplified 98.5%

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

12. Taylor expanded in n1_i around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 97.7

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

14. Simplified 97.7%

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

15. Final simplification 97.7%

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

Alternative 10: 86.0% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n0\_i, -u, n0\_i\right)\\ \mathbf{if}\;n0\_i \leq -8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(u, n1\_i, n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (fma n0_i (- u) n0_i)))
   (if (<= n0_i -8.00000002901995e-15)
     t_0
     (if (<= n0_i 5.00000006675716e-11) (fma u n1_i n0_i) t_0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = fmaf(n0_i, -u, n0_i);
	float tmp;
	if (n0_i <= -8.00000002901995e-15f) {
		tmp = t_0;
	} else if (n0_i <= 5.00000006675716e-11f) {
		tmp = fmaf(u, n1_i, n0_i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = fma(n0_i, Float32(-u), n0_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-8.00000002901995e-15))
		tmp = t_0;
	elseif (n0_i <= Float32(5.00000006675716e-11))
		tmp = fma(u, n1_i, n0_i);
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -8.00000003e-15 or 5.00000007e-11 < n0_i

   1. Initial program 98.6%

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

   3. Taylor expanded in normAngle around 0

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

      1. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 98.6

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

   5. Simplified 98.6%

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

   6. Taylor expanded in n0_i around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f32 91.0

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

   8. Simplified 91.0%

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

   if -8.00000003e-15 < n0_i < 5.00000007e-11

   1. Initial program 97.0%

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

   3. Taylor expanded in normAngle around 0

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

      1. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 96.5

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

   5. Simplified 96.5%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 86.0%

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

      1. lift-*.f32 N/A

         \[\leadsto n0\_i \cdot 1 + \color{blue}{u \cdot n1\_i} \]

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. lower-fma.f32 86.0

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

   10. Applied egg-rr 86.0%

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

Alternative 11: 85.9% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n0\_i \cdot \left(1 - u\right)\\ \mathbf{if}\;n0\_i \leq -8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(u, n1\_i, n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* n0_i (- 1.0 u))))
   (if (<= n0_i -8.00000002901995e-15)
     t_0
     (if (<= n0_i 5.00000006675716e-11) (fma u n1_i n0_i) t_0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = n0_i * (1.0f - u);
	float tmp;
	if (n0_i <= -8.00000002901995e-15f) {
		tmp = t_0;
	} else if (n0_i <= 5.00000006675716e-11f) {
		tmp = fmaf(u, n1_i, n0_i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(n0_i * Float32(Float32(1.0) - u))
	tmp = Float32(0.0)
	if (n0_i <= Float32(-8.00000002901995e-15))
		tmp = t_0;
	elseif (n0_i <= Float32(5.00000006675716e-11))
		tmp = fma(u, n1_i, n0_i);
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -8.00000003e-15 or 5.00000007e-11 < n0_i

   1. Initial program 98.6%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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. associate-/l* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. distribute-lft-neg-in N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-fma.f32 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f32 N/A

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

      15. lower-sin.f32 90.9

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

   5. Simplified 90.9%

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

   6. Taylor expanded in normAngle around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f32 90.7

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

   8. Simplified 90.7%

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

   if -8.00000003e-15 < n0_i < 5.00000007e-11

   1. Initial program 97.0%

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

   3. Taylor expanded in normAngle around 0

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

      1. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 96.5

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

   5. Simplified 96.5%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 86.0%

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

      1. lift-*.f32 N/A

         \[\leadsto n0\_i \cdot 1 + \color{blue}{u \cdot n1\_i} \]

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. lower-fma.f32 86.0

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

   10. Applied egg-rr 86.0%

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

Alternative 12: 60.7% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.499999999549897 \cdot 10^{-24}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 7.99999974612418 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -2.499999999549897e-24)
   n0_i
   (if (<= n0_i 7.99999974612418e-20) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -2.499999999549897e-24f) {
		tmp = n0_i;
	} else if (n0_i <= 7.99999974612418e-20f) {
		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 <= (-2.499999999549897e-24)) then
        tmp = n0_i
    else if (n0_i <= 7.99999974612418e-20) 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(-2.499999999549897e-24))
		tmp = n0_i;
	elseif (n0_i <= Float32(7.99999974612418e-20))
		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(-2.499999999549897e-24))
		tmp = n0_i;
	elseif (n0_i <= single(7.99999974612418e-20))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -2.5e-24 or 7.99999975e-20 < n0_i

   1. Initial program 98.2%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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. associate-/l* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. distribute-lft-neg-in N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-fma.f32 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f32 N/A

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

      15. lower-sin.f32 81.4

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

   5. Simplified 81.4%

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

   6. Taylor expanded in u around 0

      \[\leadsto n0\_i \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 61.6%

      \[\leadsto n0\_i \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 61.6

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

   10. Applied egg-rr 61.6%

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

   if -2.5e-24 < n0_i < 7.99999975e-20

   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

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

      1. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 94.9

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

   5. Simplified 94.9%

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

   6. Taylor expanded in n0_i around 0

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

      1. lower-*.f32 67.5

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

   8. Simplified 67.5%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.499999999549897 \cdot 10^{-24}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 7.99999974612418 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 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 97.5%

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

3. Taylor expanded in 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. lower-fma.f32 N/A

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

5. Simplified 99.5%

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

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. lower-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. lower--.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 14: 81.6% 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 97.5%

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

3. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 97.2

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

5. Simplified 97.2%

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

6. Taylor expanded in u around 0

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

8. Simplified 80.2%

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

   1. lift-*.f32 N/A

      \[\leadsto n0\_i \cdot 1 + \color{blue}{u \cdot n1\_i} \]

   2. *-rgt-identity N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lower-fma.f32 80.2

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

10. Applied egg-rr 80.2%

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

Alternative 15: 46.6% 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 97.5%

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

3. Taylor expanded in 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. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. *-commutative N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. distribute-lft-neg-in N/A

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

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

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

   11. mul-1-neg N/A

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

   12. lower-fma.f32 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f32 N/A

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

   15. lower-sin.f32 59.0

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

5. Simplified 59.0%

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

6. Taylor expanded in u around 0

   \[\leadsto n0\_i \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 46.4%

   \[\leadsto n0\_i \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 46.4

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

10. Applied egg-rr 46.4%

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

Reproduce

herbie shell --seed 2024208 
(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)))