Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.4%

Time: 14.3s

Alternatives: 16

Speedup: 45.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   n0_i
   (/ (* normAngle (cos normAngle)) (- (sin normAngle)))
   (* 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, ((normAngle * cosf(normAngle)) / -sinf(normAngle)), (normAngle * (n1_i / sinf(normAngle)))), n0_i);
}

Julia

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

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.4%

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

Alternative 2: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.4%

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

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

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

7. Applied egg-rr 99.3%

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

8. Final simplification 99.3%

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

Alternative 3: 99.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n0\_i, 0.05555555555555555, n0\_i \cdot 0.008333333333333333\right)\\ \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.3333333333333333, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \left(\mathsf{fma}\left(-0.16666666666666666, n0\_i \cdot 0.041666666666666664 - t\_0, \mathsf{fma}\left(n0\_i, -0.0001984126984126984, n0\_i \cdot -0.002777777777777778\right)\right) - n0\_i \cdot -0.001388888888888889\right) - \mathsf{fma}\left(n1\_i, 0.0011904761904761906, n1\_i \cdot -0.0032407407407407406\right), \mathsf{fma}\left(n1\_i, 0.019444444444444445, t\_0 - n0\_i \cdot 0.041666666666666664\right)\right), n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right), n0\_i\right) \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (fma n0_i 0.05555555555555555 (* n0_i 0.008333333333333333))))
   (fma
    u
    (fma
     (* normAngle normAngle)
     (fma
      n0_i
      0.3333333333333333
      (fma
       (* normAngle normAngle)
       (fma
        (* normAngle normAngle)
        (-
         (-
          (fma
           -0.16666666666666666
           (- (* n0_i 0.041666666666666664) t_0)
           (fma n0_i -0.0001984126984126984 (* n0_i -0.002777777777777778)))
          (* n0_i -0.001388888888888889))
         (fma n1_i 0.0011904761904761906 (* n1_i -0.0032407407407407406)))
        (fma n1_i 0.019444444444444445 (- t_0 (* n0_i 0.041666666666666664))))
       (* n1_i 0.16666666666666666)))
     (- n1_i n0_i))
    n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = fmaf(n0_i, 0.05555555555555555f, (n0_i * 0.008333333333333333f));
	return fmaf(u, fmaf((normAngle * normAngle), fmaf(n0_i, 0.3333333333333333f, fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), ((fmaf(-0.16666666666666666f, ((n0_i * 0.041666666666666664f) - t_0), fmaf(n0_i, -0.0001984126984126984f, (n0_i * -0.002777777777777778f))) - (n0_i * -0.001388888888888889f)) - fmaf(n1_i, 0.0011904761904761906f, (n1_i * -0.0032407407407407406f))), fmaf(n1_i, 0.019444444444444445f, (t_0 - (n0_i * 0.041666666666666664f)))), (n1_i * 0.16666666666666666f))), (n1_i - n0_i)), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = fma(n0_i, Float32(0.05555555555555555), Float32(n0_i * Float32(0.008333333333333333)))
	return fma(u, fma(Float32(normAngle * normAngle), fma(n0_i, Float32(0.3333333333333333), fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), Float32(Float32(fma(Float32(-0.16666666666666666), Float32(Float32(n0_i * Float32(0.041666666666666664)) - t_0), fma(n0_i, Float32(-0.0001984126984126984), Float32(n0_i * Float32(-0.002777777777777778)))) - Float32(n0_i * Float32(-0.001388888888888889))) - fma(n1_i, Float32(0.0011904761904761906), Float32(n1_i * Float32(-0.0032407407407407406)))), fma(n1_i, Float32(0.019444444444444445), Float32(t_0 - Float32(n0_i * Float32(0.041666666666666664))))), Float32(n1_i * Float32(0.16666666666666666)))), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.4%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(-1 \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(\left(-1 \cdot \left(\frac{1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \frac{1}{120} \cdot n0\_i\right)\right) + {normAngle}^{2} \cdot \left(-1 \cdot \left(\frac{-1}{720} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \frac{1}{120} \cdot n0\_i\right)\right) + \left(\frac{-1}{5040} \cdot n0\_i + \frac{1}{120} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right)\right)\right)\right) - \left(\frac{-1}{5040} \cdot n1\_i + \left(\frac{1}{720} \cdot n1\_i + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right)\right)\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)}, n0\_i\right) \]

8. Simplified 99.2%

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.3333333333333333, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, -\left(\left(n0\_i \cdot -0.001388888888888889 - \mathsf{fma}\left(-0.16666666666666666, n0\_i \cdot 0.041666666666666664 - \mathsf{fma}\left(n0\_i, 0.05555555555555555, n0\_i \cdot 0.008333333333333333\right), \mathsf{fma}\left(n0\_i, -0.0001984126984126984, n0\_i \cdot -0.002777777777777778\right)\right)\right) + \mathsf{fma}\left(n1\_i, 0.0011904761904761906, n1\_i \cdot -0.0032407407407407406\right)\right), \mathsf{fma}\left(n1\_i, 0.019444444444444445, -\left(n0\_i \cdot 0.041666666666666664 - \mathsf{fma}\left(n0\_i, 0.05555555555555555, n0\_i \cdot 0.008333333333333333\right)\right)\right)\right), n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right)}, n0\_i\right) \]

9. Final simplification 99.2%

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.3333333333333333, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \left(\mathsf{fma}\left(-0.16666666666666666, n0\_i \cdot 0.041666666666666664 - \mathsf{fma}\left(n0\_i, 0.05555555555555555, n0\_i \cdot 0.008333333333333333\right), \mathsf{fma}\left(n0\_i, -0.0001984126984126984, n0\_i \cdot -0.002777777777777778\right)\right) - n0\_i \cdot -0.001388888888888889\right) - \mathsf{fma}\left(n1\_i, 0.0011904761904761906, n1\_i \cdot -0.0032407407407407406\right), \mathsf{fma}\left(n1\_i, 0.019444444444444445, \mathsf{fma}\left(n0\_i, 0.05555555555555555, n0\_i \cdot 0.008333333333333333\right) - n0\_i \cdot 0.041666666666666664\right)\right), n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right), n0\_i\right) \]
10. Add Preprocessing

Alternative 4: 99.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.022222222222222223, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.009523809523809525, n0\_i \cdot -0.007407407407407408\right) - \mathsf{fma}\left(n1\_i, 0.0011904761904761906, n1\_i \cdot -0.0032407407407407406\right), n1\_i \cdot 0.019444444444444445\right)\right), \mathsf{fma}\left(n0\_i, 0.3333333333333333, n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (fma
    (* normAngle normAngle)
    (fma
     (* normAngle normAngle)
     (fma
      n0_i
      0.022222222222222223
      (fma
       (* normAngle normAngle)
       (-
        (fma n0_i 0.009523809523809525 (* n0_i -0.007407407407407408))
        (fma n1_i 0.0011904761904761906 (* n1_i -0.0032407407407407406)))
       (* n1_i 0.019444444444444445)))
     (fma n0_i 0.3333333333333333 (* n1_i 0.16666666666666666)))
    (- n1_i n0_i)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), fmaf(n0_i, 0.022222222222222223f, fmaf((normAngle * normAngle), (fmaf(n0_i, 0.009523809523809525f, (n0_i * -0.007407407407407408f)) - fmaf(n1_i, 0.0011904761904761906f, (n1_i * -0.0032407407407407406f))), (n1_i * 0.019444444444444445f))), fmaf(n0_i, 0.3333333333333333f, (n1_i * 0.16666666666666666f))), (n1_i - n0_i)));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), fma(n0_i, Float32(0.022222222222222223), fma(Float32(normAngle * normAngle), Float32(fma(n0_i, Float32(0.009523809523809525), Float32(n0_i * Float32(-0.007407407407407408))) - fma(n1_i, Float32(0.0011904761904761906), Float32(n1_i * Float32(-0.0032407407407407406)))), Float32(n1_i * Float32(0.019444444444444445)))), fma(n0_i, Float32(0.3333333333333333), Float32(n1_i * Float32(0.16666666666666666)))), Float32(n1_i - n0_i))))
end

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.4%

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

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

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

7. Applied egg-rr 99.3%

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

8. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{3} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{9} \cdot n0\_i + \left(\frac{2}{15} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot n0\_i + \frac{2}{15} \cdot n0\_i\right) + \left(\frac{-2}{45} \cdot n0\_i + \frac{17}{315} \cdot n0\_i\right)\right) - \left(\frac{-1}{5040} \cdot n1\_i + \left(\frac{1}{720} \cdot n1\_i + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right)} \cdot u + n0\_i \]

10. Simplified 99.1%

    \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.022222222222222223, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.009523809523809525, n0\_i \cdot -0.007407407407407408\right) - \mathsf{fma}\left(n1\_i, 0.0011904761904761906, n1\_i \cdot -0.0032407407407407406\right), n1\_i \cdot 0.019444444444444445\right)\right), \mathsf{fma}\left(n0\_i, 0.3333333333333333, n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right)} \cdot u + n0\_i \]

11. Final simplification 99.1%

    \[\leadsto n0\_i + u \cdot \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.022222222222222223, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.009523809523809525, n0\_i \cdot -0.007407407407407408\right) - \mathsf{fma}\left(n1\_i, 0.0011904761904761906, n1\_i \cdot -0.0032407407407407406\right), n1\_i \cdot 0.019444444444444445\right)\right), \mathsf{fma}\left(n0\_i, 0.3333333333333333, n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right) \]
12. Add Preprocessing

Alternative 5: 99.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.3333333333333333, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n1\_i, 0.019444444444444445, \mathsf{fma}\left(n0\_i, 0.05555555555555555, n0\_i \cdot 0.008333333333333333\right) - n0\_i \cdot 0.041666666666666664\right), n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* normAngle normAngle)
   (fma
    n0_i
    0.3333333333333333
    (fma
     (* normAngle normAngle)
     (fma
      n1_i
      0.019444444444444445
      (-
       (fma n0_i 0.05555555555555555 (* n0_i 0.008333333333333333))
       (* n0_i 0.041666666666666664)))
     (* 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((normAngle * normAngle), fmaf(n1_i, 0.019444444444444445f, (fmaf(n0_i, 0.05555555555555555f, (n0_i * 0.008333333333333333f)) - (n0_i * 0.041666666666666664f))), (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(Float32(normAngle * normAngle), fma(n1_i, Float32(0.019444444444444445), Float32(fma(n0_i, Float32(0.05555555555555555), Float32(n0_i * Float32(0.008333333333333333))) - Float32(n0_i * Float32(0.041666666666666664)))), Float32(n1_i * Float32(0.16666666666666666)))), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.4%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(-1 \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(-1 \cdot \left(\frac{1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \frac{1}{120} \cdot n0\_i\right)\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)}, n0\_i\right) \]

8. Simplified 99.0%

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.3333333333333333, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n1\_i, 0.019444444444444445, -\left(n0\_i \cdot 0.041666666666666664 - \mathsf{fma}\left(n0\_i, 0.05555555555555555, n0\_i \cdot 0.008333333333333333\right)\right)\right), n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right)}, n0\_i\right) \]

9. Final simplification 99.0%

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.3333333333333333, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n1\_i, 0.019444444444444445, \mathsf{fma}\left(n0\_i, 0.05555555555555555, n0\_i \cdot 0.008333333333333333\right) - n0\_i \cdot 0.041666666666666664\right), n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right), n0\_i\right) \]
10. Add Preprocessing

Alternative 6: 99.1% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.4%

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

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

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

7. Applied egg-rr 99.3%

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

8. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{3} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{9} \cdot n0\_i + \frac{2}{15} \cdot n0\_i\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right)} \cdot u + n0\_i \]
9. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(\left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{3} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{9} \cdot n0\_i + \frac{2}{15} \cdot n0\_i\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)} \cdot u + n0\_i \]

   2. mul-1-neg N/A

      \[\leadsto \left(\left(n1\_i + \color{blue}{\left(\mathsf{neg}\left(n0\_i\right)\right)}\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{3} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{9} \cdot n0\_i + \frac{2}{15} \cdot n0\_i\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right) \cdot u + n0\_i \]

   3. sub-neg N/A

      \[\leadsto \left(\color{blue}{\left(n1\_i - n0\_i\right)} + {normAngle}^{2} \cdot \left(\left(\frac{1}{3} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{9} \cdot n0\_i + \frac{2}{15} \cdot n0\_i\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right) \cdot u + n0\_i \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left({normAngle}^{2} \cdot \left(\left(\frac{1}{3} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{9} \cdot n0\_i + \frac{2}{15} \cdot n0\_i\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right) + \left(n1\_i - n0\_i\right)\right)} \cdot u + n0\_i \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left({normAngle}^{2}, \left(\frac{1}{3} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{9} \cdot n0\_i + \frac{2}{15} \cdot n0\_i\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i, n1\_i - n0\_i\right)} \cdot u + n0\_i \]

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 7: 99.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(-0.16666666666666666, \left(normAngle \cdot normAngle\right) \cdot \left(\mathsf{fma}\left(u, n0\_i \cdot 3, \left(-n0\_i\right) - n0\_i\right) - n1\_i\right), n1\_i - n0\_i\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   -0.16666666666666666
   (* (* normAngle normAngle) (- (fma u (* n0_i 3.0) (- (- n0_i) n0_i)) n1_i))
   (- n1_i n0_i))
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, fmaf(-0.16666666666666666f, ((normAngle * normAngle) * (fmaf(u, (n0_i * 3.0f), (-n0_i - n0_i)) - n1_i)), (n1_i - n0_i)), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, fma(Float32(-0.16666666666666666), Float32(Float32(normAngle * normAngle) * Float32(fma(u, Float32(n0_i * Float32(3.0)), Float32(Float32(-n0_i) - n0_i)) - n1_i)), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in u around 0

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

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

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

   2. *-commutative N/A

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

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

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

   4. sin-lowering-sin.f32 95.5

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

5. Simplified 95.5%

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

6. Taylor expanded in normAngle around 0

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

8. Simplified 98.9%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

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

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

11. Simplified 98.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(-0.16666666666666666, \left(normAngle \cdot normAngle\right) \cdot \left(\mathsf{fma}\left(u, 3 \cdot n0\_i, \left(-n0\_i\right) - n0\_i\right) - n1\_i\right), n1\_i - n0\_i\right), n0\_i\right)} \]

12. Final simplification 98.8%

    \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(-0.16666666666666666, \left(normAngle \cdot normAngle\right) \cdot \left(\mathsf{fma}\left(u, n0\_i \cdot 3, \left(-n0\_i\right) - n0\_i\right) - n1\_i\right), n1\_i - n0\_i\right), n0\_i\right) \]
13. Add Preprocessing

Alternative 8: 99.0% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.4%

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

6. Taylor expanded in normAngle around 0

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 9: 99.0% accurate, 14.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.4%

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

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. associate-+r+ N/A

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

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

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

   10. metadata-eval N/A

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

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

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

   12. mul-1-neg N/A

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

   13. distribute-rgt-out-- N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

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

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

   18. mul-1-neg N/A

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

   19. unsub-neg N/A

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

   20. --lowering--.f32 98.8

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

8. Simplified 98.8%

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

Alternative 10: 98.9% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in u around 0

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

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

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

   2. *-commutative N/A

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

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

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

   4. sin-lowering-sin.f32 95.5

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

5. Simplified 95.5%

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

6. Taylor expanded in normAngle around 0

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

8. Simplified 98.9%

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

9. Taylor expanded in n0_i around 0

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

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

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

    2. *-lowering-*.f32 98.7

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

11. Simplified 98.7%

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

12. Final simplification 98.7%

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

Alternative 11: 98.9% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in normAngle around 0

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\left(u + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot {u}^{3} - \frac{-1}{6} \cdot u\right)\right)} \cdot n1\_i \]
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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f32 97.5

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

5. Simplified 97.5%

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

6. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. mul-1-neg N/A

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

   8. neg-lowering-neg.f32 98.5

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

8. Simplified 98.5%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

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

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

    3. +-commutative N/A

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

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

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

    5. +-commutative N/A

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

    6. *-commutative N/A

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

    7. unpow2 N/A

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

    8. associate-*l* N/A

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

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

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

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

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

    11. mul-1-neg N/A

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

    12. neg-lowering-neg.f32 98.7

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

11. Simplified 98.7%

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

Alternative 12: 70.3% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n0\_i, -u, n0\_i\right)\\ \mathbf{if}\;n0\_i \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (fma n0_i (- u) n0_i)))
   (if (<= n0_i -1.999999936531045e-19)
     t_0
     (if (<= n0_i 2.0000000390829628e-24) (* u n1_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 <= -1.999999936531045e-19f) {
		tmp = t_0;
	} else if (n0_i <= 2.0000000390829628e-24f) {
		tmp = u * n1_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(-1.999999936531045e-19))
		tmp = t_0;
	elseif (n0_i <= Float32(2.0000000390829628e-24))
		tmp = Float32(u * n1_i);
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.99999994e-19 or 2.00000004e-24 < n0_i

   1. Initial program 97.2%

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

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

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

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

      4. sin-lowering-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. accelerator-lowering-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. neg-lowering-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. sin-lowering-sin.f32 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in normAngle around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f32 75.4

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

   8. Simplified 75.4%

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

   if -1.99999994e-19 < n0_i < 2.00000004e-24

   1. Initial program 94.1%

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

   3. Taylor expanded in normAngle around 0

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\left(u + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot {u}^{3} - \frac{-1}{6} \cdot u\right)\right)} \cdot n1\_i \]
   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f32 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in n0_i around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f32 65.7

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

   8. Simplified 65.7%

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

   9. Taylor expanded in normAngle around 0

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

       1. *-lowering-*.f32 63.9

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

   11. Simplified 63.9%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(n0\_i, -u, n0\_i\right)\\ \mathbf{elif}\;n0\_i \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n0\_i, -u, n0\_i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.0% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n0\_i \cdot \left(1 - u\right)\\ \mathbf{if}\;n0\_i \leq -6.000000068087077 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* n0_i (- 1.0 u))))
   (if (<= n0_i -6.000000068087077e-19)
     t_0
     (if (<= n0_i 2.0000000390829628e-24) (* u n1_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 <= -6.000000068087077e-19f) {
		tmp = t_0;
	} else if (n0_i <= 2.0000000390829628e-24f) {
		tmp = u * n1_i;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = n0_i * (1.0e0 - u)
    if (n0_i <= (-6.000000068087077e-19)) then
        tmp = t_0
    else if (n0_i <= 2.0000000390829628e-24) then
        tmp = u * n1_i
    else
        tmp = t_0
    end if
    code = tmp
end function

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(-6.000000068087077e-19))
		tmp = t_0;
	elseif (n0_i <= Float32(2.0000000390829628e-24))
		tmp = Float32(u * n1_i);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	t_0 = n0_i * (single(1.0) - u);
	tmp = single(0.0);
	if (n0_i <= single(-6.000000068087077e-19))
		tmp = t_0;
	elseif (n0_i <= single(2.0000000390829628e-24))
		tmp = u * n1_i;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -6.00000007e-19 or 2.00000004e-24 < n0_i

   1. Initial program 97.1%

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

   3. Taylor expanded in 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. *-lowering-*.f32 N/A

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

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

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

      4. sin-lowering-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. accelerator-lowering-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. neg-lowering-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. sin-lowering-sin.f32 75.4

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

   5. Simplified 75.4%

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

   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. --lowering--.f32 75.5

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

   8. Simplified 75.5%

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

   if -6.00000007e-19 < n0_i < 2.00000004e-24

   1. Initial program 94.2%

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

   3. Taylor expanded in 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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f32 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in n0_i around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f32 65.6

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

   8. Simplified 65.6%

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

   9. Taylor expanded in normAngle around 0

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

       1. *-lowering-*.f32 63.8

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

   11. Simplified 63.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -6.000000068087077 \cdot 10^{-19}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{elif}\;n0\_i \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.2% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 4.000000014509975 \cdot 10^{-15}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -4.99999991225835e-15)
   n0_i
   (if (<= n0_i 4.000000014509975e-15) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -4.99999991225835e-15f) {
		tmp = n0_i;
	} else if (n0_i <= 4.000000014509975e-15f) {
		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 <= (-4.99999991225835e-15)) then
        tmp = n0_i
    else if (n0_i <= 4.000000014509975e-15) 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(-4.99999991225835e-15))
		tmp = n0_i;
	elseif (n0_i <= Float32(4.000000014509975e-15))
		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(-4.99999991225835e-15))
		tmp = n0_i;
	elseif (n0_i <= single(4.000000014509975e-15))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.99999991e-15 or 4.00000001e-15 < n0_i

   1. Initial program 98.5%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 61.9%

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

   if -4.99999991e-15 < n0_i < 4.00000001e-15

   1. Initial program 93.9%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f32 96.7

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

   5. Simplified 96.7%

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

   6. Taylor expanded in n0_i around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f32 61.7

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

   8. Simplified 61.7%

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

   9. Taylor expanded in normAngle around 0

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

       1. *-lowering-*.f32 60.3

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

   11. Simplified 60.3%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 4.000000014509975 \cdot 10^{-15}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 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 95.6%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.4%

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

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. accelerator-lowering-fma.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. --lowering--.f32 97.4

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

8. Simplified 97.4%

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

Alternative 16: 47.1% accurate, 459.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in u around 0

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

5. Simplified 40.4%

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

Reproduce

herbie shell --seed 2024198 
(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)))