Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 99.4%

Time: 16.5s

Alternatives: 10

Speedup: 45.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \left(\mathsf{fma}\left(u \cdot u, \mathsf{fma}\left(-0.027777777777777776, n1\_i - n0\_i, n0\_i \cdot -0.08333333333333333\right), n0\_i \cdot -0.041666666666666664\right) - \mathsf{fma}\left(0.008333333333333333, n1\_i - n0\_i, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right) \cdot 0.027777777777777776\right)\right), -0.16666666666666666 \cdot \mathsf{fma}\left(1 - u, n0\_i \cdot \mathsf{fma}\left(1 - u, 1 - u, -1\right), \mathsf{fma}\left(u, u, -1\right) \cdot \left(u \cdot n1\_i\right)\right)\right), \mathsf{fma}\left(n1\_i, u, \mathsf{fma}\left(n0\_i, -u, n0\_i\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (* normAngle normAngle)
  (fma
   (* normAngle normAngle)
   (*
    u
    (-
     (fma
      (* u u)
      (fma -0.027777777777777776 (- n1_i n0_i) (* n0_i -0.08333333333333333))
      (* n0_i -0.041666666666666664))
     (fma
      0.008333333333333333
      (- n1_i n0_i)
      (* (fma n0_i -2.0 (- n1_i)) 0.027777777777777776))))
   (*
    -0.16666666666666666
    (fma
     (- 1.0 u)
     (* n0_i (fma (- 1.0 u) (- 1.0 u) -1.0))
     (* (fma u u -1.0) (* u n1_i)))))
  (fma n1_i u (fma n0_i (- u) n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), (u * (fmaf((u * u), fmaf(-0.027777777777777776f, (n1_i - n0_i), (n0_i * -0.08333333333333333f)), (n0_i * -0.041666666666666664f)) - fmaf(0.008333333333333333f, (n1_i - n0_i), (fmaf(n0_i, -2.0f, -n1_i) * 0.027777777777777776f)))), (-0.16666666666666666f * fmaf((1.0f - u), (n0_i * fmaf((1.0f - u), (1.0f - u), -1.0f)), (fmaf(u, u, -1.0f) * (u * n1_i))))), fmaf(n1_i, u, fmaf(n0_i, -u, n0_i)));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), Float32(u * Float32(fma(Float32(u * u), fma(Float32(-0.027777777777777776), Float32(n1_i - n0_i), Float32(n0_i * Float32(-0.08333333333333333))), Float32(n0_i * Float32(-0.041666666666666664))) - fma(Float32(0.008333333333333333), Float32(n1_i - n0_i), Float32(fma(n0_i, Float32(-2.0), Float32(-n1_i)) * Float32(0.027777777777777776))))), Float32(Float32(-0.16666666666666666) * fma(Float32(Float32(1.0) - u), Float32(n0_i * fma(Float32(Float32(1.0) - u), Float32(Float32(1.0) - u), Float32(-1.0))), Float32(fma(u, u, Float32(-1.0)) * Float32(u * n1_i))))), fma(n1_i, u, fma(n0_i, Float32(-u), n0_i)))
end

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) + \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) + {normAngle}^{2} \cdot \left(\left(\frac{-1}{5040} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{7}\right) + \frac{-1}{5040} \cdot \left(n1\_i \cdot {u}^{7}\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right) + \left(\frac{-1}{5040} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \left(\frac{-1}{5040} \cdot \left(n1\_i \cdot u\right) + \left(\frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \left(\frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(0.008333333333333333, \mathsf{fma}\left(n0\_i, {\left(1 - u\right)}^{5}, n1\_i \cdot {u}^{5}\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(-0.0001984126984126984, \mathsf{fma}\left(n0\_i, {\left(1 - u\right)}^{7}, n1\_i \cdot {u}^{7}\right), -\mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(n1\_i, 0.008333333333333333 \cdot \left({u}^{5} - u\right), -0.027777777777777776 \cdot \left(n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right)\right) + \mathsf{fma}\left(n0\_i, 0.008333333333333333 \cdot \left({\left(1 - u\right)}^{5} - \left(1 - u\right)\right), -0.027777777777777776 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right)\right)\right), \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), -0.0001984126984126984 \cdot \mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right)\right)\right)\right), -\mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right) \cdot 0.008333333333333333\right)\right)\right)\right)\right)\right)} \]

6. Taylor expanded in u around 0

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(u \cdot \left(\left(\frac{-1}{24} \cdot n0\_i + \left(u \cdot \left(\left(\frac{1}{12} \cdot n0\_i + \left(u \cdot \left(\left(\frac{-1}{12} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{1}{144} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{12} \cdot n0\_i + \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{36} \cdot n0\_i\right)\right) + \frac{-1}{720} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) - \frac{1}{36} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right) + {normAngle}^{2} \cdot \left(\frac{-1}{240} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \left(n0\_i + 2 \cdot n0\_i\right) + \frac{1}{12} \cdot n0\_i\right) + \frac{-1}{720} \cdot \left(n0\_i + 2 \cdot n0\_i\right)\right)\right)\right)\right) - \frac{1}{36} \cdot \left(n0\_i + 2 \cdot n0\_i\right)\right) + {normAngle}^{2} \cdot \left(\frac{1}{720} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{30} \cdot n0\_i + \left(\frac{-1}{120} \cdot n1\_i + \left(\frac{1}{36} \cdot n1\_i + \frac{1}{18} \cdot n0\_i\right)\right)\right) + \left(\frac{-1}{720} \cdot \left(-2 \cdot n0\_i + -1 \cdot n1\_i\right) + \frac{-1}{5040} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right)\right)\right) - \left(\frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right) + \frac{1}{36} \cdot \left(-2 \cdot n0\_i + -1 \cdot n1\_i\right)\right)\right)\right)}\right)\right)\right) \]

8. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(u \cdot \mathsf{fma}\left(n0\_i, -0.041666666666666664, \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, 0.08333333333333333, \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, -0.08333333333333333, \mathsf{fma}\left(normAngle \cdot normAngle, n0\_i \cdot 0.006944444444444444 - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.08333333333333333, \mathsf{fma}\left(n0\_i, 0.027777777777777776, -0.027777777777777776 \cdot n1\_i\right)\right), -0.16666666666666666, -0.001388888888888889 \cdot \left(n1\_i - n0\_i\right)\right), -0.027777777777777776 \cdot \left(n1\_i - n0\_i\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(\mathsf{fma}\left(n0\_i, -0.004166666666666667, 0.16666666666666666 \cdot \mathsf{fma}\left(-0.027777777777777776, 3 \cdot n0\_i, n0\_i \cdot 0.08333333333333333\right)\right) + 0.001388888888888889 \cdot \left(3 \cdot n0\_i\right)\right)\right)\right) + -0.027777777777777776 \cdot \left(3 \cdot n0\_i\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.001388888888888889 - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.03333333333333333, \mathsf{fma}\left(n1\_i, 0.019444444444444445, n0\_i \cdot 0.05555555555555555\right)\right), -0.16666666666666666, \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), -0.0001984126984126984 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)}\right)\right)\right) \]

9. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{\frac{-1}{12} \cdot n0\_i + \left(\frac{1}{12} \cdot n0\_i + u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right)\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{\left(\frac{1}{12} \cdot n0\_i + u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right)\right) + \frac{-1}{12} \cdot n0\_i}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    2. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{\left(u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + \frac{1}{12} \cdot n0\_i\right)} + \frac{-1}{12} \cdot n0\_i, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    3. associate-+l+ N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + \left(\frac{1}{12} \cdot n0\_i + \frac{-1}{12} \cdot n0\_i\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    4. distribute-rgt-out N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + \color{blue}{n0\_i \cdot \left(\frac{1}{12} + \frac{-1}{12}\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + n0\_i \cdot \color{blue}{0}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    6. mul0-rgt N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + \color{blue}{0}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    7. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right), 0\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    8. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{\frac{-1}{36} \cdot \left(n1\_i - n0\_i\right) + \frac{-1}{12} \cdot n0\_i}, 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    9. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \frac{-1}{36} \cdot \color{blue}{\left(n1\_i + \left(\mathsf{neg}\left(n0\_i\right)\right)\right)} + \frac{-1}{12} \cdot n0\_i, 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    10. mul-1-neg N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \frac{-1}{36} \cdot \left(n1\_i + \color{blue}{-1 \cdot n0\_i}\right) + \frac{-1}{12} \cdot n0\_i, 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    11. lower-fma.f32 N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(\frac{-1}{36}, n1\_i + -1 \cdot n0\_i, \frac{-1}{12} \cdot n0\_i\right)}, 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    12. mul-1-neg N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-1}{36}, n1\_i + \color{blue}{\left(\mathsf{neg}\left(n0\_i\right)\right)}, \frac{-1}{12} \cdot n0\_i\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    13. sub-neg N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-1}{36}, \color{blue}{n1\_i - n0\_i}, \frac{-1}{12} \cdot n0\_i\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    14. lower--.f32 N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-1}{36}, \color{blue}{n1\_i - n0\_i}, \frac{-1}{12} \cdot n0\_i\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-1}{36}, n1\_i - n0\_i, \color{blue}{n0\_i \cdot \frac{-1}{12}}\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    16. lower-*.f32 99.0

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, -0.041666666666666664, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(-0.027777777777777776, n1\_i - n0\_i, \color{blue}{n0\_i \cdot -0.08333333333333333}\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.001388888888888889 - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.03333333333333333, \mathsf{fma}\left(n1\_i, 0.019444444444444445, n0\_i \cdot 0.05555555555555555\right)\right), -0.16666666666666666, \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), -0.0001984126984126984 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

11. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, -0.041666666666666664, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(-0.027777777777777776, n1\_i - n0\_i, n0\_i \cdot -0.08333333333333333\right), 0\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.001388888888888889 - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.03333333333333333, \mathsf{fma}\left(n1\_i, 0.019444444444444445, n0\_i \cdot 0.05555555555555555\right)\right), -0.16666666666666666, \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), -0.0001984126984126984 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

12. Taylor expanded in normAngle around 0

    \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot \left(u \cdot \left({u}^{2} - 1\right)\right) + \left(1 - u\right) \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{2} - n0\_i\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\left(\frac{-1}{24} \cdot n0\_i + {u}^{2} \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right)\right) - \left(\frac{1}{120} \cdot \left(n1\_i - n0\_i\right) + \frac{1}{36} \cdot \left(-2 \cdot n0\_i - n1\_i\right)\right)\right)\right)\right)\right)} \]

14. Applied rewrites 99.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \left(\mathsf{fma}\left(u \cdot u, \mathsf{fma}\left(-0.027777777777777776, n1\_i - n0\_i, n0\_i \cdot -0.08333333333333333\right), n0\_i \cdot -0.041666666666666664\right) - \mathsf{fma}\left(0.008333333333333333, n1\_i - n0\_i, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right) \cdot 0.027777777777777776\right)\right), -0.16666666666666666 \cdot \mathsf{fma}\left(1 - u, n0\_i \cdot \mathsf{fma}\left(1 - u, 1 - u, -1\right), \left(n1\_i \cdot u\right) \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \mathsf{fma}\left(n1\_i, u, \mathsf{fma}\left(n0\_i, -u, n0\_i\right)\right)\right)} \]

15. Final simplification 99.2%

    \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \left(\mathsf{fma}\left(u \cdot u, \mathsf{fma}\left(-0.027777777777777776, n1\_i - n0\_i, n0\_i \cdot -0.08333333333333333\right), n0\_i \cdot -0.041666666666666664\right) - \mathsf{fma}\left(0.008333333333333333, n1\_i - n0\_i, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right) \cdot 0.027777777777777776\right)\right), -0.16666666666666666 \cdot \mathsf{fma}\left(1 - u, n0\_i \cdot \mathsf{fma}\left(1 - u, 1 - u, -1\right), \mathsf{fma}\left(u, u, -1\right) \cdot \left(u \cdot n1\_i\right)\right)\right), \mathsf{fma}\left(n1\_i, u, \mathsf{fma}\left(n0\_i, -u, n0\_i\right)\right)\right) \]
16. Add Preprocessing

Alternative 2: 99.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n0\_i, -2, -n1\_i\right)\\ \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, -0.041666666666666664, \mathsf{fma}\left(normAngle \cdot normAngle, n0\_i \cdot 0.001388888888888889 - \mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, n1\_i, n0\_i \cdot 0.022222222222222223\right), -0.16666666666666666, \mathsf{fma}\left(-0.0001984126984126984, n1\_i - n0\_i, t\_0 \cdot -0.001388888888888889\right)\right), -\mathsf{fma}\left(0.008333333333333333, n1\_i - n0\_i, t\_0 \cdot 0.027777777777777776\right)\right)\right), t\_0 \cdot -0.16666666666666666\right), n1\_i - n0\_i\right), n0\_i\right) \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (fma n0_i -2.0 (- n1_i))))
   (fma
    u
    (fma
     (* normAngle normAngle)
     (fma
      (* normAngle normAngle)
      (fma
       n0_i
       -0.041666666666666664
       (fma
        (* normAngle normAngle)
        (-
         (* n0_i 0.001388888888888889)
         (fma
          (fma 0.019444444444444445 n1_i (* n0_i 0.022222222222222223))
          -0.16666666666666666
          (fma
           -0.0001984126984126984
           (- n1_i n0_i)
           (* t_0 -0.001388888888888889))))
        (-
         (fma
          0.008333333333333333
          (- n1_i n0_i)
          (* t_0 0.027777777777777776)))))
      (* t_0 -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, -2.0f, -n1_i);
	return fmaf(u, fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), fmaf(n0_i, -0.041666666666666664f, fmaf((normAngle * normAngle), ((n0_i * 0.001388888888888889f) - fmaf(fmaf(0.019444444444444445f, n1_i, (n0_i * 0.022222222222222223f)), -0.16666666666666666f, fmaf(-0.0001984126984126984f, (n1_i - n0_i), (t_0 * -0.001388888888888889f)))), -fmaf(0.008333333333333333f, (n1_i - n0_i), (t_0 * 0.027777777777777776f)))), (t_0 * -0.16666666666666666f)), (n1_i - n0_i)), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = fma(n0_i, Float32(-2.0), Float32(-n1_i))
	return fma(u, fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), fma(n0_i, Float32(-0.041666666666666664), fma(Float32(normAngle * normAngle), Float32(Float32(n0_i * Float32(0.001388888888888889)) - fma(fma(Float32(0.019444444444444445), n1_i, Float32(n0_i * Float32(0.022222222222222223))), Float32(-0.16666666666666666), fma(Float32(-0.0001984126984126984), Float32(n1_i - n0_i), Float32(t_0 * Float32(-0.001388888888888889))))), Float32(-fma(Float32(0.008333333333333333), Float32(n1_i - n0_i), Float32(t_0 * Float32(0.027777777777777776)))))), Float32(t_0 * Float32(-0.16666666666666666))), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) + \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) + {normAngle}^{2} \cdot \left(\left(\frac{-1}{5040} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{7}\right) + \frac{-1}{5040} \cdot \left(n1\_i \cdot {u}^{7}\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right) + \left(\frac{-1}{5040} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \left(\frac{-1}{5040} \cdot \left(n1\_i \cdot u\right) + \left(\frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \left(\frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(0.008333333333333333, \mathsf{fma}\left(n0\_i, {\left(1 - u\right)}^{5}, n1\_i \cdot {u}^{5}\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(-0.0001984126984126984, \mathsf{fma}\left(n0\_i, {\left(1 - u\right)}^{7}, n1\_i \cdot {u}^{7}\right), -\mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(n1\_i, 0.008333333333333333 \cdot \left({u}^{5} - u\right), -0.027777777777777776 \cdot \left(n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right)\right) + \mathsf{fma}\left(n0\_i, 0.008333333333333333 \cdot \left({\left(1 - u\right)}^{5} - \left(1 - u\right)\right), -0.027777777777777776 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right)\right)\right), \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), -0.0001984126984126984 \cdot \mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right)\right)\right)\right), -\mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right) \cdot 0.008333333333333333\right)\right)\right)\right)\right)\right)} \]

6. Taylor expanded in u around 0

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(u \cdot \left(\left(\frac{-1}{24} \cdot n0\_i + \left(u \cdot \left(\left(\frac{1}{12} \cdot n0\_i + \left(u \cdot \left(\left(\frac{-1}{12} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{1}{144} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{12} \cdot n0\_i + \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{36} \cdot n0\_i\right)\right) + \frac{-1}{720} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) - \frac{1}{36} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right) + {normAngle}^{2} \cdot \left(\frac{-1}{240} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \left(n0\_i + 2 \cdot n0\_i\right) + \frac{1}{12} \cdot n0\_i\right) + \frac{-1}{720} \cdot \left(n0\_i + 2 \cdot n0\_i\right)\right)\right)\right)\right) - \frac{1}{36} \cdot \left(n0\_i + 2 \cdot n0\_i\right)\right) + {normAngle}^{2} \cdot \left(\frac{1}{720} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{30} \cdot n0\_i + \left(\frac{-1}{120} \cdot n1\_i + \left(\frac{1}{36} \cdot n1\_i + \frac{1}{18} \cdot n0\_i\right)\right)\right) + \left(\frac{-1}{720} \cdot \left(-2 \cdot n0\_i + -1 \cdot n1\_i\right) + \frac{-1}{5040} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right)\right)\right) - \left(\frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right) + \frac{1}{36} \cdot \left(-2 \cdot n0\_i + -1 \cdot n1\_i\right)\right)\right)\right)}\right)\right)\right) \]

8. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(u \cdot \mathsf{fma}\left(n0\_i, -0.041666666666666664, \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, 0.08333333333333333, \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, -0.08333333333333333, \mathsf{fma}\left(normAngle \cdot normAngle, n0\_i \cdot 0.006944444444444444 - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.08333333333333333, \mathsf{fma}\left(n0\_i, 0.027777777777777776, -0.027777777777777776 \cdot n1\_i\right)\right), -0.16666666666666666, -0.001388888888888889 \cdot \left(n1\_i - n0\_i\right)\right), -0.027777777777777776 \cdot \left(n1\_i - n0\_i\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(\mathsf{fma}\left(n0\_i, -0.004166666666666667, 0.16666666666666666 \cdot \mathsf{fma}\left(-0.027777777777777776, 3 \cdot n0\_i, n0\_i \cdot 0.08333333333333333\right)\right) + 0.001388888888888889 \cdot \left(3 \cdot n0\_i\right)\right)\right)\right) + -0.027777777777777776 \cdot \left(3 \cdot n0\_i\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.001388888888888889 - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.03333333333333333, \mathsf{fma}\left(n1\_i, 0.019444444444444445, n0\_i \cdot 0.05555555555555555\right)\right), -0.16666666666666666, \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), -0.0001984126984126984 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)}\right)\right)\right) \]

9. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{\frac{-1}{12} \cdot n0\_i + \left(\frac{1}{12} \cdot n0\_i + u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right)\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{\left(\frac{1}{12} \cdot n0\_i + u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right)\right) + \frac{-1}{12} \cdot n0\_i}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    2. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{\left(u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + \frac{1}{12} \cdot n0\_i\right)} + \frac{-1}{12} \cdot n0\_i, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    3. associate-+l+ N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + \left(\frac{1}{12} \cdot n0\_i + \frac{-1}{12} \cdot n0\_i\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    4. distribute-rgt-out N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + \color{blue}{n0\_i \cdot \left(\frac{1}{12} + \frac{-1}{12}\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + n0\_i \cdot \color{blue}{0}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    6. mul0-rgt N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + \color{blue}{0}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    7. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right), 0\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    8. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{\frac{-1}{36} \cdot \left(n1\_i - n0\_i\right) + \frac{-1}{12} \cdot n0\_i}, 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    9. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \frac{-1}{36} \cdot \color{blue}{\left(n1\_i + \left(\mathsf{neg}\left(n0\_i\right)\right)\right)} + \frac{-1}{12} \cdot n0\_i, 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    10. mul-1-neg N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \frac{-1}{36} \cdot \left(n1\_i + \color{blue}{-1 \cdot n0\_i}\right) + \frac{-1}{12} \cdot n0\_i, 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    11. lower-fma.f32 N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(\frac{-1}{36}, n1\_i + -1 \cdot n0\_i, \frac{-1}{12} \cdot n0\_i\right)}, 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    12. mul-1-neg N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-1}{36}, n1\_i + \color{blue}{\left(\mathsf{neg}\left(n0\_i\right)\right)}, \frac{-1}{12} \cdot n0\_i\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    13. sub-neg N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-1}{36}, \color{blue}{n1\_i - n0\_i}, \frac{-1}{12} \cdot n0\_i\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    14. lower--.f32 N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-1}{36}, \color{blue}{n1\_i - n0\_i}, \frac{-1}{12} \cdot n0\_i\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-1}{36}, n1\_i - n0\_i, \color{blue}{n0\_i \cdot \frac{-1}{12}}\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    16. lower-*.f32 99.0

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, -0.041666666666666664, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(-0.027777777777777776, n1\_i - n0\_i, \color{blue}{n0\_i \cdot -0.08333333333333333}\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.001388888888888889 - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.03333333333333333, \mathsf{fma}\left(n1\_i, 0.019444444444444445, n0\_i \cdot 0.05555555555555555\right)\right), -0.16666666666666666, \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), -0.0001984126984126984 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

11. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, -0.041666666666666664, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(-0.027777777777777776, n1\_i - n0\_i, n0\_i \cdot -0.08333333333333333\right), 0\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.001388888888888889 - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.03333333333333333, \mathsf{fma}\left(n1\_i, 0.019444444444444445, n0\_i \cdot 0.05555555555555555\right)\right), -0.16666666666666666, \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), -0.0001984126984126984 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

12. Taylor expanded in u around 0

    \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(-2 \cdot n0\_i + -1 \cdot n1\_i\right) + {normAngle}^{2} \cdot \left(\left(\frac{-1}{24} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{1}{720} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{30} \cdot n0\_i + \left(\frac{7}{360} \cdot n1\_i + \frac{1}{18} \cdot n0\_i\right)\right) + \left(\frac{-1}{720} \cdot \left(-2 \cdot n0\_i - n1\_i\right) + \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \left(\frac{1}{120} \cdot \left(n1\_i - n0\_i\right) + \frac{1}{36} \cdot \left(-2 \cdot n0\_i - n1\_i\right)\right)\right)\right)\right)\right)} \]

14. Applied rewrites 98.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, -0.041666666666666664, \mathsf{fma}\left(normAngle \cdot normAngle, n0\_i \cdot 0.001388888888888889 - \mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, n1\_i, 0.022222222222222223 \cdot n0\_i\right), -0.16666666666666666, \mathsf{fma}\left(-0.0001984126984126984, n1\_i - n0\_i, -0.001388888888888889 \cdot \mathsf{fma}\left(n0\_i, -2, -n1\_i\right)\right)\right), -\mathsf{fma}\left(0.008333333333333333, n1\_i - n0\_i, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right) \cdot 0.027777777777777776\right)\right)\right), -0.16666666666666666 \cdot \mathsf{fma}\left(n0\_i, -2, -n1\_i\right)\right), n1\_i - n0\_i\right), n0\_i\right)} \]

15. Final simplification 98.9%

    \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, -0.041666666666666664, \mathsf{fma}\left(normAngle \cdot normAngle, n0\_i \cdot 0.001388888888888889 - \mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445, n1\_i, n0\_i \cdot 0.022222222222222223\right), -0.16666666666666666, \mathsf{fma}\left(-0.0001984126984126984, n1\_i - n0\_i, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right) \cdot -0.001388888888888889\right)\right), -\mathsf{fma}\left(0.008333333333333333, n1\_i - n0\_i, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right) \cdot 0.027777777777777776\right)\right)\right), \mathsf{fma}\left(n0\_i, -2, -n1\_i\right) \cdot -0.16666666666666666\right), n1\_i - n0\_i\right), n0\_i\right) \]
16. Add Preprocessing

Alternative 3: 99.1% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-sin.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-sin.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lower-fma.f32 98.1

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in normAngle around 0

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in n1_i around 0

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

10. Applied rewrites 98.8%

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

Alternative 4: 99.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) + \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) + {normAngle}^{2} \cdot \left(\left(\frac{-1}{5040} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{7}\right) + \frac{-1}{5040} \cdot \left(n1\_i \cdot {u}^{7}\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right) + \left(\frac{-1}{5040} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \left(\frac{-1}{5040} \cdot \left(n1\_i \cdot u\right) + \left(\frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \left(\frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(0.008333333333333333, \mathsf{fma}\left(n0\_i, {\left(1 - u\right)}^{5}, n1\_i \cdot {u}^{5}\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(-0.0001984126984126984, \mathsf{fma}\left(n0\_i, {\left(1 - u\right)}^{7}, n1\_i \cdot {u}^{7}\right), -\mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(n1\_i, 0.008333333333333333 \cdot \left({u}^{5} - u\right), -0.027777777777777776 \cdot \left(n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right)\right) + \mathsf{fma}\left(n0\_i, 0.008333333333333333 \cdot \left({\left(1 - u\right)}^{5} - \left(1 - u\right)\right), -0.027777777777777776 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right)\right)\right), \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), -0.0001984126984126984 \cdot \mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right)\right)\right)\right), -\mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right) \cdot 0.008333333333333333\right)\right)\right)\right)\right)\right)} \]

6. Taylor expanded in u around 0

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(u \cdot \left(\left(\frac{-1}{24} \cdot n0\_i + \left(u \cdot \left(\left(\frac{1}{12} \cdot n0\_i + \left(u \cdot \left(\left(\frac{-1}{12} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{1}{144} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{12} \cdot n0\_i + \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{36} \cdot n0\_i\right)\right) + \frac{-1}{720} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) - \frac{1}{36} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right) + {normAngle}^{2} \cdot \left(\frac{-1}{240} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \left(n0\_i + 2 \cdot n0\_i\right) + \frac{1}{12} \cdot n0\_i\right) + \frac{-1}{720} \cdot \left(n0\_i + 2 \cdot n0\_i\right)\right)\right)\right)\right) - \frac{1}{36} \cdot \left(n0\_i + 2 \cdot n0\_i\right)\right) + {normAngle}^{2} \cdot \left(\frac{1}{720} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{30} \cdot n0\_i + \left(\frac{-1}{120} \cdot n1\_i + \left(\frac{1}{36} \cdot n1\_i + \frac{1}{18} \cdot n0\_i\right)\right)\right) + \left(\frac{-1}{720} \cdot \left(-2 \cdot n0\_i + -1 \cdot n1\_i\right) + \frac{-1}{5040} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right)\right)\right) - \left(\frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right) + \frac{1}{36} \cdot \left(-2 \cdot n0\_i + -1 \cdot n1\_i\right)\right)\right)\right)}\right)\right)\right) \]

8. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(u \cdot \mathsf{fma}\left(n0\_i, -0.041666666666666664, \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, 0.08333333333333333, \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, -0.08333333333333333, \mathsf{fma}\left(normAngle \cdot normAngle, n0\_i \cdot 0.006944444444444444 - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.08333333333333333, \mathsf{fma}\left(n0\_i, 0.027777777777777776, -0.027777777777777776 \cdot n1\_i\right)\right), -0.16666666666666666, -0.001388888888888889 \cdot \left(n1\_i - n0\_i\right)\right), -0.027777777777777776 \cdot \left(n1\_i - n0\_i\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(\mathsf{fma}\left(n0\_i, -0.004166666666666667, 0.16666666666666666 \cdot \mathsf{fma}\left(-0.027777777777777776, 3 \cdot n0\_i, n0\_i \cdot 0.08333333333333333\right)\right) + 0.001388888888888889 \cdot \left(3 \cdot n0\_i\right)\right)\right)\right) + -0.027777777777777776 \cdot \left(3 \cdot n0\_i\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.001388888888888889 - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.03333333333333333, \mathsf{fma}\left(n1\_i, 0.019444444444444445, n0\_i \cdot 0.05555555555555555\right)\right), -0.16666666666666666, \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), -0.0001984126984126984 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)}\right)\right)\right) \]

9. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{\frac{-1}{12} \cdot n0\_i + \left(\frac{1}{12} \cdot n0\_i + u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right)\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{\left(\frac{1}{12} \cdot n0\_i + u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right)\right) + \frac{-1}{12} \cdot n0\_i}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    2. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{\left(u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + \frac{1}{12} \cdot n0\_i\right)} + \frac{-1}{12} \cdot n0\_i, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    3. associate-+l+ N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + \left(\frac{1}{12} \cdot n0\_i + \frac{-1}{12} \cdot n0\_i\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    4. distribute-rgt-out N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + \color{blue}{n0\_i \cdot \left(\frac{1}{12} + \frac{-1}{12}\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + n0\_i \cdot \color{blue}{0}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    6. mul0-rgt N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, u \cdot \left(\frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right)\right) + \color{blue}{0}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    7. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \frac{-1}{12} \cdot n0\_i + \frac{-1}{36} \cdot \left(n1\_i - n0\_i\right), 0\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    8. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{\frac{-1}{36} \cdot \left(n1\_i - n0\_i\right) + \frac{-1}{12} \cdot n0\_i}, 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    9. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \frac{-1}{36} \cdot \color{blue}{\left(n1\_i + \left(\mathsf{neg}\left(n0\_i\right)\right)\right)} + \frac{-1}{12} \cdot n0\_i, 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    10. mul-1-neg N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \frac{-1}{36} \cdot \left(n1\_i + \color{blue}{-1 \cdot n0\_i}\right) + \frac{-1}{12} \cdot n0\_i, 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    11. lower-fma.f32 N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(\frac{-1}{36}, n1\_i + -1 \cdot n0\_i, \frac{-1}{12} \cdot n0\_i\right)}, 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    12. mul-1-neg N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-1}{36}, n1\_i + \color{blue}{\left(\mathsf{neg}\left(n0\_i\right)\right)}, \frac{-1}{12} \cdot n0\_i\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    13. sub-neg N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-1}{36}, \color{blue}{n1\_i - n0\_i}, \frac{-1}{12} \cdot n0\_i\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    14. lower--.f32 N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-1}{36}, \color{blue}{n1\_i - n0\_i}, \frac{-1}{12} \cdot n0\_i\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), \mathsf{neg}\left(n0\_i\right)\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, \frac{-1}{24}, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-1}{36}, n1\_i - n0\_i, \color{blue}{n0\_i \cdot \frac{-1}{12}}\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \frac{1}{720} - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, \frac{-1}{30}, \mathsf{fma}\left(n1\_i, \frac{7}{360}, n0\_i \cdot \frac{1}{18}\right)\right), \frac{-1}{6}, \mathsf{fma}\left(\frac{-1}{720}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{-1}{5040} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(\frac{1}{36}, \mathsf{fma}\left(n0\_i, -2, \mathsf{neg}\left(n1\_i\right)\right), \frac{1}{120} \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

    16. lower-*.f32 99.0

        \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, -0.041666666666666664, \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(-0.027777777777777776, n1\_i - n0\_i, \color{blue}{n0\_i \cdot -0.08333333333333333}\right), 0\right), \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.001388888888888889 - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.03333333333333333, \mathsf{fma}\left(n1\_i, 0.019444444444444445, n0\_i \cdot 0.05555555555555555\right)\right), -0.16666666666666666, \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), -0.0001984126984126984 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

11. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \mathsf{fma}\left(n0\_i, -0.041666666666666664, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(-0.027777777777777776, n1\_i - n0\_i, n0\_i \cdot -0.08333333333333333\right), 0\right)}, \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.001388888888888889 - \mathsf{fma}\left(\mathsf{fma}\left(n0\_i, -0.03333333333333333, \mathsf{fma}\left(n1\_i, 0.019444444444444445, n0\_i \cdot 0.05555555555555555\right)\right), -0.16666666666666666, \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), -0.0001984126984126984 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)\right) \]

12. Taylor expanded in normAngle around 0

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

    1. associate-*r* N/A

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

    2. lower-fma.f32 N/A

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

14. Applied rewrites 98.7%

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

15. Final simplification 98.7%

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

Alternative 5: 99.0% accurate, 12.4× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* (* normAngle normAngle) -0.16666666666666666)
   (+ n0_i (fma n0_i -3.0 (- 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(((normAngle * normAngle) * -0.16666666666666666f), (n0_i + fmaf(n0_i, -3.0f, -n1_i)), (n1_i - n0_i)), n0_i);
}

Julia

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

Derivation

1. Initial program 98.0%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-sin.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-sin.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lower-fma.f32 98.1

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in normAngle around 0

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

10. Applied rewrites 98.6%

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

11. Final simplification 98.6%

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

Alternative 6: 71.0% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u, -n0\_i, n0\_i\right)\\ \mathbf{if}\;n0\_i \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (fma u (- n0_i) n0_i)))
   (if (<= n0_i -4.999999999099794e-24)
     t_0
     (if (<= n0_i 2.00000006274879e-22) (* u n1_i) t_0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = fmaf(u, -n0_i, n0_i);
	float tmp;
	if (n0_i <= -4.999999999099794e-24f) {
		tmp = t_0;
	} else if (n0_i <= 2.00000006274879e-22f) {
		tmp = u * n1_i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = fma(u, Float32(-n0_i), n0_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-4.999999999099794e-24))
		tmp = t_0;
	elseif (n0_i <= Float32(2.00000006274879e-22))
		tmp = Float32(u * n1_i);
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -5e-24 or 2.00000006e-22 < n0_i

   1. Initial program 98.7%

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

   3. Taylor expanded in n0_i around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. lower-fma.f32 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f32 N/A

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

      15. lower-sin.f32 78.0

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in normAngle around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

   8. Applied rewrites 78.0%

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

   9. Taylor expanded in normAngle around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. *-lft-identity N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. lower-fma.f32 N/A

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

       9. mul-1-neg N/A

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

       10. lower-neg.f32 78.1

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

   11. Applied rewrites 78.1%

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

   if -5e-24 < n0_i < 2.00000006e-22

   1. Initial program 96.9%

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

   3. Taylor expanded in normAngle around 0

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

      1. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 96.2

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in n0_i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 68.1

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

   8. Applied rewrites 68.1%

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

Alternative 7: 70.8% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n0\_i \cdot \left(1 - u\right)\\ \mathbf{if}\;n0\_i \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* n0_i (- 1.0 u))))
   (if (<= n0_i -4.999999999099794e-24)
     t_0
     (if (<= n0_i 2.00000006274879e-22) (* 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 <= -4.999999999099794e-24f) {
		tmp = t_0;
	} else if (n0_i <= 2.00000006274879e-22f) {
		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 <= (-4.999999999099794e-24)) then
        tmp = t_0
    else if (n0_i <= 2.00000006274879e-22) 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(-4.999999999099794e-24))
		tmp = t_0;
	elseif (n0_i <= Float32(2.00000006274879e-22))
		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(-4.999999999099794e-24))
		tmp = t_0;
	elseif (n0_i <= single(2.00000006274879e-22))
		tmp = u * n1_i;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -5e-24 or 2.00000006e-22 < n0_i

   1. Initial program 98.7%

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

   3. Taylor expanded in normAngle around 0

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

      1. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in n0_i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 77.7

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

   8. Applied rewrites 77.7%

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

   if -5e-24 < n0_i < 2.00000006e-22

   1. Initial program 96.9%

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

   3. Taylor expanded in normAngle around 0

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

      1. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 96.2

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in n0_i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 68.1

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

   8. Applied rewrites 68.1%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{elif}\;n0\_i \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.9% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.5000000786160286e-23)
   n0_i
   (if (<= n0_i 2.00000006274879e-22) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.5000000786160286e-23f) {
		tmp = n0_i;
	} else if (n0_i <= 2.00000006274879e-22f) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-1.5000000786160286e-23)) then
        tmp = n0_i
    else if (n0_i <= 2.00000006274879e-22) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-1.5000000786160286e-23))
		tmp = n0_i;
	elseif (n0_i <= Float32(2.00000006274879e-22))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-1.5000000786160286e-23))
		tmp = n0_i;
	elseif (n0_i <= single(2.00000006274879e-22))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.50000008e-23 or 2.00000006e-22 < n0_i

   1. Initial program 98.8%

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

   3. Taylor expanded in n0_i around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. lower-fma.f32 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f32 N/A

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

      15. lower-sin.f32 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 59.6%

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

      1. *-rgt-identity 59.6

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

   10. Applied rewrites 59.6%

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

   if -1.50000008e-23 < n0_i < 2.00000006e-22

   1. Initial program 96.9%

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

   3. Taylor expanded in normAngle around 0

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

      1. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in n0_i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 67.7

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

   8. Applied rewrites 67.7%

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

Alternative 9: 98.2% accurate, 45.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 97.4

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

5. Applied rewrites 97.4%

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

6. Taylor expanded in u 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. lower-fma.f32 N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. lower--.f32 97.8

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

8. Applied rewrites 97.8%

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

Alternative 10: 46.6% accurate, 459.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in n0_i around inf

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. *-commutative N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-lft1-in N/A

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

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

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

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

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

   11. mul-1-neg N/A

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

   12. lower-fma.f32 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f32 N/A

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

   15. lower-sin.f32 58.9

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

5. Applied rewrites 58.9%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 46.6%

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

   1. *-rgt-identity 46.6

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

10. Applied rewrites 46.6%

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

Reproduce

herbie shell --seed 2024219 
(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)))