Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 99.3%

Time: 18.8s

Alternatives: 11

Speedup: 30.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u + \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right)\\ \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, {\left(1 - u\right)}^{5} \cdot \left(0.008333333333333333 \cdot n0\_i\right) - \mathsf{fma}\left(0.027777777777777776, n0\_i \cdot t\_0, n0\_i \cdot \mathsf{fma}\left(u, -0.008333333333333333, 0.008333333333333333\right)\right), t\_0 \cdot \left(n0\_i \cdot -0.16666666666666666\right)\right), \mathsf{fma}\left(u, -n0\_i, n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (+ u (fma (- 1.0 u) (* (- 1.0 u) (- 1.0 u)) -1.0))))
   (+
    (fma
     (* normAngle normAngle)
     (fma
      (* normAngle normAngle)
      (-
       (* (pow (- 1.0 u) 5.0) (* 0.008333333333333333 n0_i))
       (fma
        0.027777777777777776
        (* n0_i t_0)
        (* n0_i (fma u -0.008333333333333333 0.008333333333333333))))
      (* t_0 (* n0_i -0.16666666666666666)))
     (fma u (- n0_i) n0_i))
    (* (* normAngle (/ u (sin normAngle))) n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = u + fmaf((1.0f - u), ((1.0f - u) * (1.0f - u)), -1.0f);
	return fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), ((powf((1.0f - u), 5.0f) * (0.008333333333333333f * n0_i)) - fmaf(0.027777777777777776f, (n0_i * t_0), (n0_i * fmaf(u, -0.008333333333333333f, 0.008333333333333333f)))), (t_0 * (n0_i * -0.16666666666666666f))), fmaf(u, -n0_i, n0_i)) + ((normAngle * (u / sinf(normAngle))) * n1_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(u + fma(Float32(Float32(1.0) - u), Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), Float32(-1.0)))
	return Float32(fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), Float32(Float32((Float32(Float32(1.0) - u) ^ Float32(5.0)) * Float32(Float32(0.008333333333333333) * n0_i)) - fma(Float32(0.027777777777777776), Float32(n0_i * t_0), Float32(n0_i * fma(u, Float32(-0.008333333333333333), Float32(0.008333333333333333))))), Float32(t_0 * Float32(n0_i * Float32(-0.16666666666666666)))), fma(u, Float32(-n0_i), n0_i)) + Float32(Float32(normAngle * Float32(u / sin(normAngle))) * n1_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 u around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in normAngle around 0

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

   \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, {\left(1 - u\right)}^{5} \cdot \left(0.008333333333333333 \cdot n0\_i\right) - \mathsf{fma}\left(0.027777777777777776, n0\_i \cdot \left(u + \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right)\right), n0\_i \cdot \mathsf{fma}\left(u, -0.008333333333333333, 0.008333333333333333\right)\right), \left(u + \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right)\right) \cdot \left(n0\_i \cdot -0.16666666666666666\right)\right), \mathsf{fma}\left(u, -n0\_i, n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]
10. Add Preprocessing

Alternative 2: 99.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, n0\_i \cdot -0.05555555555555555, n0\_i \cdot 0.022222222222222223\right), \left(u + \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right)\right) \cdot \left(n0\_i \cdot -0.16666666666666666\right)\right), \mathsf{fma}\left(u, -n0\_i, n0\_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* (* normAngle (/ u (sin normAngle))) n1_i)
  (fma
   (* normAngle normAngle)
   (fma
    (* normAngle normAngle)
    (*
     u
     (fma (* u u) (* n0_i -0.05555555555555555) (* n0_i 0.022222222222222223)))
    (*
     (+ u (fma (- 1.0 u) (* (- 1.0 u) (- 1.0 u)) -1.0))
     (* n0_i -0.16666666666666666)))
   (fma u (- n0_i) n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((normAngle * (u / sinf(normAngle))) * n1_i) + fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), (u * fmaf((u * u), (n0_i * -0.05555555555555555f), (n0_i * 0.022222222222222223f))), ((u + fmaf((1.0f - u), ((1.0f - u) * (1.0f - u)), -1.0f)) * (n0_i * -0.16666666666666666f))), fmaf(u, -n0_i, n0_i));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(Float32(normAngle * Float32(u / sin(normAngle))) * n1_i) + fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), Float32(u * fma(Float32(u * u), Float32(n0_i * Float32(-0.05555555555555555)), Float32(n0_i * Float32(0.022222222222222223)))), Float32(Float32(u + fma(Float32(Float32(1.0) - u), Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), Float32(-1.0))) * Float32(n0_i * Float32(-0.16666666666666666)))), 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. Taylor expanded in u around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in normAngle around 0

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

8. Simplified 99.3%

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

9. Taylor expanded in u around 0

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

    1. lower-*.f32 N/A

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

    2. +-commutative N/A

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

    3. associate--l+ N/A

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

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

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

    5. metadata-eval N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f32 N/A

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

    8. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(\color{blue}{u \cdot u}, \frac{-1}{18} \cdot n0\_i, \frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{18} \cdot n0\_i + \frac{-1}{120} \cdot n0\_i\right)\right), \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    9. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(\color{blue}{u \cdot u}, \frac{-1}{18} \cdot n0\_i, \frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{18} \cdot n0\_i + \frac{-1}{120} \cdot n0\_i\right)\right), \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    10. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, \color{blue}{n0\_i \cdot \frac{-1}{18}}, \frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{18} \cdot n0\_i + \frac{-1}{120} \cdot n0\_i\right)\right), \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    11. lower-*.f32 N/A

        \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, \color{blue}{n0\_i \cdot \frac{-1}{18}}, \frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{18} \cdot n0\_i + \frac{-1}{120} \cdot n0\_i\right)\right), \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    12. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, n0\_i \cdot \frac{-1}{18}, \color{blue}{n0\_i \cdot \frac{-1}{24}} - \left(\frac{-1}{18} \cdot n0\_i + \frac{-1}{120} \cdot n0\_i\right)\right), \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    13. distribute-rgt-out N/A

        \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, n0\_i \cdot \frac{-1}{18}, n0\_i \cdot \frac{-1}{24} - \color{blue}{n0\_i \cdot \left(\frac{-1}{18} + \frac{-1}{120}\right)}\right), \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    14. distribute-lft-out-- N/A

        \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, n0\_i \cdot \frac{-1}{18}, \color{blue}{n0\_i \cdot \left(\frac{-1}{24} - \left(\frac{-1}{18} + \frac{-1}{120}\right)\right)}\right), \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    15. lower-*.f32 N/A

        \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, n0\_i \cdot \frac{-1}{18}, \color{blue}{n0\_i \cdot \left(\frac{-1}{24} - \left(\frac{-1}{18} + \frac{-1}{120}\right)\right)}\right), \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    16. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, n0\_i \cdot \frac{-1}{18}, n0\_i \cdot \left(\frac{-1}{24} - \color{blue}{\frac{-23}{360}}\right)\right), \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    17. metadata-eval 99.3

        \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, n0\_i \cdot -0.05555555555555555, n0\_i \cdot \color{blue}{0.022222222222222223}\right), \left(-0.16666666666666666 \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, -n0\_i, n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

11. Simplified 99.3%

    \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{u \cdot \mathsf{fma}\left(u \cdot u, n0\_i \cdot -0.05555555555555555, n0\_i \cdot 0.022222222222222223\right)}, \left(-0.16666666666666666 \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, -n0\_i, n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

12. Final simplification 99.3%

    \[\leadsto \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, n0\_i \cdot -0.05555555555555555, n0\_i \cdot 0.022222222222222223\right), \left(u + \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right)\right) \cdot \left(n0\_i \cdot -0.16666666666666666\right)\right), \mathsf{fma}\left(u, -n0\_i, n0\_i\right)\right) \]
13. Add Preprocessing

Alternative 3: 99.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \left(n0\_i \cdot 0.022222222222222223\right), \left(u + \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right)\right) \cdot \left(n0\_i \cdot -0.16666666666666666\right)\right), \mathsf{fma}\left(u, -n0\_i, n0\_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* (* normAngle (/ u (sin normAngle))) n1_i)
  (fma
   (* normAngle normAngle)
   (fma
    (* normAngle normAngle)
    (* u (* n0_i 0.022222222222222223))
    (*
     (+ u (fma (- 1.0 u) (* (- 1.0 u) (- 1.0 u)) -1.0))
     (* n0_i -0.16666666666666666)))
   (fma u (- n0_i) n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((normAngle * (u / sinf(normAngle))) * n1_i) + fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), (u * (n0_i * 0.022222222222222223f)), ((u + fmaf((1.0f - u), ((1.0f - u) * (1.0f - u)), -1.0f)) * (n0_i * -0.16666666666666666f))), fmaf(u, -n0_i, n0_i));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(Float32(normAngle * Float32(u / sin(normAngle))) * n1_i) + fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), Float32(u * Float32(n0_i * Float32(0.022222222222222223))), Float32(Float32(u + fma(Float32(Float32(1.0) - u), Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), Float32(-1.0))) * Float32(n0_i * Float32(-0.16666666666666666)))), 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. Taylor expanded in u around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in normAngle around 0

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

8. Simplified 99.3%

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

9. Taylor expanded in u around 0

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

    1. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{u \cdot \left(\frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{18} \cdot n0\_i + \frac{-1}{120} \cdot n0\_i\right)\right)}, \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    2. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \left(\color{blue}{n0\_i \cdot \frac{-1}{24}} - \left(\frac{-1}{18} \cdot n0\_i + \frac{-1}{120} \cdot n0\_i\right)\right), \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    3. distribute-rgt-out N/A

       \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \left(n0\_i \cdot \frac{-1}{24} - \color{blue}{n0\_i \cdot \left(\frac{-1}{18} + \frac{-1}{120}\right)}\right), \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    4. distribute-lft-out-- N/A

       \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \color{blue}{\left(n0\_i \cdot \left(\frac{-1}{24} - \left(\frac{-1}{18} + \frac{-1}{120}\right)\right)\right)}, \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    5. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \color{blue}{\left(n0\_i \cdot \left(\frac{-1}{24} - \left(\frac{-1}{18} + \frac{-1}{120}\right)\right)\right)}, \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \left(n0\_i \cdot \left(\frac{-1}{24} - \color{blue}{\frac{-23}{360}}\right)\right), \left(\frac{-1}{6} \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, \mathsf{neg}\left(n0\_i\right), n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

    7. metadata-eval 99.3

       \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \left(n0\_i \cdot \color{blue}{0.022222222222222223}\right), \left(-0.16666666666666666 \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, -n0\_i, n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

11. Simplified 99.3%

    \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{u \cdot \left(n0\_i \cdot 0.022222222222222223\right)}, \left(-0.16666666666666666 \cdot n0\_i\right) \cdot \left(\mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right) + u\right)\right), \mathsf{fma}\left(u, -n0\_i, n0\_i\right)\right) + \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i \]

12. Final simplification 99.3%

    \[\leadsto \left(normAngle \cdot \frac{u}{\sin normAngle}\right) \cdot n1\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \left(n0\_i \cdot 0.022222222222222223\right), \left(u + \mathsf{fma}\left(1 - u, \left(1 - u\right) \cdot \left(1 - u\right), -1\right)\right) \cdot \left(n0\_i \cdot -0.16666666666666666\right)\right), \mathsf{fma}\left(u, -n0\_i, n0\_i\right)\right) \]
13. Add Preprocessing

Alternative 4: 99.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(Float32(normAngle * Float32(u / sin(normAngle))) * n1_i) + fma(Float32(normAngle * normAngle), Float32(Float32(u + fma(Float32(Float32(1.0) - u), Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), Float32(-1.0))) * Float32(n0_i * Float32(-0.16666666666666666))), 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. Taylor expanded in u around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

Alternative 5: 99.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-fma.f32 N/A

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

5. Simplified 98.1%

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

6. Taylor expanded in u around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.8

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 6: 99.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(fma(u, Float32(-n0_i), n0_i) + Float32(Float32(normAngle * Float32(u / sin(normAngle))) * n1_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 u around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. distribute-lft-neg-out N/A

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

   5. distribute-rgt-neg-out N/A

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

   6. mul-1-neg N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f32 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f32 98.7

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

8. Simplified 98.7%

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

Alternative 7: 98.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-fma.f32 N/A

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

5. Simplified 98.1%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. distribute-lft-out-- N/A

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

   7. lower-*.f32 N/A

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

   8. sub-neg N/A

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

   9. cube-mult N/A

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

   10. unpow2 N/A

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

   11. mul-1-neg N/A

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

   12. lower-fma.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

   15. mul-1-neg N/A

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

   16. lower-neg.f32 98.3

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

8. Simplified 98.3%

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

9. Final simplification 98.3%

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

Alternative 8: 98.4% accurate, 7.8× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (* normAngle normAngle)
  (*
   (* n0_i -0.16666666666666666)
   (fma (- 1.0 u) (* (- 1.0 u) (- 1.0 u)) (+ u -1.0)))
  (fma u n1_i (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 * -0.16666666666666666f) * fmaf((1.0f - u), ((1.0f - u) * (1.0f - u)), (u + -1.0f))), fmaf(u, n1_i, fmaf(u, -n0_i, n0_i)));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(normAngle * normAngle), Float32(Float32(n0_i * Float32(-0.16666666666666666)) * fma(Float32(Float32(1.0) - u), Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), Float32(u + Float32(-1.0)))), fma(u, n1_i, 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. Taylor expanded in u around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in normAngle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 97.6

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

8. Simplified 97.6%

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

9. Taylor expanded in normAngle around 0

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

    1. associate-+r+ N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f32 N/A

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

11. Simplified 98.1%

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

Alternative 9: 98.1% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, n1_i, 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. Taylor expanded in u around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in normAngle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 97.6

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

8. Simplified 97.6%

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

9. Taylor expanded in normAngle around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. sub-neg N/A

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

    5. mul-1-neg N/A

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

    6. +-commutative N/A

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

    7. distribute-rgt-in N/A

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

    8. *-lft-identity N/A

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

    9. mul-1-neg N/A

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

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

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

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

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

    12. mul-1-neg N/A

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

    13. lower-fma.f32 N/A

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

    14. mul-1-neg N/A

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

    15. lower-neg.f32 97.8

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

11. Simplified 97.8%

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

Alternative 10: 81.9% accurate, 65.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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 u around 0

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

5. Simplified 81.0%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i + n1\_i \cdot u} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 80.7

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

8. Simplified 80.7%

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

Alternative 11: 39.0% accurate, 76.5× speedup

Math

\[\begin{array}{l} \\ u \cdot n1\_i \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = u * n1_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 u around 0

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

5. Simplified 81.0%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i + n1\_i \cdot u} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 80.7

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

8. Simplified 80.7%

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

9. Taylor expanded in u around inf

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

    1. *-commutative N/A

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

    2. lower-*.f32 37.9

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

11. Simplified 37.9%

    \[\leadsto \color{blue}{u \cdot n1\_i} \]
12. 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)))