Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.0% → 99.3%

Time: 13.6s

Alternatives: 5

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: 97.0% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.4%

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

Alternative 2: 99.2% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

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

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

   12. metadata-eval N/A

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

   13. distribute-lft-out N/A

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

   14. lower-*.f32 N/A

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

   15. lower-fma.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in normAngle around 0

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

8. Applied rewrites 99.2%

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

Alternative 3: 98.9% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in normAngle around 0

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

5. Applied rewrites 98.9%

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

Alternative 4: 98.8% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

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

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

   12. metadata-eval N/A

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

   13. distribute-lft-out N/A

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

   14. lower-*.f32 N/A

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

   15. lower-fma.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

7. Applied rewrites 83.2%

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

8. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f32 98.9

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

10. Applied rewrites 98.9%

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

11. Final simplification 98.9%

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

Alternative 5: 98.2% accurate, 45.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

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

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

   12. metadata-eval N/A

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

   13. distribute-lft-out N/A

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

   14. lower-*.f32 N/A

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

   15. lower-fma.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-+l+ N/A

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

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

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

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

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

   9. mul-1-neg N/A

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

   10. distribute-rgt-in N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. mul-1-neg N/A

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

   14. unsub-neg N/A

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

   15. lower--.f32 98.4

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

8. Applied rewrites 98.4%

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

Reproduce

herbie shell --seed 2024212 
(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)))