Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 99.1%

Time: 15.8s

Alternatives: 9

Speedup: 60.1×

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: 99.1% accurate, 9.4× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (+
   (*
    (* normAngle normAngle)
    (+
     (* (+ (* n0_i 0.3333333333333333) (* n1_i 0.16666666666666666)) u)
     (*
      (* (* normAngle normAngle) u)
      (+
       (* -0.16666666666666666 (* n0_i -0.3333333333333333))
       (-
        (* n0_i 0.008333333333333333)
        (+ (* n1_i -0.019444444444444445) (* n0_i 0.041666666666666664)))))))
   (* u (- n1_i n0_i)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (((normAngle * normAngle) * ((((n0_i * 0.3333333333333333f) + (n1_i * 0.16666666666666666f)) * u) + (((normAngle * normAngle) * u) * ((-0.16666666666666666f * (n0_i * -0.3333333333333333f)) + ((n0_i * 0.008333333333333333f) - ((n1_i * -0.019444444444444445f) + (n0_i * 0.041666666666666664f))))))) + (u * (n1_i - n0_i)));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i + (((normangle * normangle) * ((((n0_i * 0.3333333333333333e0) + (n1_i * 0.16666666666666666e0)) * u) + (((normangle * normangle) * u) * (((-0.16666666666666666e0) * (n0_i * (-0.3333333333333333e0))) + ((n0_i * 0.008333333333333333e0) - ((n1_i * (-0.019444444444444445e0)) + (n0_i * 0.041666666666666664e0))))))) + (u * (n1_i - n0_i)))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(Float32(Float32(normAngle * normAngle) * Float32(Float32(Float32(Float32(n0_i * Float32(0.3333333333333333)) + Float32(n1_i * Float32(0.16666666666666666))) * u) + Float32(Float32(Float32(normAngle * normAngle) * u) * Float32(Float32(Float32(-0.16666666666666666) * Float32(n0_i * Float32(-0.3333333333333333))) + Float32(Float32(n0_i * Float32(0.008333333333333333)) - Float32(Float32(n1_i * Float32(-0.019444444444444445)) + Float32(n0_i * Float32(0.041666666666666664)))))))) + Float32(u * Float32(n1_i - n0_i))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (((normAngle * normAngle) * ((((n0_i * single(0.3333333333333333)) + (n1_i * single(0.16666666666666666))) * u) + (((normAngle * normAngle) * u) * ((single(-0.16666666666666666) * (n0_i * single(-0.3333333333333333))) + ((n0_i * single(0.008333333333333333)) - ((n1_i * single(-0.019444444444444445)) + (n0_i * single(0.041666666666666664)))))))) + (u * (n1_i - n0_i)));
end

Derivation

1. Initial program 97.2%

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

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \color{blue}{\left(u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} + \color{blue}{-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} + \left(\mathsf{neg}\left(\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)\right)\right)\right)\right) \]

   5. unsub-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} - \color{blue}{\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(\frac{n1\_i \cdot normAngle}{\sin normAngle}\right), \color{blue}{\left(\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(\frac{normAngle \cdot n1\_i}{\sin normAngle}\right), \left(\frac{\color{blue}{n0\_i} \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), \left(\frac{\color{blue}{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \left(\frac{n1\_i}{\sin normAngle}\right)\right), \left(\frac{\color{blue}{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \sin normAngle\right)\right), \left(\frac{n0\_i \cdot \color{blue}{\left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

   11. sin-lowering-sin.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\frac{n0\_i \cdot \left(normAngle \cdot \color{blue}{\cos normAngle}\right)}{\sin normAngle}\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\frac{\left(normAngle \cdot \cos normAngle\right) \cdot n0\_i}{\sin \color{blue}{normAngle}}\right)\right)\right)\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\left(normAngle \cdot \cos normAngle\right) \cdot \color{blue}{\frac{n0\_i}{\sin normAngle}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\left(normAngle \cdot \cos normAngle\right), \color{blue}{\left(\frac{n0\_i}{\sin normAngle}\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \cos normAngle\right), \left(\frac{\color{blue}{n0\_i}}{\sin normAngle}\right)\right)\right)\right)\right) \]

   16. cos-lowering-cos.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{cos.f32}\left(normAngle\right)\right), \left(\frac{n0\_i}{\sin normAngle}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{cos.f32}\left(normAngle\right)\right), \mathsf{/.f32}\left(n0\_i, \color{blue}{\sin normAngle}\right)\right)\right)\right)\right) \]

5. Simplified 99.2%

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

6. Taylor expanded in normAngle around 0

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

8. Simplified 99.4%

   \[\leadsto \color{blue}{n0\_i + \left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(n0\_i \cdot 0.3333333333333333 + n1\_i \cdot 0.16666666666666666\right) \cdot u + \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(-0.16666666666666666 \cdot \left(n0\_i \cdot -0.3333333333333333\right) + \left(n0\_i \cdot 0.008333333333333333 - \left(n1\_i \cdot -0.019444444444444445 + n0\_i \cdot 0.041666666666666664\right)\right)\right)\right) + u \cdot \left(n1\_i - n0\_i\right)\right)} \]
9. Add Preprocessing

Alternative 2: 98.9% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot \left(n1\_i - n0\_i\right)\\ n0\_i + \left(t\_0 + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \left(u \cdot \left(n0\_i \cdot -0.5\right) + n0\_i \cdot 0.5\right) + \left(n0\_i \cdot -0.16666666666666666 + 0.16666666666666666 \cdot \left(n0\_i + t\_0\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* u (- n1_i n0_i))))
   (+
    n0_i
    (+
     t_0
     (*
      (* normAngle normAngle)
      (+
       (* u (+ (* u (* n0_i -0.5)) (* n0_i 0.5)))
       (+
        (* n0_i -0.16666666666666666)
        (* 0.16666666666666666 (+ n0_i t_0)))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = u * (n1_i - n0_i);
	return n0_i + (t_0 + ((normAngle * normAngle) * ((u * ((u * (n0_i * -0.5f)) + (n0_i * 0.5f))) + ((n0_i * -0.16666666666666666f) + (0.16666666666666666f * (n0_i + t_0))))));
}

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 = u * (n1_i - n0_i)
    code = n0_i + (t_0 + ((normangle * normangle) * ((u * ((u * (n0_i * (-0.5e0))) + (n0_i * 0.5e0))) + ((n0_i * (-0.16666666666666666e0)) + (0.16666666666666666e0 * (n0_i + t_0))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(u * Float32(n1_i - n0_i))
	return Float32(n0_i + Float32(t_0 + Float32(Float32(normAngle * normAngle) * Float32(Float32(u * Float32(Float32(u * Float32(n0_i * Float32(-0.5))) + Float32(n0_i * Float32(0.5)))) + Float32(Float32(n0_i * Float32(-0.16666666666666666)) + Float32(Float32(0.16666666666666666) * Float32(n0_i + t_0)))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	t_0 = u * (n1_i - n0_i);
	tmp = n0_i + (t_0 + ((normAngle * normAngle) * ((u * ((u * (n0_i * single(-0.5))) + (n0_i * single(0.5)))) + ((n0_i * single(-0.16666666666666666)) + (single(0.16666666666666666) * (n0_i + t_0))))));
end

Derivation

1. Initial program 97.2%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-out N/A

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

      \[\leadsto \mathsf{/.f32}\left(\left(n1\_i \cdot \sin \left(u \cdot normAngle\right) + n0\_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right), \color{blue}{\sin normAngle}\right) \]

3. Simplified 75.6%

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

5. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(normAngle \cdot \left(n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right)\right)\right)\right)}, \mathsf{sin.f32}\left(normAngle\right)\right) \]
6. Step-by-step derivation

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

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

   2. associate-+r+ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(normAngle, \left(\left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(\left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right), \left({normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(n0\_i \cdot \left(1 - u\right)\right), \left(n1\_i \cdot u\right)\right), \left({normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \left(1 - u\right)\right), \left(n1\_i \cdot u\right)\right), \left({normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(1, u\right)\right), \left(n1\_i \cdot u\right)\right), \left({normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(1, u\right)\right), \left(u \cdot n1\_i\right)\right), \left({normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{*.f32}\left(u, n1\_i\right)\right), \left({normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{*.f32}\left(u, n1\_i\right)\right), \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right) \cdot {normAngle}^{2}\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

   10. distribute-lft-out N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{*.f32}\left(u, n1\_i\right)\right), \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3} + n1\_i \cdot {u}^{3}\right)\right) \cdot {normAngle}^{2}\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{*.f32}\left(u, n1\_i\right)\right), \left(\frac{-1}{6} \cdot \left(\left(n0\_i \cdot {\left(1 - u\right)}^{3} + n1\_i \cdot {u}^{3}\right) \cdot {normAngle}^{2}\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{*.f32}\left(u, n1\_i\right)\right), \mathsf{*.f32}\left(\frac{-1}{6}, \left(\left(n0\_i \cdot {\left(1 - u\right)}^{3} + n1\_i \cdot {u}^{3}\right) \cdot {normAngle}^{2}\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{*.f32}\left(u, n1\_i\right)\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(\left(n0\_i \cdot {\left(1 - u\right)}^{3} + n1\_i \cdot {u}^{3}\right), \left({normAngle}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

7. Simplified 75.3%

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

8. Taylor expanded in u around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(normAngle \cdot \left(n0\_i + \frac{-1}{6} \cdot \left(n0\_i \cdot {normAngle}^{2}\right)\right) + u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{3} \cdot u\right)\right) + normAngle \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \frac{1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right)\right)\right)\right)\right)}, \mathsf{sin.f32}\left(normAngle\right)\right) \]
9. Step-by-step derivation

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

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

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(normAngle, \left(n0\_i + \frac{-1}{6} \cdot \left(n0\_i \cdot {normAngle}^{2}\right)\right)\right), \left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{3} \cdot u\right)\right) + normAngle \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \frac{1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(n0\_i, \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {normAngle}^{2}\right)\right)\right)\right), \left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{3} \cdot u\right)\right) + normAngle \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \frac{1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(\frac{-1}{6}, \left(n0\_i \cdot {normAngle}^{2}\right)\right)\right)\right), \left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{3} \cdot u\right)\right) + normAngle \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \frac{1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(n0\_i, \left({normAngle}^{2}\right)\right)\right)\right)\right), \left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{3} \cdot u\right)\right) + normAngle \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \frac{1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(n0\_i, \left(normAngle \cdot normAngle\right)\right)\right)\right)\right), \left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{3} \cdot u\right)\right) + normAngle \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \frac{1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(n0\_i, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right)\right)\right), \left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{3} \cdot u\right)\right) + normAngle \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \frac{1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(n0\_i, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right)\right)\right), \mathsf{*.f32}\left(u, \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{3} \cdot u\right)\right) + normAngle \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \frac{1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(n0\_i, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right)\right)\right), \mathsf{*.f32}\left(u, \left(normAngle \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \frac{1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right)\right)\right) + \frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{3} \cdot u\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(n0\_i, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right)\right)\right), \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(normAngle \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \frac{1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right)\right)\right)\right), \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{3} \cdot u\right)\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(normAngle\right)\right) \]

10. Simplified 75.0%

    \[\leadsto \frac{\color{blue}{normAngle \cdot \left(n0\_i + -0.16666666666666666 \cdot \left(n0\_i \cdot \left(normAngle \cdot normAngle\right)\right)\right) + u \cdot \left(normAngle \cdot \left(\left(n1\_i - n0\_i\right) + \left(n0\_i \cdot \left(normAngle \cdot normAngle\right)\right) \cdot 0.5\right) + -0.5 \cdot \left(n0\_i \cdot \left(u \cdot \left(normAngle \cdot \left(normAngle \cdot normAngle\right)\right)\right)\right)\right)}}{\sin normAngle} \]

11. Taylor expanded in normAngle around 0

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

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \color{blue}{\left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot n0\_i + u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot u\right) + \frac{1}{2} \cdot n0\_i\right)\right) - \frac{-1}{6} \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)}\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{+.f32}\left(\left(u \cdot \left(n1\_i - n0\_i\right)\right), \color{blue}{\left({normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot n0\_i + u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot u\right) + \frac{1}{2} \cdot n0\_i\right)\right) - \frac{-1}{6} \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)}\right)\right) \]

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

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

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right), \left({normAngle}^{\color{blue}{2}} \cdot \left(\left(\frac{-1}{6} \cdot n0\_i + u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot u\right) + \frac{1}{2} \cdot n0\_i\right)\right) - \frac{-1}{6} \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right), \mathsf{*.f32}\left(\left({normAngle}^{2}\right), \color{blue}{\left(\left(\frac{-1}{6} \cdot n0\_i + u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot u\right) + \frac{1}{2} \cdot n0\_i\right)\right) - \frac{-1}{6} \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)\right)}\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right), \mathsf{*.f32}\left(\left(normAngle \cdot normAngle\right), \left(\color{blue}{\left(\frac{-1}{6} \cdot n0\_i + u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot u\right) + \frac{1}{2} \cdot n0\_i\right)\right)} - \frac{-1}{6} \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\color{blue}{\left(\frac{-1}{6} \cdot n0\_i + u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot u\right) + \frac{1}{2} \cdot n0\_i\right)\right)} - \frac{-1}{6} \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right) \]

    8. sub-neg N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\left(\frac{-1}{6} \cdot n0\_i + u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot u\right) + \frac{1}{2} \cdot n0\_i\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6} \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)}\right)\right)\right)\right) \]

    9. +-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot u\right) + \frac{1}{2} \cdot n0\_i\right) + \frac{-1}{6} \cdot n0\_i\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{6} \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)}\right)\right)\right)\right)\right)\right) \]

    10. associate-+l+ N/A

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot u\right) + \frac{1}{2} \cdot n0\_i\right) + \color{blue}{\left(\frac{-1}{6} \cdot n0\_i + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot u\right) + \frac{1}{2} \cdot n0\_i\right)\right), \color{blue}{\left(\frac{-1}{6} \cdot n0\_i + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)}\right)\right)\right)\right) \]

13. Simplified 99.3%

    \[\leadsto \color{blue}{n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \left(u \cdot \left(n0\_i \cdot -0.5\right) + n0\_i \cdot 0.5\right) + \left(n0\_i \cdot -0.16666666666666666 + 0.16666666666666666 \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)} \]
14. Add Preprocessing

Alternative 3: 98.9% accurate, 22.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i + (u * (n1_i + (((normangle * normangle) * ((n0_i * 0.3333333333333333e0) + (n1_i * 0.16666666666666666e0))) - n0_i)))
end function

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * (n1_i + (((normAngle * normAngle) * ((n0_i * single(0.3333333333333333)) + (n1_i * single(0.16666666666666666)))) - n0_i)));
end

Derivation

1. Initial program 97.2%

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

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \color{blue}{\left(u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} + \color{blue}{-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} + \left(\mathsf{neg}\left(\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)\right)\right)\right)\right) \]

   5. unsub-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} - \color{blue}{\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(\frac{n1\_i \cdot normAngle}{\sin normAngle}\right), \color{blue}{\left(\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(\frac{normAngle \cdot n1\_i}{\sin normAngle}\right), \left(\frac{\color{blue}{n0\_i} \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), \left(\frac{\color{blue}{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \left(\frac{n1\_i}{\sin normAngle}\right)\right), \left(\frac{\color{blue}{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \sin normAngle\right)\right), \left(\frac{n0\_i \cdot \color{blue}{\left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

   11. sin-lowering-sin.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\frac{n0\_i \cdot \left(normAngle \cdot \color{blue}{\cos normAngle}\right)}{\sin normAngle}\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\frac{\left(normAngle \cdot \cos normAngle\right) \cdot n0\_i}{\sin \color{blue}{normAngle}}\right)\right)\right)\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\left(normAngle \cdot \cos normAngle\right) \cdot \color{blue}{\frac{n0\_i}{\sin normAngle}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\left(normAngle \cdot \cos normAngle\right), \color{blue}{\left(\frac{n0\_i}{\sin normAngle}\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \cos normAngle\right), \left(\frac{\color{blue}{n0\_i}}{\sin normAngle}\right)\right)\right)\right)\right) \]

   16. cos-lowering-cos.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{cos.f32}\left(normAngle\right)\right), \left(\frac{n0\_i}{\sin normAngle}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{cos.f32}\left(normAngle\right)\right), \mathsf{/.f32}\left(n0\_i, \color{blue}{\sin normAngle}\right)\right)\right)\right)\right) \]

5. Simplified 99.2%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(\left(n1\_i + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot n0\_i - \left(\frac{-1}{2} \cdot n0\_i + \frac{-1}{6} \cdot n1\_i\right)\right)\right) - n0\_i\right)}\right)\right) \]
7. Step-by-step derivation

   1. associate--l+ N/A

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

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

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

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\left({normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot n0\_i - \left(\frac{-1}{2} \cdot n0\_i + \frac{-1}{6} \cdot n1\_i\right)\right)\right), \color{blue}{n0\_i}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left({normAngle}^{2}\right), \left(\frac{-1}{6} \cdot n0\_i - \left(\frac{-1}{2} \cdot n0\_i + \frac{-1}{6} \cdot n1\_i\right)\right)\right), n0\_i\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left(normAngle \cdot normAngle\right), \left(\frac{-1}{6} \cdot n0\_i - \left(\frac{-1}{2} \cdot n0\_i + \frac{-1}{6} \cdot n1\_i\right)\right)\right), n0\_i\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\frac{-1}{6} \cdot n0\_i - \left(\frac{-1}{2} \cdot n0\_i + \frac{-1}{6} \cdot n1\_i\right)\right)\right), n0\_i\right)\right)\right)\right) \]

   7. associate--r+ N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\left(\frac{-1}{6} \cdot n0\_i - \frac{-1}{2} \cdot n0\_i\right) - \frac{-1}{6} \cdot n1\_i\right)\right), n0\_i\right)\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\left(\frac{-1}{6} \cdot n0\_i - \frac{-1}{2} \cdot n0\_i\right) + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\left(\frac{-1}{6} \cdot n0\_i - \frac{-1}{2} \cdot n0\_i\right), \left(\mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\left(n0\_i \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \left(\frac{-1}{6} - \frac{-1}{2}\right)\right), \left(\mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \left(\mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \left(\mathsf{neg}\left(n1\_i \cdot \frac{-1}{6}\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \left(n1\_i \cdot \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \left(n1\_i \cdot \frac{1}{6}\right)\right)\right), n0\_i\right)\right)\right)\right) \]

   16. *-lowering-*.f32 99.3%

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right), n0\_i\right)\right)\right)\right) \]

8. Simplified 99.3%

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

Alternative 4: 69.7% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -3.499999942646603 \cdot 10^{-11}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{elif}\;n1\_i \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -3.499999942646603e-11)
   (* n1_i u)
   (if (<= n1_i 5.00000006675716e-11) (* n0_i (- 1.0 u)) (* n1_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -3.499999942646603e-11f) {
		tmp = n1_i * u;
	} else if (n1_i <= 5.00000006675716e-11f) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = n1_i * u;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n1_i <= (-3.499999942646603e-11)) then
        tmp = n1_i * u
    else if (n1_i <= 5.00000006675716e-11) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = n1_i * u
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-3.499999942646603e-11))
		tmp = Float32(n1_i * u);
	elseif (n1_i <= Float32(5.00000006675716e-11))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(n1_i * u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n1_i <= single(-3.499999942646603e-11))
		tmp = n1_i * u;
	elseif (n1_i <= single(5.00000006675716e-11))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = n1_i * u;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -3.49999994e-11 or 5.00000007e-11 < n1_i

   1. Initial program 97.2%

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

   3. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(\color{blue}{n0\_i}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(u, normAngle\right)\right), \mathsf{/.f32}\left(1, \mathsf{sin.f32}\left(normAngle\right)\right)\right), n1\_i\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 91.4%

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

   6. Taylor expanded in normAngle around 0

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(\color{blue}{u}, n1\_i\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 92.6%

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

   9. Taylor expanded in n0_i around 0

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

       1. *-lowering-*.f32 75.8%

          \[\leadsto \mathsf{*.f32}\left(n1\_i, \color{blue}{u}\right) \]

   11. Simplified 75.8%

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

   if -3.49999994e-11 < n1_i < 5.00000007e-11

   1. Initial program 97.2%

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

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

         \[\leadsto \mathsf{+.f32}\left(n0\_i, \color{blue}{\left(u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} + \color{blue}{-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} + \left(\mathsf{neg}\left(\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} - \color{blue}{\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(\frac{n1\_i \cdot normAngle}{\sin normAngle}\right), \color{blue}{\left(\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(\frac{normAngle \cdot n1\_i}{\sin normAngle}\right), \left(\frac{\color{blue}{n0\_i} \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), \left(\frac{\color{blue}{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \left(\frac{n1\_i}{\sin normAngle}\right)\right), \left(\frac{\color{blue}{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \sin normAngle\right)\right), \left(\frac{n0\_i \cdot \color{blue}{\left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

      11. sin-lowering-sin.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\frac{n0\_i \cdot \left(normAngle \cdot \color{blue}{\cos normAngle}\right)}{\sin normAngle}\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\frac{\left(normAngle \cdot \cos normAngle\right) \cdot n0\_i}{\sin \color{blue}{normAngle}}\right)\right)\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\left(normAngle \cdot \cos normAngle\right) \cdot \color{blue}{\frac{n0\_i}{\sin normAngle}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\left(normAngle \cdot \cos normAngle\right), \color{blue}{\left(\frac{n0\_i}{\sin normAngle}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \cos normAngle\right), \left(\frac{\color{blue}{n0\_i}}{\sin normAngle}\right)\right)\right)\right)\right) \]

      16. cos-lowering-cos.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{cos.f32}\left(normAngle\right)\right), \left(\frac{n0\_i}{\sin normAngle}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{cos.f32}\left(normAngle\right)\right), \mathsf{/.f32}\left(n0\_i, \color{blue}{\sin normAngle}\right)\right)\right)\right)\right) \]

   5. Simplified 99.2%

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

   6. Taylor expanded in normAngle around 0

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \color{blue}{n0\_i}\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 99.0%

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

   9. Taylor expanded in n0_i around inf

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

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

          \[\leadsto \mathsf{*.f32}\left(n0\_i, \color{blue}{\left(1 + -1 \cdot u\right)}\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{*.f32}\left(n0\_i, \left(1 + \left(\mathsf{neg}\left(u\right)\right)\right)\right) \]

       3. unsub-neg N/A

          \[\leadsto \mathsf{*.f32}\left(n0\_i, \left(1 - \color{blue}{u}\right)\right) \]

       4. --lowering--.f32 73.9%

          \[\leadsto \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(1, \color{blue}{u}\right)\right) \]

   11. Simplified 73.9%

       \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.7% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i + (u * ((n1_i * (1.0e0 + ((normangle * normangle) * 0.16666666666666666e0))) - n0_i))
end function

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * ((n1_i * (single(1.0) + ((normAngle * normAngle) * single(0.16666666666666666)))) - n0_i));
end

Derivation

1. Initial program 97.2%

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

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \color{blue}{\left(u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} + \color{blue}{-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} + \left(\mathsf{neg}\left(\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)\right)\right)\right)\right) \]

   5. unsub-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} - \color{blue}{\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(\frac{n1\_i \cdot normAngle}{\sin normAngle}\right), \color{blue}{\left(\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(\frac{normAngle \cdot n1\_i}{\sin normAngle}\right), \left(\frac{\color{blue}{n0\_i} \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), \left(\frac{\color{blue}{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \left(\frac{n1\_i}{\sin normAngle}\right)\right), \left(\frac{\color{blue}{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \sin normAngle\right)\right), \left(\frac{n0\_i \cdot \color{blue}{\left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

   11. sin-lowering-sin.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\frac{n0\_i \cdot \left(normAngle \cdot \color{blue}{\cos normAngle}\right)}{\sin normAngle}\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\frac{\left(normAngle \cdot \cos normAngle\right) \cdot n0\_i}{\sin \color{blue}{normAngle}}\right)\right)\right)\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\left(normAngle \cdot \cos normAngle\right) \cdot \color{blue}{\frac{n0\_i}{\sin normAngle}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\left(normAngle \cdot \cos normAngle\right), \color{blue}{\left(\frac{n0\_i}{\sin normAngle}\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \cos normAngle\right), \left(\frac{\color{blue}{n0\_i}}{\sin normAngle}\right)\right)\right)\right)\right) \]

   16. cos-lowering-cos.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{cos.f32}\left(normAngle\right)\right), \left(\frac{n0\_i}{\sin normAngle}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{cos.f32}\left(normAngle\right)\right), \mathsf{/.f32}\left(n0\_i, \color{blue}{\sin normAngle}\right)\right)\right)\right)\right) \]

5. Simplified 99.2%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \color{blue}{n0\_i}\right)\right)\right) \]
7. Step-by-step derivation

8. Simplified 99.1%

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

9. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(\left(n1\_i + \frac{1}{6} \cdot \left(n1\_i \cdot {normAngle}^{2}\right)\right) - n0\_i\right)}\right)\right) \]
10. Step-by-step derivation

    1. *-lft-identity N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\left(1 \cdot n1\_i + \frac{1}{6} \cdot \left(n1\_i \cdot {normAngle}^{2}\right)\right) - n0\_i\right)\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\left(1 \cdot n1\_i + \frac{1}{6} \cdot \left({normAngle}^{2} \cdot n1\_i\right)\right) - n0\_i\right)\right)\right) \]

    3. associate-*r* N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\left(1 \cdot n1\_i + \left(\frac{1}{6} \cdot {normAngle}^{2}\right) \cdot n1\_i\right) - n0\_i\right)\right)\right) \]

    4. distribute-rgt-in N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(n1\_i \cdot \left(1 + \frac{1}{6} \cdot {normAngle}^{2}\right) - n0\_i\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(n1\_i \cdot \left(1 + \frac{1}{6} \cdot {normAngle}^{2}\right)\right), \color{blue}{n0\_i}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(n1\_i, \left(1 + \frac{1}{6} \cdot {normAngle}^{2}\right)\right), n0\_i\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(n1\_i, \mathsf{+.f32}\left(1, \left(\frac{1}{6} \cdot {normAngle}^{2}\right)\right)\right), n0\_i\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(n1\_i, \mathsf{+.f32}\left(1, \left({normAngle}^{2} \cdot \frac{1}{6}\right)\right)\right), n0\_i\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(n1\_i, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({normAngle}^{2}\right), \frac{1}{6}\right)\right)\right), n0\_i\right)\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(n1\_i, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(normAngle \cdot normAngle\right), \frac{1}{6}\right)\right)\right), n0\_i\right)\right)\right) \]

    11. *-lowering-*.f32 99.2%

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(n1\_i, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \frac{1}{6}\right)\right)\right), n0\_i\right)\right)\right) \]

11. Simplified 99.2%

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

Alternative 6: 59.4% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.000000045813705 \cdot 10^{-18}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{elif}\;n1\_i \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;n0\_i\\ \mathbf{else}:\\ \;\;\;\;n1\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -1.000000045813705e-18)
   (* n1_i u)
   (if (<= n1_i 5.00000006675716e-11) n0_i (* n1_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -1.000000045813705e-18f) {
		tmp = n1_i * u;
	} else if (n1_i <= 5.00000006675716e-11f) {
		tmp = n0_i;
	} else {
		tmp = n1_i * u;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n1_i <= (-1.000000045813705e-18)) then
        tmp = n1_i * u
    else if (n1_i <= 5.00000006675716e-11) then
        tmp = n0_i
    else
        tmp = n1_i * u
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-1.000000045813705e-18))
		tmp = Float32(n1_i * u);
	elseif (n1_i <= Float32(5.00000006675716e-11))
		tmp = n0_i;
	else
		tmp = Float32(n1_i * u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n1_i <= single(-1.000000045813705e-18))
		tmp = n1_i * u;
	elseif (n1_i <= single(5.00000006675716e-11))
		tmp = n0_i;
	else
		tmp = n1_i * u;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.00000005e-18 or 5.00000007e-11 < n1_i

   1. Initial program 97.3%

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

   3. Taylor expanded in u around 0

      \[\leadsto \mathsf{+.f32}\left(\color{blue}{n0\_i}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(u, normAngle\right)\right), \mathsf{/.f32}\left(1, \mathsf{sin.f32}\left(normAngle\right)\right)\right), n1\_i\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 88.4%

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

   6. Taylor expanded in normAngle around 0

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(\color{blue}{u}, n1\_i\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 89.6%

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

   9. Taylor expanded in n0_i around 0

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

       1. *-lowering-*.f32 66.6%

          \[\leadsto \mathsf{*.f32}\left(n1\_i, \color{blue}{u}\right) \]

   11. Simplified 66.6%

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

   if -1.00000005e-18 < n1_i < 5.00000007e-11

   1. Initial program 97.2%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 61.3%

      \[\leadsto \color{blue}{n0\_i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.0% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(n1\_i - n0\_i\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

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

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \color{blue}{\left(u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} + \color{blue}{-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} + \left(\mathsf{neg}\left(\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)\right)\right)\right)\right) \]

   5. unsub-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\frac{n1\_i \cdot normAngle}{\sin normAngle} - \color{blue}{\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(\frac{n1\_i \cdot normAngle}{\sin normAngle}\right), \color{blue}{\left(\frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(\frac{normAngle \cdot n1\_i}{\sin normAngle}\right), \left(\frac{\color{blue}{n0\_i} \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\left(normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), \left(\frac{\color{blue}{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \left(\frac{n1\_i}{\sin normAngle}\right)\right), \left(\frac{\color{blue}{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \sin normAngle\right)\right), \left(\frac{n0\_i \cdot \color{blue}{\left(normAngle \cdot \cos normAngle\right)}}{\sin normAngle}\right)\right)\right)\right) \]

   11. sin-lowering-sin.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\frac{n0\_i \cdot \left(normAngle \cdot \color{blue}{\cos normAngle}\right)}{\sin normAngle}\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\frac{\left(normAngle \cdot \cos normAngle\right) \cdot n0\_i}{\sin \color{blue}{normAngle}}\right)\right)\right)\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \left(\left(normAngle \cdot \cos normAngle\right) \cdot \color{blue}{\frac{n0\_i}{\sin normAngle}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\left(normAngle \cdot \cos normAngle\right), \color{blue}{\left(\frac{n0\_i}{\sin normAngle}\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \cos normAngle\right), \left(\frac{\color{blue}{n0\_i}}{\sin normAngle}\right)\right)\right)\right)\right) \]

   16. cos-lowering-cos.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{cos.f32}\left(normAngle\right)\right), \left(\frac{n0\_i}{\sin normAngle}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{/.f32}\left(n1\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{cos.f32}\left(normAngle\right)\right), \mathsf{/.f32}\left(n0\_i, \color{blue}{\sin normAngle}\right)\right)\right)\right)\right) \]

5. Simplified 99.2%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(n1\_i - n0\_i\right)}\right)\right) \]
7. Step-by-step derivation

   1. --lowering--.f32 98.4%

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(n1\_i, \color{blue}{n0\_i}\right)\right)\right) \]

8. Simplified 98.4%

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

Alternative 8: 81.5% accurate, 84.2× speedup

Math

\[\begin{array}{l} \\ n0\_i + n1\_i \cdot u \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

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

3. Taylor expanded in u around 0

   \[\leadsto \mathsf{+.f32}\left(\color{blue}{n0\_i}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(u, normAngle\right)\right), \mathsf{/.f32}\left(1, \mathsf{sin.f32}\left(normAngle\right)\right)\right), n1\_i\right)\right) \]
4. Step-by-step derivation

5. Simplified 81.5%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(\color{blue}{u}, n1\_i\right)\right) \]
7. Step-by-step derivation

8. Simplified 82.6%

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

9. Final simplification 82.6%

   \[\leadsto n0\_i + n1\_i \cdot u \]
10. Add Preprocessing

Alternative 9: 46.2% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

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

3. Taylor expanded in u around 0

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

5. Simplified 46.9%

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

Reproduce

herbie shell --seed 2024288 
(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)))