Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.3%

Time: 15.4s

Alternatives: 16

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: 97.3% 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

\[\begin{array}{l} \\ n0\_i + u \cdot \left(normAngle \cdot \frac{n1\_i}{\sin normAngle} + n0\_i \cdot \left(u \cdot \left(-0.5 \cdot \left(normAngle \cdot normAngle\right)\right) - \frac{normAngle \cdot \cos normAngle}{\sin normAngle}\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    (* normAngle (/ n1_i (sin normAngle)))
    (*
     n0_i
     (-
      (* u (* -0.5 (* normAngle normAngle)))
      (/ (* normAngle (cos normAngle)) (sin normAngle))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((normAngle * (n1_i / sinf(normAngle))) + (n0_i * ((u * (-0.5f * (normAngle * normAngle))) - ((normAngle * cosf(normAngle)) / sinf(normAngle))))));
}

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 * ((normangle * (n1_i / sin(normangle))) + (n0_i * ((u * ((-0.5e0) * (normangle * normangle))) - ((normangle * cos(normangle)) / sin(normangle))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(Float32(normAngle * Float32(n1_i / sin(normAngle))) + Float32(n0_i * Float32(Float32(u * Float32(Float32(-0.5) * Float32(normAngle * normAngle))) - Float32(Float32(normAngle * cos(normAngle)) / sin(normAngle)))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * ((normAngle * (n1_i / sin(normAngle))) + (n0_i * ((u * (single(-0.5) * (normAngle * normAngle))) - ((normAngle * cos(normAngle)) / sin(normAngle))))));
end

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u around 0

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

5. Simplified 99.6%

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

Alternative 2: 99.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(n1\_i + \left(normAngle \cdot \left(normAngle \cdot \left(normAngle \cdot \left(normAngle \cdot \left(n1\_i \cdot 0.019444444444444445\right)\right) + n1\_i \cdot 0.16666666666666666\right)\right) + n0\_i \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(u \cdot -0.5\right) - \frac{normAngle}{\tan normAngle}\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    n1_i
    (+
     (*
      normAngle
      (*
       normAngle
       (+
        (* normAngle (* normAngle (* n1_i 0.019444444444444445)))
        (* n1_i 0.16666666666666666))))
     (*
      n0_i
      (-
       (* (* normAngle normAngle) (* u -0.5))
       (/ normAngle (tan normAngle)))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * (n1_i + ((normAngle * (normAngle * ((normAngle * (normAngle * (n1_i * 0.019444444444444445f))) + (n1_i * 0.16666666666666666f)))) + (n0_i * (((normAngle * normAngle) * (u * -0.5f)) - (normAngle / tanf(normAngle)))))));
}

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 * ((normangle * (normangle * (n1_i * 0.019444444444444445e0))) + (n1_i * 0.16666666666666666e0)))) + (n0_i * (((normangle * normangle) * (u * (-0.5e0))) - (normangle / tan(normangle)))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(n1_i + Float32(Float32(normAngle * Float32(normAngle * Float32(Float32(normAngle * Float32(normAngle * Float32(n1_i * Float32(0.019444444444444445)))) + Float32(n1_i * Float32(0.16666666666666666))))) + Float32(n0_i * Float32(Float32(Float32(normAngle * normAngle) * Float32(u * Float32(-0.5))) - Float32(normAngle / tan(normAngle))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * (n1_i + ((normAngle * (normAngle * ((normAngle * (normAngle * (n1_i * single(0.019444444444444445)))) + (n1_i * single(0.16666666666666666))))) + (n0_i * (((normAngle * normAngle) * (u * single(-0.5))) - (normAngle / tan(normAngle)))))));
end

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u around 0

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(normAngle \cdot \frac{n1\_i}{\sin normAngle} + n0\_i \cdot \left(u \cdot \left(-0.5 \cdot \left(normAngle \cdot normAngle\right)\right) - \frac{normAngle \cdot \cos normAngle}{\sin normAngle}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

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

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

7. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{u \cdot \left(\frac{normAngle}{\frac{\sin normAngle}{n1\_i}} + n0\_i \cdot \left(u \cdot \left(-0.5 \cdot \left(normAngle \cdot normAngle\right)\right) - \frac{1}{\frac{\tan normAngle}{normAngle}}\right)\right) + n0\_i} \]

8. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\color{blue}{\left(n1\_i + {normAngle}^{2} \cdot \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)}, \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \left({normAngle}^{2} \cdot \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \mathsf{*.f32}\left(\left({normAngle}^{2}\right), \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \mathsf{*.f32}\left(\left(normAngle \cdot normAngle\right), \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot n1\_i\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) + \frac{1}{6} \cdot n1\_i\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right), \left(\frac{1}{6} \cdot n1\_i\right)\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

10. Simplified 99.6%

    \[\leadsto u \cdot \left(\color{blue}{\left(n1\_i + \left(normAngle \cdot normAngle\right) \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot 0.019444444444444445\right) + n1\_i \cdot 0.16666666666666666\right)\right)} + n0\_i \cdot \left(u \cdot \left(-0.5 \cdot \left(normAngle \cdot normAngle\right)\right) - \frac{1}{\frac{\tan normAngle}{normAngle}}\right)\right) + n0\_i \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(\left(\left(\left(n1\_i + \left(normAngle \cdot normAngle\right) \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot \frac{7}{360}\right) + n1\_i \cdot \frac{1}{6}\right)\right) + n0\_i \cdot \left(u \cdot \left(\frac{-1}{2} \cdot \left(normAngle \cdot normAngle\right)\right) - \frac{1}{\frac{\tan normAngle}{normAngle}}\right)\right) \cdot u\right), n0\_i\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(\left(n1\_i + \left(normAngle \cdot normAngle\right) \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot \frac{7}{360}\right) + n1\_i \cdot \frac{1}{6}\right)\right) + n0\_i \cdot \left(u \cdot \left(\frac{-1}{2} \cdot \left(normAngle \cdot normAngle\right)\right) - \frac{1}{\frac{\tan normAngle}{normAngle}}\right)\right), u\right), n0\_i\right) \]

12. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{\left(n1\_i + \left(normAngle \cdot \left(normAngle \cdot \left(normAngle \cdot \left(normAngle \cdot \left(n1\_i \cdot 0.019444444444444445\right)\right) + n1\_i \cdot 0.16666666666666666\right)\right) + n0\_i \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(u \cdot -0.5\right) - \frac{normAngle}{\tan normAngle}\right)\right)\right) \cdot u} + n0\_i \]

13. Final simplification 99.6%

    \[\leadsto n0\_i + u \cdot \left(n1\_i + \left(normAngle \cdot \left(normAngle \cdot \left(normAngle \cdot \left(normAngle \cdot \left(n1\_i \cdot 0.019444444444444445\right)\right) + n1\_i \cdot 0.16666666666666666\right)\right) + n0\_i \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(u \cdot -0.5\right) - \frac{normAngle}{\tan normAngle}\right)\right)\right) \]
14. Add Preprocessing

Alternative 3: 99.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(n1\_i + \left(normAngle \cdot \left(normAngle \cdot \left(n0\_i \cdot \left(0.3333333333333333 + u \cdot -0.5\right) + \left(n1\_i \cdot 0.16666666666666666 + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.022222222222222223 + \left(n1\_i \cdot 0.019444444444444445 + normAngle \cdot \left(normAngle \cdot \left(n0\_i \cdot 0.0021164021164021165 - \left(n1\_i \cdot -0.0032407407407407406 + n1\_i \cdot 0.0011904761904761906\right)\right)\right)\right)\right)\right)\right)\right) - n0\_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    n1_i
    (-
     (*
      normAngle
      (*
       normAngle
       (+
        (* n0_i (+ 0.3333333333333333 (* u -0.5)))
        (+
         (* n1_i 0.16666666666666666)
         (*
          (* normAngle normAngle)
          (+
           (* n0_i 0.022222222222222223)
           (+
            (* n1_i 0.019444444444444445)
            (*
             normAngle
             (*
              normAngle
              (-
               (* n0_i 0.0021164021164021165)
               (+
                (* n1_i -0.0032407407407407406)
                (* n1_i 0.0011904761904761906))))))))))))
     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 + (u * -0.5f))) + ((n1_i * 0.16666666666666666f) + ((normAngle * normAngle) * ((n0_i * 0.022222222222222223f) + ((n1_i * 0.019444444444444445f) + (normAngle * (normAngle * ((n0_i * 0.0021164021164021165f) - ((n1_i * -0.0032407407407407406f) + (n1_i * 0.0011904761904761906f)))))))))))) - 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 + (u * (-0.5e0)))) + ((n1_i * 0.16666666666666666e0) + ((normangle * normangle) * ((n0_i * 0.022222222222222223e0) + ((n1_i * 0.019444444444444445e0) + (normangle * (normangle * ((n0_i * 0.0021164021164021165e0) - ((n1_i * (-0.0032407407407407406e0)) + (n1_i * 0.0011904761904761906e0)))))))))))) - n0_i)))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(n1_i + Float32(Float32(normAngle * Float32(normAngle * Float32(Float32(n0_i * Float32(Float32(0.3333333333333333) + Float32(u * Float32(-0.5)))) + Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(Float32(normAngle * normAngle) * Float32(Float32(n0_i * Float32(0.022222222222222223)) + Float32(Float32(n1_i * Float32(0.019444444444444445)) + Float32(normAngle * Float32(normAngle * Float32(Float32(n0_i * Float32(0.0021164021164021165)) - Float32(Float32(n1_i * Float32(-0.0032407407407407406)) + Float32(n1_i * Float32(0.0011904761904761906))))))))))))) - 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) + (u * single(-0.5)))) + ((n1_i * single(0.16666666666666666)) + ((normAngle * normAngle) * ((n0_i * single(0.022222222222222223)) + ((n1_i * single(0.019444444444444445)) + (normAngle * (normAngle * ((n0_i * single(0.0021164021164021165)) - ((n1_i * single(-0.0032407407407407406)) + (n1_i * single(0.0011904761904761906))))))))))))) - n0_i)));
end

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u around 0

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

5. Simplified 99.6%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{45} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{2}{945} \cdot n0\_i - \left(\frac{-1}{5040} \cdot n1\_i + \left(\frac{1}{720} \cdot n1\_i + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right)\right)\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right)}\right)\right) \]

8. Simplified 99.5%

   \[\leadsto n0\_i + u \cdot \color{blue}{\left(n1\_i + \left(normAngle \cdot \left(normAngle \cdot \left(n0\_i \cdot \left(0.3333333333333333 + -0.5 \cdot u\right) + \left(\left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.022222222222222223 + \left(normAngle \cdot \left(normAngle \cdot \left(n0\_i \cdot 0.0021164021164021165 - \left(n1\_i \cdot -0.0032407407407407406 + n1\_i \cdot 0.0011904761904761906\right)\right)\right) + n1\_i \cdot 0.019444444444444445\right)\right) + n1\_i \cdot 0.16666666666666666\right)\right)\right) - n0\_i\right)\right)} \]

9. Final simplification 99.5%

   \[\leadsto n0\_i + u \cdot \left(n1\_i + \left(normAngle \cdot \left(normAngle \cdot \left(n0\_i \cdot \left(0.3333333333333333 + u \cdot -0.5\right) + \left(n1\_i \cdot 0.16666666666666666 + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.022222222222222223 + \left(n1\_i \cdot 0.019444444444444445 + normAngle \cdot \left(normAngle \cdot \left(n0\_i \cdot 0.0021164021164021165 - \left(n1\_i \cdot -0.0032407407407407406 + n1\_i \cdot 0.0011904761904761906\right)\right)\right)\right)\right)\right)\right)\right) - n0\_i\right)\right) \]
10. Add Preprocessing

Alternative 4: 99.1% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \left(0.3333333333333333 + u \cdot -0.5\right) + \left(n1\_i \cdot 0.16666666666666666 + \left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot 0.019444444444444445 + n0\_i \cdot 0.022222222222222223\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    (- n1_i n0_i)
    (*
     (* normAngle normAngle)
     (+
      (* n0_i (+ 0.3333333333333333 (* u -0.5)))
      (+
       (* n1_i 0.16666666666666666)
       (*
        (* normAngle normAngle)
        (+ (* n1_i 0.019444444444444445) (* n0_i 0.022222222222222223))))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((n1_i - n0_i) + ((normAngle * normAngle) * ((n0_i * (0.3333333333333333f + (u * -0.5f))) + ((n1_i * 0.16666666666666666f) + ((normAngle * normAngle) * ((n1_i * 0.019444444444444445f) + (n0_i * 0.022222222222222223f))))))));
}

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) + ((normangle * normangle) * ((n0_i * (0.3333333333333333e0 + (u * (-0.5e0)))) + ((n1_i * 0.16666666666666666e0) + ((normangle * normangle) * ((n1_i * 0.019444444444444445e0) + (n0_i * 0.022222222222222223e0))))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(Float32(n1_i - n0_i) + Float32(Float32(normAngle * normAngle) * Float32(Float32(n0_i * Float32(Float32(0.3333333333333333) + Float32(u * Float32(-0.5)))) + Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(Float32(normAngle * normAngle) * Float32(Float32(n1_i * Float32(0.019444444444444445)) + Float32(n0_i * Float32(0.022222222222222223))))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * ((n1_i - n0_i) + ((normAngle * normAngle) * ((n0_i * (single(0.3333333333333333) + (u * single(-0.5)))) + ((n1_i * single(0.16666666666666666)) + ((normAngle * normAngle) * ((n1_i * single(0.019444444444444445)) + (n0_i * single(0.022222222222222223)))))))));
end

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u around 0

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

5. Simplified 99.6%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right)}\right)\right) \]
7. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\left(n1\_i + -1 \cdot n0\_i\right) + \color{blue}{{normAngle}^{2} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i + -1 \cdot n0\_i\right), \color{blue}{\left({normAngle}^{2} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)}\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i + \left(\mathsf{neg}\left(n0\_i\right)\right)\right), \left({normAngle}^{\color{blue}{2}} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i - n0\_i\right), \left(\color{blue}{{normAngle}^{2}} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \left(\color{blue}{{normAngle}^{2}} \cdot \left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\left({normAngle}^{2}\right), \color{blue}{\left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\left(normAngle \cdot normAngle\right), \left(\color{blue}{\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right)} - \frac{-1}{6} \cdot n1\_i\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\color{blue}{\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + {normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right)} - \frac{-1}{6} \cdot n1\_i\right)\right)\right)\right)\right) \]

   9. associate--l+ N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right) + \color{blue}{\left({normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\left(n0\_i \cdot \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right)\right), \color{blue}{\left({normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \left(\frac{1}{3} + \frac{-1}{2} \cdot u\right)\right), \left(\color{blue}{{normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)} - \frac{-1}{6} \cdot n1\_i\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{-1}{2} \cdot u\right)\right)\right), \left({normAngle}^{2} \cdot \color{blue}{\left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)} - \frac{-1}{6} \cdot n1\_i\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\frac{-1}{2}, u\right)\right)\right), \left({normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \color{blue}{\left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)}\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right)\right)\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(\frac{-1}{2}, u\right)\right)\right), \left({normAngle}^{2} \cdot \left(\frac{1}{45} \cdot n0\_i - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6} \cdot n1\_i\right)\right)}\right)\right)\right)\right)\right)\right) \]

8. Simplified 99.4%

   \[\leadsto n0\_i + u \cdot \color{blue}{\left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \left(0.3333333333333333 + -0.5 \cdot u\right) + \left(\left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.022222222222222223 + n1\_i \cdot 0.019444444444444445\right) + n1\_i \cdot 0.16666666666666666\right)\right)\right)} \]

9. Final simplification 99.4%

   \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \left(0.3333333333333333 + u \cdot -0.5\right) + \left(n1\_i \cdot 0.16666666666666666 + \left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot 0.019444444444444445 + n0\_i \cdot 0.022222222222222223\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 5: 99.1% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot \left(0.019444444444444445 + \left(normAngle \cdot normAngle\right) \cdot 0.00205026455026455\right)\right) + \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    (- n1_i n0_i)
    (*
     (* normAngle normAngle)
     (+
      (*
       (* normAngle normAngle)
       (*
        n1_i
        (+
         0.019444444444444445
         (* (* normAngle normAngle) 0.00205026455026455))))
      (+ (* n1_i 0.16666666666666666) (* n0_i 0.3333333333333333))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((n1_i - n0_i) + ((normAngle * normAngle) * (((normAngle * normAngle) * (n1_i * (0.019444444444444445f + ((normAngle * normAngle) * 0.00205026455026455f)))) + ((n1_i * 0.16666666666666666f) + (n0_i * 0.3333333333333333f))))));
}

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) + ((normangle * normangle) * (((normangle * normangle) * (n1_i * (0.019444444444444445e0 + ((normangle * normangle) * 0.00205026455026455e0)))) + ((n1_i * 0.16666666666666666e0) + (n0_i * 0.3333333333333333e0))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(Float32(n1_i - n0_i) + Float32(Float32(normAngle * normAngle) * Float32(Float32(Float32(normAngle * normAngle) * Float32(n1_i * Float32(Float32(0.019444444444444445) + Float32(Float32(normAngle * normAngle) * Float32(0.00205026455026455))))) + Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.3333333333333333))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * ((n1_i - n0_i) + ((normAngle * normAngle) * (((normAngle * normAngle) * (n1_i * (single(0.019444444444444445) + ((normAngle * normAngle) * single(0.00205026455026455))))) + ((n1_i * single(0.16666666666666666)) + (n0_i * single(0.3333333333333333)))))));
end

Derivation

1. Initial program 98.1%

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

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

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

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

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

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

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

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

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

   16. sin-lowering-sin.f32 96.3%

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

5. Simplified 96.3%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(\left(n1\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \left(\frac{1}{120} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(\frac{1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \frac{1}{120} \cdot n0\_i\right)\right) + \left(\frac{-1}{5040} \cdot n0\_i + \frac{1}{120} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right)\right)\right) - \left(\frac{-1}{720} \cdot n0\_i + \left(\frac{-1}{5040} \cdot n1\_i + \left(\frac{1}{720} \cdot n1\_i + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{36} \cdot n1\_i + \left(\frac{1}{120} \cdot n1\_i + \frac{1}{24} \cdot n0\_i\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot n0\_i + \frac{-1}{6} \cdot n1\_i\right)\right)\right) - n0\_i\right)}\right)\right) \]

8. Simplified 99.5%

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

9. Taylor expanded in n0_i around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \color{blue}{\left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-7}{2160} \cdot n1\_i + \frac{1}{840} \cdot n1\_i\right)\right) - \frac{-7}{360} \cdot n1\_i\right)}\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-7}{2160} \cdot n1\_i + \frac{1}{840} \cdot n1\_i\right)\right) + \left(\mathsf{neg}\left(\frac{-7}{360}\right)\right) \cdot n1\_i\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    2. mul-1-neg N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\left(\mathsf{neg}\left({normAngle}^{2} \cdot \left(\frac{-7}{2160} \cdot n1\_i + \frac{1}{840} \cdot n1\_i\right)\right)\right) + \left(\mathsf{neg}\left(\frac{-7}{360}\right)\right) \cdot n1\_i\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left({normAngle}^{2} \cdot \left(\mathsf{neg}\left(\left(\frac{-7}{2160} \cdot n1\_i + \frac{1}{840} \cdot n1\_i\right)\right)\right) + \left(\mathsf{neg}\left(\frac{-7}{360}\right)\right) \cdot n1\_i\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    4. distribute-rgt-out N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left({normAngle}^{2} \cdot \left(\mathsf{neg}\left(n1\_i \cdot \left(\frac{-7}{2160} + \frac{1}{840}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{-7}{360}\right)\right) \cdot n1\_i\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left({normAngle}^{2} \cdot \left(\mathsf{neg}\left(n1\_i \cdot \frac{-31}{15120}\right)\right) + \left(\mathsf{neg}\left(\frac{-7}{360}\right)\right) \cdot n1\_i\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left({normAngle}^{2} \cdot \left(\mathsf{neg}\left(\frac{-31}{15120} \cdot n1\_i\right)\right) + \left(\mathsf{neg}\left(\frac{-7}{360}\right)\right) \cdot n1\_i\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left({normAngle}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{-31}{15120}\right)\right) \cdot n1\_i\right) + \left(\mathsf{neg}\left(\frac{-7}{360}\right)\right) \cdot n1\_i\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left({normAngle}^{2} \cdot \left(\frac{31}{15120} \cdot n1\_i\right) + \left(\mathsf{neg}\left(\frac{-7}{360}\right)\right) \cdot n1\_i\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    9. associate-*l* N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\left({normAngle}^{2} \cdot \frac{31}{15120}\right) \cdot n1\_i + \left(\mathsf{neg}\left(\frac{-7}{360}\right)\right) \cdot n1\_i\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\left(\frac{31}{15120} \cdot {normAngle}^{2}\right) \cdot n1\_i + \left(\mathsf{neg}\left(\frac{-7}{360}\right)\right) \cdot n1\_i\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\left(\frac{31}{15120} \cdot {normAngle}^{2}\right) \cdot n1\_i + \frac{7}{360} \cdot n1\_i\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    12. distribute-rgt-in N/A

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(n1\_i \cdot \left(\frac{31}{15120} \cdot {normAngle}^{2} + \frac{7}{360}\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    13. +-commutative N/A

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(n1\_i \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {normAngle}^{2}\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(n1\_i, \left(\frac{7}{360} + \frac{31}{15120} \cdot {normAngle}^{2}\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(n1\_i, \mathsf{+.f32}\left(\frac{7}{360}, \left(\frac{31}{15120} \cdot {normAngle}^{2}\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    16. *-commutative N/A

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(n1\_i, \mathsf{+.f32}\left(\frac{7}{360}, \left({normAngle}^{2} \cdot \frac{31}{15120}\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(n1\_i, \mathsf{+.f32}\left(\frac{7}{360}, \mathsf{*.f32}\left(\left({normAngle}^{2}\right), \frac{31}{15120}\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    18. unpow2 N/A

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(n1\_i, \mathsf{+.f32}\left(\frac{7}{360}, \mathsf{*.f32}\left(\left(normAngle \cdot normAngle\right), \frac{31}{15120}\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    19. *-lowering-*.f32 99.4%

        \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(n1\_i, \mathsf{+.f32}\left(\frac{7}{360}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \frac{31}{15120}\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

11. Simplified 99.4%

    \[\leadsto n0\_i + u \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(n1\_i \cdot \left(0.019444444444444445 + \left(normAngle \cdot normAngle\right) \cdot 0.00205026455026455\right)\right)} + \left(n0\_i \cdot 0.3333333333333333 + n1\_i \cdot 0.16666666666666666\right)\right) + \left(n1\_i - n0\_i\right)\right) \]

12. Final simplification 99.4%

    \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot \left(0.019444444444444445 + \left(normAngle \cdot normAngle\right) \cdot 0.00205026455026455\right)\right) + \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right)\right)\right) \]
13. Add Preprocessing

Alternative 6: 98.8% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \left(0.022222222222222223 + \left(normAngle \cdot normAngle\right) \cdot 0.0021164021164021165\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    (- n1_i n0_i)
    (*
     (* normAngle normAngle)
     (+
      (+ (* n1_i 0.16666666666666666) (* n0_i 0.3333333333333333))
      (*
       (* normAngle normAngle)
       (*
        n0_i
        (+
         0.022222222222222223
         (* (* normAngle normAngle) 0.0021164021164021165))))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((n1_i - n0_i) + ((normAngle * normAngle) * (((n1_i * 0.16666666666666666f) + (n0_i * 0.3333333333333333f)) + ((normAngle * normAngle) * (n0_i * (0.022222222222222223f + ((normAngle * normAngle) * 0.0021164021164021165f))))))));
}

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) + ((normangle * normangle) * (((n1_i * 0.16666666666666666e0) + (n0_i * 0.3333333333333333e0)) + ((normangle * normangle) * (n0_i * (0.022222222222222223e0 + ((normangle * normangle) * 0.0021164021164021165e0))))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(Float32(n1_i - n0_i) + Float32(Float32(normAngle * normAngle) * Float32(Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.3333333333333333))) + Float32(Float32(normAngle * normAngle) * Float32(n0_i * Float32(Float32(0.022222222222222223) + Float32(Float32(normAngle * normAngle) * Float32(0.0021164021164021165))))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * ((n1_i - n0_i) + ((normAngle * normAngle) * (((n1_i * single(0.16666666666666666)) + (n0_i * single(0.3333333333333333))) + ((normAngle * normAngle) * (n0_i * (single(0.022222222222222223) + ((normAngle * normAngle) * single(0.0021164021164021165)))))))));
end

Derivation

1. Initial program 98.1%

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

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

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

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

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

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

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

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

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

   16. sin-lowering-sin.f32 96.3%

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

5. Simplified 96.3%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(\left(n1\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \left(\frac{1}{120} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(\frac{1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \frac{1}{120} \cdot n0\_i\right)\right) + \left(\frac{-1}{5040} \cdot n0\_i + \frac{1}{120} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right)\right)\right) - \left(\frac{-1}{720} \cdot n0\_i + \left(\frac{-1}{5040} \cdot n1\_i + \left(\frac{1}{720} \cdot n1\_i + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{36} \cdot n1\_i + \left(\frac{1}{120} \cdot n1\_i + \frac{1}{24} \cdot n0\_i\right)\right)\right)\right) - \left(\frac{-1}{2} \cdot n0\_i + \frac{-1}{6} \cdot n1\_i\right)\right)\right) - n0\_i\right)}\right)\right) \]

8. Simplified 99.5%

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

9. Taylor expanded in n0_i around inf

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \color{blue}{\left(n0\_i \cdot \left(\frac{1}{45} + \frac{2}{945} \cdot {normAngle}^{2}\right)\right)}\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(n0\_i, \left(\frac{1}{45} + \frac{2}{945} \cdot {normAngle}^{2}\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(n0\_i, \mathsf{+.f32}\left(\frac{1}{45}, \left(\frac{2}{945} \cdot {normAngle}^{2}\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(n0\_i, \mathsf{+.f32}\left(\frac{1}{45}, \left({normAngle}^{2} \cdot \frac{2}{945}\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(n0\_i, \mathsf{+.f32}\left(\frac{1}{45}, \mathsf{*.f32}\left(\left({normAngle}^{2}\right), \frac{2}{945}\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(n0\_i, \mathsf{+.f32}\left(\frac{1}{45}, \mathsf{*.f32}\left(\left(normAngle \cdot normAngle\right), \frac{2}{945}\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

    6. *-lowering-*.f32 99.3%

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(n0\_i, \mathsf{+.f32}\left(\frac{1}{45}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \frac{2}{945}\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right), \mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f32}\left(n1\_i, n0\_i\right)\right)\right)\right) \]

11. Simplified 99.3%

    \[\leadsto n0\_i + u \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(n0\_i \cdot \left(0.022222222222222223 + \left(normAngle \cdot normAngle\right) \cdot 0.0021164021164021165\right)\right)} + \left(n0\_i \cdot 0.3333333333333333 + n1\_i \cdot 0.16666666666666666\right)\right) + \left(n1\_i - n0\_i\right)\right) \]

12. Final simplification 99.3%

    \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \left(0.022222222222222223 + \left(normAngle \cdot normAngle\right) \cdot 0.0021164021164021165\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 7: 98.8% accurate, 16.8× speedup

Math

\[\begin{array}{l} \\ \left(u \cdot \left(normAngle \cdot normAngle\right)\right) \cdot \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot \left(0.3333333333333333 + u \cdot -0.5\right)\right) + \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (*
   (* u (* normAngle normAngle))
   (+ (* n1_i 0.16666666666666666) (* n0_i (+ 0.3333333333333333 (* u -0.5)))))
  (+ n0_i (* u (- n1_i n0_i)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((u * (normAngle * normAngle)) * ((n1_i * 0.16666666666666666f) + (n0_i * (0.3333333333333333f + (u * -0.5f))))) + (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 = ((u * (normangle * normangle)) * ((n1_i * 0.16666666666666666e0) + (n0_i * (0.3333333333333333e0 + (u * (-0.5e0)))))) + (n0_i + (u * (n1_i - n0_i)))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(Float32(u * Float32(normAngle * normAngle)) * Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(Float32(0.3333333333333333) + Float32(u * Float32(-0.5)))))) + Float32(n0_i + Float32(u * Float32(n1_i - n0_i))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = ((u * (normAngle * normAngle)) * ((n1_i * single(0.16666666666666666)) + (n0_i * (single(0.3333333333333333) + (u * single(-0.5)))))) + (n0_i + (u * (n1_i - n0_i)));
end

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u around 0

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(normAngle \cdot \frac{n1\_i}{\sin normAngle} + n0\_i \cdot \left(u \cdot \left(-0.5 \cdot \left(normAngle \cdot normAngle\right)\right) - \frac{normAngle \cdot \cos normAngle}{\sin normAngle}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

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

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

7. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{u \cdot \left(\frac{normAngle}{\frac{\sin normAngle}{n1\_i}} + n0\_i \cdot \left(u \cdot \left(-0.5 \cdot \left(normAngle \cdot normAngle\right)\right) - \frac{1}{\frac{\tan normAngle}{normAngle}}\right)\right) + n0\_i} \]

8. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\color{blue}{\left(n1\_i + {normAngle}^{2} \cdot \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)}, \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \left({normAngle}^{2} \cdot \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \mathsf{*.f32}\left(\left({normAngle}^{2}\right), \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \mathsf{*.f32}\left(\left(normAngle \cdot normAngle\right), \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot n1\_i\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) + \frac{1}{6} \cdot n1\_i\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{+.f32}\left(n1\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right), \left(\frac{1}{6} \cdot n1\_i\right)\right)\right)\right), \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(normAngle, normAngle\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{tan.f32}\left(normAngle\right), normAngle\right)\right)\right)\right)\right)\right), n0\_i\right) \]

10. Simplified 99.6%

    \[\leadsto u \cdot \left(\color{blue}{\left(n1\_i + \left(normAngle \cdot normAngle\right) \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot 0.019444444444444445\right) + n1\_i \cdot 0.16666666666666666\right)\right)} + n0\_i \cdot \left(u \cdot \left(-0.5 \cdot \left(normAngle \cdot normAngle\right)\right) - \frac{1}{\frac{\tan normAngle}{normAngle}}\right)\right) + n0\_i \]

11. Taylor expanded in normAngle around 0

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

    1. associate-+r+ N/A

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

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

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

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

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

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

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

    5. mul-1-neg N/A

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

    6. unsub-neg N/A

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

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

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

    8. associate-*r* N/A

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

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

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

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

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

    11. unpow2 N/A

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

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

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

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

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

    14. *-commutative N/A

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

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

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

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

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

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

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

    18. *-lowering-*.f32 99.3%

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

13. Simplified 99.3%

    \[\leadsto \color{blue}{\left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right) + \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot \left(0.3333333333333333 + -0.5 \cdot u\right)\right)} \]

14. Final simplification 99.3%

    \[\leadsto \left(u \cdot \left(normAngle \cdot normAngle\right)\right) \cdot \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot \left(0.3333333333333333 + u \cdot -0.5\right)\right) + \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right) \]
15. Add Preprocessing

Alternative 8: 98.9% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + normAngle \cdot \left(normAngle \cdot \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot \left(0.3333333333333333 + u \cdot -0.5\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    (- n1_i n0_i)
    (*
     normAngle
     (*
      normAngle
      (+
       (* n1_i 0.16666666666666666)
       (* n0_i (+ 0.3333333333333333 (* u -0.5))))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((n1_i - n0_i) + (normAngle * (normAngle * ((n1_i * 0.16666666666666666f) + (n0_i * (0.3333333333333333f + (u * -0.5f))))))));
}

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) + (normangle * (normangle * ((n1_i * 0.16666666666666666e0) + (n0_i * (0.3333333333333333e0 + (u * (-0.5e0)))))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(Float32(n1_i - n0_i) + Float32(normAngle * Float32(normAngle * Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(Float32(0.3333333333333333) + Float32(u * Float32(-0.5))))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * ((n1_i - n0_i) + (normAngle * (normAngle * ((n1_i * single(0.16666666666666666)) + (n0_i * (single(0.3333333333333333) + (u * single(-0.5)))))))));
end

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u around 0

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

5. Simplified 99.6%

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

6. Taylor expanded in normAngle around 0

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

   1. associate-+r+ N/A

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

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

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

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

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

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

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

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

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

   10. sub-neg N/A

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

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

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

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

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

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

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

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

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

   15. *-commutative N/A

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

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

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

   17. metadata-eval N/A

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

   18. *-lowering-*.f32 99.3%

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

8. Simplified 99.3%

   \[\leadsto n0\_i + u \cdot \color{blue}{\left(\left(n1\_i - n0\_i\right) + normAngle \cdot \left(normAngle \cdot \left(n0\_i \cdot \left(0.3333333333333333 + -0.5 \cdot u\right) + n1\_i \cdot 0.16666666666666666\right)\right)\right)} \]

9. Final simplification 99.3%

   \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + normAngle \cdot \left(normAngle \cdot \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot \left(0.3333333333333333 + u \cdot -0.5\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 9: 98.8% accurate, 22.2× speedup

Math

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

FPCore

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

C

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

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) + ((normangle * normangle) * ((n1_i * 0.16666666666666666e0) + (n0_i * 0.3333333333333333e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

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

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

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

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

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

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

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

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

   16. sin-lowering-sin.f32 96.3%

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

5. Simplified 96.3%

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

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. sub-neg N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. associate-+l+ N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. associate--r+ N/A

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

   10. sub-neg N/A

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

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

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

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

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

   13. metadata-eval N/A

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

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

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

   15. *-commutative N/A

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

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

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

   17. metadata-eval N/A

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

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

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

   19. mul-1-neg N/A

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

   20. sub-neg N/A

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

   \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right)\right) \]
10. Add Preprocessing

Alternative 10: 98.6% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

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))) + ((n1_i * 0.16666666666666666e0) * (u * (normangle * normangle)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

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

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

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

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

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

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

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

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

   16. sin-lowering-sin.f32 96.3%

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

5. Simplified 96.3%

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

6. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. associate-+r+ N/A

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

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

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

   11. associate-*r* N/A

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

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

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

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

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

   14. unpow2 N/A

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

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

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

   16. associate--r+ N/A

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

8. Simplified 99.3%

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

9. Taylor expanded in n0_i around 0

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

    1. *-commutative N/A

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

    2. *-lowering-*.f32 99.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

    \[\leadsto \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right) + \left(n1\_i \cdot 0.16666666666666666\right) \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right) \]
13. Add Preprocessing

Alternative 11: 98.0% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

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))) + ((u * (normangle * normangle)) * (n0_i * 0.3333333333333333e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

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

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

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

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

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

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

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

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

   16. sin-lowering-sin.f32 96.3%

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

5. Simplified 96.3%

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

6. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. associate-+r+ N/A

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

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

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

   11. associate-*r* N/A

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

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

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

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

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

   14. unpow2 N/A

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

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

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

   16. associate--r+ N/A

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

8. Simplified 99.3%

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

9. Taylor expanded in n0_i around inf

   \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), u\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), u\right), \color{blue}{\left(\frac{1}{3} \cdot n0\_i\right)}\right)\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f32 98.9%

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

    \[\leadsto \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right) + \left(u \cdot \left(normAngle \cdot normAngle\right)\right) \cdot \left(n0\_i \cdot 0.3333333333333333\right) \]
13. Add Preprocessing

Alternative 12: 71.0% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -3.7999998655537104 \cdot 10^{-13}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{elif}\;n1\_i \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -3.7999998655537104e-13)
   (* u n1_i)
   (if (<= n1_i 4.99999991225835e-15) (* n0_i (- 1.0 u)) (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -3.7999998655537104e-13f) {
		tmp = u * n1_i;
	} else if (n1_i <= 4.99999991225835e-15f) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = u * n1_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n1_i <= (-3.7999998655537104e-13)) then
        tmp = u * n1_i
    else if (n1_i <= 4.99999991225835e-15) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-3.7999998655537104e-13))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(4.99999991225835e-15))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n1_i <= single(-3.7999998655537104e-13))
		tmp = u * n1_i;
	elseif (n1_i <= single(4.99999991225835e-15))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -3.79999987e-13 or 4.99999991e-15 < n1_i

   1. Initial program 97.7%

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

   3. Taylor expanded in normAngle around 0

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

   5. Simplified 98.4%

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

   6. Taylor expanded in u around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f32 66.7%

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

   8. Simplified 66.7%

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

   if -3.79999987e-13 < n1_i < 4.99999991e-15

   1. Initial program 98.3%

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

   3. Taylor expanded in n0_i around inf

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. sin-lowering-sin.f32 77.9%

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

   5. Simplified 77.9%

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

   6. Taylor expanded in normAngle around 0

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

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

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

      2. --lowering--.f32 77.9%

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

   8. Simplified 77.9%

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

Alternative 13: 60.6% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -3.7999998655537104 \cdot 10^{-13}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{elif}\;n1\_i \leq 9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -3.7999998655537104e-13)
   (* u n1_i)
   (if (<= n1_i 9.99999983775159e-18) n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -3.7999998655537104e-13f) {
		tmp = u * n1_i;
	} else if (n1_i <= 9.99999983775159e-18f) {
		tmp = n0_i;
	} else {
		tmp = u * n1_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n1_i <= (-3.7999998655537104e-13)) then
        tmp = u * n1_i
    else if (n1_i <= 9.99999983775159e-18) then
        tmp = n0_i
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-3.7999998655537104e-13))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(9.99999983775159e-18))
		tmp = n0_i;
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n1_i <= single(-3.7999998655537104e-13))
		tmp = u * n1_i;
	elseif (n1_i <= single(9.99999983775159e-18))
		tmp = n0_i;
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -3.79999987e-13 or 9.99999984e-18 < n1_i

   1. Initial program 97.9%

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

   3. Taylor expanded in normAngle around 0

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

   5. Simplified 98.6%

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

   6. Taylor expanded in u around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f32 63.6%

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

   8. Simplified 63.6%

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

   if -3.79999987e-13 < n1_i < 9.99999984e-18

   1. Initial program 98.3%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 63.4%

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

Alternative 14: 84.0% accurate, 42.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.1999999518369497 \cdot 10^{-12}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.1999999518369497e-12) (* n0_i (- 1.0 u)) (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.1999999518369497e-12f) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = n0_i + (u * n1_i);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-1.1999999518369497e-12)) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = n0_i + (u * n1_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-1.1999999518369497e-12))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(n0_i + Float32(u * n1_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-1.1999999518369497e-12))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.19999995e-12

   1. Initial program 98.7%

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

   3. Taylor expanded in n0_i around inf

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. sin-lowering-sin.f32 94.4%

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

   5. Simplified 94.4%

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

   6. Taylor expanded in normAngle around 0

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

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

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

      2. --lowering--.f32 94.0%

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

   8. Simplified 94.0%

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

   if -1.19999995e-12 < n0_i

   1. Initial program 98.0%

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

   3. Taylor expanded in normAngle around 0

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

   5. Simplified 98.5%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 83.7%

      \[\leadsto \color{blue}{n0\_i} + u \cdot n1\_i \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 97.9% 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 98.1%

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

3. Taylor expanded in u around 0

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

5. Simplified 99.6%

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

6. Taylor expanded in normAngle around 0

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

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

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

   5. sub-neg N/A

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

   6. --lowering--.f32 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

   \[\leadsto n0\_i + u \cdot \left(n1\_i - n0\_i\right) \]
10. Add Preprocessing

Alternative 16: 47.1% 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 98.1%

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

3. Taylor expanded in u around 0

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

5. Simplified 50.3%

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

Reproduce

herbie shell --seed 2024192 
(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)))