Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.0%

Time: 16.4s

Alternatives: 11

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: 97.2% 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: 99.0% accurate, 15.6× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * (n1_i + (((normAngle * normAngle) * (((n1_i * 0.16666666666666666f) + (n0_i * 0.3333333333333333f)) + (0.019444444444444445f * (n1_i * (normAngle * normAngle))))) - 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) * (((n1_i * 0.16666666666666666e0) + (n0_i * 0.3333333333333333e0)) + (0.019444444444444445e0 * (n1_i * (normangle * normangle))))) - n0_i)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

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

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

5. Simplified 97.8%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right)\right), \mathsf{\_.f32}\left(1, u\right)\right)\right)\right), \mathsf{\_.f32}\left(1, u\right)\right), n0\_i\right), \mathsf{*.f32}\left(\color{blue}{\left(u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot {u}^{3} + {normAngle}^{2} \cdot \left(\frac{1}{120} \cdot {u}^{5} - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot {u}^{3} - \frac{-1}{6} \cdot u\right) + \frac{1}{120} \cdot u\right)\right)\right) - \frac{-1}{6} \cdot u\right)\right)}, n1\_i\right)\right) \]

8. Simplified 99.2%

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

9. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

    5. sub-neg N/A

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

    6. metadata-eval N/A

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

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

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

    8. *-commutative N/A

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

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

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

    10. unpow2 N/A

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

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

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

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

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

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

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

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

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

11. Simplified 99.2%

    \[\leadsto \color{blue}{n0\_i + u \cdot \left(n0\_i \cdot \left(\left(normAngle \cdot normAngle\right) \cdot 0.3333333333333333 + -1\right) + n1\_i \cdot \left(1 + \left(normAngle \cdot normAngle\right) \cdot \left(0.16666666666666666 + \left(normAngle \cdot normAngle\right) \cdot 0.019444444444444445\right)\right)\right)} \]

12. Taylor expanded in normAngle around 0

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

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

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

    2. mul-1-neg N/A

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

    3. +-commutative N/A

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

    4. unsub-neg N/A

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

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

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

14. Simplified 99.3%

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

Alternative 2: 98.8% accurate, 20.0× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (+ n0_i (* u (- n1_i n0_i)))
  (*
   (+ (* n1_i 0.16666666666666666) (* n0_i 0.3333333333333333))
   (* 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) + (n0_i * 0.3333333333333333f)) * (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) + (n0_i * 0.3333333333333333e0)) * (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(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.3333333333333333))) * 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)) + (n0_i * single(0.3333333333333333))) * (u * (normAngle * normAngle)));
end

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

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

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

5. Simplified 97.8%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right)\right), \mathsf{\_.f32}\left(1, u\right)\right)\right)\right), \mathsf{\_.f32}\left(1, u\right)\right), n0\_i\right), \mathsf{*.f32}\left(\color{blue}{\left(u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot {u}^{3} + {normAngle}^{2} \cdot \left(\frac{1}{120} \cdot {u}^{5} - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot {u}^{3} - \frac{-1}{6} \cdot u\right) + \frac{1}{120} \cdot u\right)\right)\right) - \frac{-1}{6} \cdot u\right)\right)}, n1\_i\right)\right) \]

8. Simplified 99.2%

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

9. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

    5. sub-neg N/A

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

    6. metadata-eval N/A

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

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

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

    8. *-commutative N/A

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

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

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

    10. unpow2 N/A

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

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

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

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

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

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

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

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

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

11. Simplified 99.2%

    \[\leadsto \color{blue}{n0\_i + u \cdot \left(n0\_i \cdot \left(\left(normAngle \cdot normAngle\right) \cdot 0.3333333333333333 + -1\right) + n1\_i \cdot \left(1 + \left(normAngle \cdot normAngle\right) \cdot \left(0.16666666666666666 + \left(normAngle \cdot normAngle\right) \cdot 0.019444444444444445\right)\right)\right)} \]

12. Taylor expanded in normAngle around 0

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

    1. associate-+r+ N/A

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

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

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

    3. mul-1-neg N/A

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

    4. sub-neg N/A

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

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

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

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

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

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

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

    8. associate-*r* N/A

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

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

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

    10. *-commutative N/A

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

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

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

    12. unpow2 N/A

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

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

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

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

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

    15. *-commutative N/A

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

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

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

    17. *-commutative N/A

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

    18. *-lowering-*.f32 99.1%

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

14. Simplified 99.1%

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

15. Final simplification 99.1%

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

Alternative 3: 98.8% accurate, 20.0× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (+
   (* u (- n1_i n0_i))
   (*
    (+ (* n1_i 0.16666666666666666) (* n0_i 0.3333333333333333))
    (* 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) + (n0_i * 0.3333333333333333f)) * (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) + (n0_i * 0.3333333333333333e0)) * (u * (normangle * normangle))))
end function

Julia

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

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

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

5. Simplified 97.8%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right)\right), \mathsf{\_.f32}\left(1, u\right)\right)\right)\right), \mathsf{\_.f32}\left(1, u\right)\right), n0\_i\right), \mathsf{*.f32}\left(\color{blue}{\left(u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot {u}^{3} + {normAngle}^{2} \cdot \left(\frac{1}{120} \cdot {u}^{5} - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot {u}^{3} - \frac{-1}{6} \cdot u\right) + \frac{1}{120} \cdot u\right)\right)\right) - \frac{-1}{6} \cdot u\right)\right)}, n1\_i\right)\right) \]

8. Simplified 99.2%

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

9. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

    5. sub-neg N/A

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

    6. metadata-eval N/A

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

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

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

    8. *-commutative N/A

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

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

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

    10. unpow2 N/A

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

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

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

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

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

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

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

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

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

11. Simplified 99.2%

    \[\leadsto \color{blue}{n0\_i + u \cdot \left(n0\_i \cdot \left(\left(normAngle \cdot normAngle\right) \cdot 0.3333333333333333 + -1\right) + n1\_i \cdot \left(1 + \left(normAngle \cdot normAngle\right) \cdot \left(0.16666666666666666 + \left(normAngle \cdot normAngle\right) \cdot 0.019444444444444445\right)\right)\right)} \]

12. Taylor expanded in normAngle around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. sub-neg N/A

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

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

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

    5. associate-*r* N/A

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

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

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

    7. *-commutative N/A

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

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

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

    9. unpow2 N/A

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

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

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

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

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

    12. *-commutative N/A

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

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

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

    14. *-commutative N/A

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

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

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

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

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

    17. --lowering--.f32 99.1%

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

14. Simplified 99.1%

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

15. Final simplification 99.1%

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

Alternative 4: 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 97.4%

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

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

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

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

5. Simplified 97.8%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right)\right), \mathsf{\_.f32}\left(1, u\right)\right)\right)\right), \mathsf{\_.f32}\left(1, u\right)\right), n0\_i\right), \mathsf{*.f32}\left(\color{blue}{\left(u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot {u}^{3} + {normAngle}^{2} \cdot \left(\frac{1}{120} \cdot {u}^{5} - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot {u}^{3} - \frac{-1}{6} \cdot u\right) + \frac{1}{120} \cdot u\right)\right)\right) - \frac{-1}{6} \cdot u\right)\right)}, n1\_i\right)\right) \]

8. Simplified 99.2%

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

9. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

    5. sub-neg N/A

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

    6. metadata-eval N/A

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

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

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

    8. *-commutative N/A

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

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

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

    10. unpow2 N/A

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

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

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

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

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

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

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

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

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

11. Simplified 99.2%

    \[\leadsto \color{blue}{n0\_i + u \cdot \left(n0\_i \cdot \left(\left(normAngle \cdot normAngle\right) \cdot 0.3333333333333333 + -1\right) + n1\_i \cdot \left(1 + \left(normAngle \cdot normAngle\right) \cdot \left(0.16666666666666666 + \left(normAngle \cdot normAngle\right) \cdot 0.019444444444444445\right)\right)\right)} \]

12. Taylor expanded in normAngle around 0

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

    1. mul-1-neg N/A

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

    2. associate-+r+ N/A

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

    3. sub-neg N/A

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

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

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

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

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

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

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

    7. unpow2 N/A

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

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

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

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

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

    10. *-commutative N/A

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

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

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

    12. *-commutative N/A

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

    13. *-lowering-*.f32 99.1%

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

14. Simplified 99.1%

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

Alternative 5: 98.7% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 99.0%

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

6. Taylor expanded in n0_i around 0

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

   1. associate-*r* N/A

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

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

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

   3. *-commutative N/A

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

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

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

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

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f32 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in u around 0

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

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

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

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

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

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

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

    4. +-commutative N/A

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

    5. mul-1-neg N/A

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

    6. unsub-neg N/A

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

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

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

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

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

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

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

    10. unpow2 N/A

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

    11. *-lowering-*.f32 98.9%

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

11. Simplified 98.9%

    \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(0.16666666666666666 \cdot \left(n1\_i \cdot \left(normAngle \cdot normAngle\right)\right) - n0\_i\right)\right)} \]
12. Step-by-step derivation

    1. +-commutative N/A

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

    2. distribute-rgt-in N/A

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

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

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

    4. *-commutative N/A

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

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

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

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

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

    7. associate-*r* N/A

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

    8. *-commutative N/A

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

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

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

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

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

    11. *-commutative N/A

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

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

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

    13. *-commutative N/A

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

    14. *-lowering-*.f32 98.9%

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

13. Applied egg-rr 98.9%

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

Alternative 6: 98.7% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 99.0%

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

6. Taylor expanded in n0_i around 0

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

   1. associate-*r* N/A

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

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

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

   3. *-commutative N/A

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

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

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

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

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f32 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in u around 0

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

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

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

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

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

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

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

    4. +-commutative N/A

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

    5. mul-1-neg N/A

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

    6. unsub-neg N/A

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

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

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

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

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

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

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

    10. unpow2 N/A

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

    11. *-lowering-*.f32 98.9%

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

11. Simplified 98.9%

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

Alternative 7: 83.8% accurate, 42.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq 9.999999960041972 \cdot 10^{-13}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i 9.999999960041972e-13) (+ n0_i (* u n1_i)) (- n0_i (* n0_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= 9.999999960041972e-13f) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i - (n0_i * u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= 9.999999960041972e-13) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i - (n0_i * u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(9.999999960041972e-13))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i - Float32(n0_i * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(9.999999960041972e-13))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i - (n0_i * u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < 9.99999996e-13

   1. Initial program 97.3%

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

   3. Taylor expanded in normAngle around 0

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

   5. Simplified 98.9%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 84.6%

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

   if 9.99999996e-13 < n0_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. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

   3. Simplified 90.8%

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

   5. Taylor expanded in n0_i around inf

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

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

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

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

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

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

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

      4. --lowering--.f32 82.9%

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

   7. Simplified 82.9%

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

   8. Taylor expanded in normAngle around 0

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
   9. 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 89.4%

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

   10. Simplified 89.4%

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

   11. Taylor expanded in u around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

       4. *-lowering-*.f32 89.8%

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

   13. Simplified 89.8%

       \[\leadsto \color{blue}{n0\_i - n0\_i \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq 9.999999960041972 \cdot 10^{-13}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.8% accurate, 42.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq 9.999999960041972 \cdot 10^{-13}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i 9.999999960041972e-13) (+ n0_i (* u n1_i)) (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= 9.999999960041972e-13f) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= 9.999999960041972e-13) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(9.999999960041972e-13))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(9.999999960041972e-13))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < 9.99999996e-13

   1. Initial program 97.3%

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

   3. Taylor expanded in normAngle around 0

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

   5. Simplified 98.9%

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

   6. Taylor expanded in u around 0

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

   8. Simplified 84.6%

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

   if 9.99999996e-13 < n0_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. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

   3. Simplified 90.8%

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

   5. Taylor expanded in n0_i around inf

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

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

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

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

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

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

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

      4. --lowering--.f32 82.9%

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

   7. Simplified 82.9%

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

   8. Taylor expanded in normAngle around 0

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
   9. 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 89.4%

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

   10. Simplified 89.4%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq 9.999999960041972 \cdot 10^{-13}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.0% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

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

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

5. Simplified 97.8%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, u\right)\right)\right), \mathsf{\_.f32}\left(1, u\right)\right)\right)\right), \mathsf{\_.f32}\left(1, u\right)\right), n0\_i\right), \mathsf{*.f32}\left(\color{blue}{\left(u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot {u}^{3} + {normAngle}^{2} \cdot \left(\frac{1}{120} \cdot {u}^{5} - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot {u}^{3} - \frac{-1}{6} \cdot u\right) + \frac{1}{120} \cdot u\right)\right)\right) - \frac{-1}{6} \cdot u\right)\right)}, n1\_i\right)\right) \]

8. Simplified 99.2%

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

9. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

    5. sub-neg N/A

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

    6. metadata-eval N/A

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

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

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

    8. *-commutative N/A

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

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

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

    10. unpow2 N/A

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

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

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

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

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

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

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

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

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

11. Simplified 99.2%

    \[\leadsto \color{blue}{n0\_i + u \cdot \left(n0\_i \cdot \left(\left(normAngle \cdot normAngle\right) \cdot 0.3333333333333333 + -1\right) + n1\_i \cdot \left(1 + \left(normAngle \cdot normAngle\right) \cdot \left(0.16666666666666666 + \left(normAngle \cdot normAngle\right) \cdot 0.019444444444444445\right)\right)\right)} \]

12. Taylor expanded in normAngle around 0

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

    1. mul-1-neg N/A

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

    2. sub-neg N/A

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

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

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

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

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

    5. --lowering--.f32 98.4%

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

14. Simplified 98.4%

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

Alternative 10: 58.4% accurate, 84.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-out N/A

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

3. Simplified 72.2%

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

5. Taylor expanded in n0_i around inf

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

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

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

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

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

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

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

   4. --lowering--.f32 44.6%

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

7. Simplified 44.6%

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

8. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
9. 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 59.0%

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

10. Simplified 59.0%

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

Alternative 11: 46.7% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in u around 0

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

5. Simplified 46.6%

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

Reproduce

herbie shell --seed 2024158 
(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)))