Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 98.9%

Time: 13.9s

Alternatives: 9

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 20.0× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
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 98.6%

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

6. Taylor expanded in u around 0

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

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

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

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

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

   3. associate-+r+ N/A

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

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

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

   11. associate--l+ N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. metadata-eval 98.8%

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

8. Simplified 98.8%

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

   1. distribute-rgt-in N/A

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

   2. associate-+r+ N/A

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

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

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

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

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

   5. *-commutative N/A

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

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

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

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

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

   8. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f32 98.8%

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

10. Applied egg-rr 98.8%

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

Alternative 2: 98.9% 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 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
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 98.6%

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

6. Taylor expanded in u around 0

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

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

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

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

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

   3. associate-+r+ N/A

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

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

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

   11. associate--l+ N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. metadata-eval 98.8%

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

8. Simplified 98.8%

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

Alternative 3: 98.8% accurate, 24.8× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n0_i + (u * (n1_i - n0_i))) + (u * (0.16666666666666666f * (n1_i * (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))) + (u * (0.16666666666666666e0 * (n1_i * (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(u * Float32(Float32(0.16666666666666666) * Float32(n1_i * Float32(normAngle * normAngle)))))
end

MATLAB

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

Derivation

1. Initial program 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
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 98.6%

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

6. Taylor expanded in u around 0

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

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

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

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

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

   3. associate-+r+ N/A

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

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

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

   11. associate--l+ N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. metadata-eval 98.8%

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

8. Simplified 98.8%

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

   1. distribute-rgt-in N/A

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

   2. associate-+r+ N/A

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

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

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

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

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

   5. *-commutative N/A

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

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

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

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

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

   8. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f32 98.8%

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

10. Applied egg-rr 98.8%

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

11. Taylor expanded in n1_i around inf

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

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

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

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

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

    3. unpow2 N/A

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

    4. *-lowering-*.f32 98.6%

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

13. Simplified 98.6%

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

Alternative 4: 68.6% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n0\_i \cdot \left(1 - u\right)\\ \mathbf{if}\;n0\_i \leq -1.5000000170217692 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* n0_i (- 1.0 u))))
   (if (<= n0_i -1.5000000170217692e-18)
     t_0
     (if (<= n0_i 4.0000000126843074e-29) (* u n1_i) t_0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = n0_i * (1.0f - u);
	float tmp;
	if (n0_i <= -1.5000000170217692e-18f) {
		tmp = t_0;
	} else if (n0_i <= 4.0000000126843074e-29f) {
		tmp = u * n1_i;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = n0_i * (1.0e0 - u)
    if (n0_i <= (-1.5000000170217692e-18)) then
        tmp = t_0
    else if (n0_i <= 4.0000000126843074e-29) then
        tmp = u * n1_i
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(n0_i * Float32(Float32(1.0) - u))
	tmp = Float32(0.0)
	if (n0_i <= Float32(-1.5000000170217692e-18))
		tmp = t_0;
	elseif (n0_i <= Float32(4.0000000126843074e-29))
		tmp = Float32(u * n1_i);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	t_0 = n0_i * (single(1.0) - u);
	tmp = single(0.0);
	if (n0_i <= single(-1.5000000170217692e-18))
		tmp = t_0;
	elseif (n0_i <= single(4.0000000126843074e-29))
		tmp = u * n1_i;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.50000002e-18 or 4.00000001e-29 < n0_i

   1. Initial program 98.2%

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

   3. Taylor expanded in 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%

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

   6. Taylor expanded in n1_i around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. associate-*r* N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

   8. Simplified 82.0%

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

   9. Taylor expanded in normAngle around 0

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
   10. 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 81.8%

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

   11. Simplified 81.8%

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

   if -1.50000002e-18 < n0_i < 4.00000001e-29

   1. Initial program 96.4%

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

   3. Taylor expanded in n0_i around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. sin-lowering-sin.f32 66.1%

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

   5. Simplified 66.1%

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

   6. Taylor expanded in normAngle around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f32 66.1%

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

   8. Simplified 66.1%

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

Alternative 5: 98.8% accurate, 28.1× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((n1_i - n0_i) + (0.16666666666666666f * (n1_i * (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) + (0.16666666666666666e0 * (n1_i * (normangle * normangle)))))
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(0.16666666666666666) * Float32(n1_i * Float32(normAngle * normAngle))))))
end

MATLAB

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

Derivation

1. Initial program 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
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 98.6%

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

6. Taylor expanded in u around 0

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

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

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

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

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

   3. associate-+r+ N/A

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

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

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

   11. associate--l+ N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. metadata-eval 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in n1_i around inf

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

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

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

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

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

    3. unpow2 N/A

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

    4. *-lowering-*.f32 98.6%

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

11. Simplified 98.6%

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

Alternative 6: 59.6% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.5000000170217692 \cdot 10^{-18}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 5.000000097707407 \cdot 10^{-25}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.5000000170217692e-18)
   n0_i
   (if (<= n0_i 5.000000097707407e-25) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.5000000170217692e-18f) {
		tmp = n0_i;
	} else if (n0_i <= 5.000000097707407e-25f) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-1.5000000170217692e-18)) then
        tmp = n0_i
    else if (n0_i <= 5.000000097707407e-25) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-1.5000000170217692e-18))
		tmp = n0_i;
	elseif (n0_i <= Float32(5.000000097707407e-25))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-1.5000000170217692e-18))
		tmp = n0_i;
	elseif (n0_i <= single(5.000000097707407e-25))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.50000002e-18 or 5.0000001e-25 < n0_i

   1. Initial program 98.5%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 68.4%

      \[\leadsto \color{blue}{n0\_i} \]

   if -1.50000002e-18 < n0_i < 5.0000001e-25

   1. Initial program 96.2%

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

   3. Taylor expanded in n0_i around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. sin-lowering-sin.f32 63.0%

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

   5. Simplified 63.0%

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

   6. Taylor expanded in normAngle around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f32 62.9%

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

   8. Simplified 62.9%

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

Alternative 7: 83.5% accurate, 42.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq 1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i 1.99999996490334e-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 <= 1.99999996490334e-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 <= 1.99999996490334e-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(1.99999996490334e-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(1.99999996490334e-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 < 1.99999996e-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 98.4%

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

   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 83.8%

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

   if 1.99999996e-13 < n0_i

   1. Initial program 98.9%

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

   3. Taylor expanded in normAngle around 0

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

   5. Simplified 99.6%

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

   6. Taylor expanded in n1_i around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. associate-*r* N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

   8. Simplified 96.8%

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

   9. Taylor expanded in normAngle around 0

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
   10. 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 96.8%

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

   11. Simplified 96.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq 1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.1% 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 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
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 98.6%

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

6. Taylor expanded in u around 0

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

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

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

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

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

   3. associate-+r+ N/A

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

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

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

   11. associate--l+ N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. metadata-eval 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in normAngle around 0

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

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

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

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

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

    3. --lowering--.f32 98.1%

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

11. Simplified 98.1%

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

Alternative 9: 47.6% 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 97.5%

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

3. Taylor expanded in u around 0

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

5. Simplified 50.7%

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

Reproduce

herbie shell --seed 2024139 
(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)))