Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.1%

Time: 15.9s

Alternatives: 10

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.1% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i + (u * ((n1_i - n0_i) + (((((n1_i * 0.16666666666666666e0) + (n0_i * 0.5e0)) + (n0_i * (-0.16666666666666666e0))) * (normangle * normangle)) + (u * (((-0.5e0) * (n0_i * (normangle * normangle))) + ((normangle * normangle) * (u * ((n0_i * 0.16666666666666666e0) + (n1_i * (-0.16666666666666666e0))))))))))
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(Float32(Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.5))) + Float32(n0_i * Float32(-0.16666666666666666))) * Float32(normAngle * normAngle)) + Float32(u * Float32(Float32(Float32(-0.5) * Float32(n0_i * Float32(normAngle * normAngle))) + Float32(Float32(normAngle * normAngle) * Float32(u * Float32(Float32(n0_i * Float32(0.16666666666666666)) + Float32(n1_i * Float32(-0.16666666666666666)))))))))))
end

MATLAB

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

Derivation

1. Initial program 97.0%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.6%

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

6. Taylor expanded in u around 0

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 2: 98.9% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ n0\_i \cdot \left(1 - u\right) + \left(u \cdot n1\_i + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \left(n1\_i \cdot 0.16666666666666666 + \left(n0\_i \cdot -0.16666666666666666 + \left(n0\_i \cdot 0.5 + u \cdot \left(u \cdot \left(n0\_i \cdot 0.16666666666666666 + n1\_i \cdot -0.16666666666666666\right) + n0\_i \cdot -0.5\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* n0_i (- 1.0 u))
  (+
   (* u n1_i)
   (*
    (* normAngle normAngle)
    (*
     u
     (+
      (* n1_i 0.16666666666666666)
      (+
       (* n0_i -0.16666666666666666)
       (+
        (* n0_i 0.5)
        (*
         u
         (+
          (* u (+ (* n0_i 0.16666666666666666) (* n1_i -0.16666666666666666)))
          (* n0_i -0.5)))))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n0_i * (1.0f - u)) + ((u * n1_i) + ((normAngle * normAngle) * (u * ((n1_i * 0.16666666666666666f) + ((n0_i * -0.16666666666666666f) + ((n0_i * 0.5f) + (u * ((u * ((n0_i * 0.16666666666666666f) + (n1_i * -0.16666666666666666f))) + (n0_i * -0.5f)))))))));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = (n0_i * (1.0e0 - u)) + ((u * n1_i) + ((normangle * normangle) * (u * ((n1_i * 0.16666666666666666e0) + ((n0_i * (-0.16666666666666666e0)) + ((n0_i * 0.5e0) + (u * ((u * ((n0_i * 0.16666666666666666e0) + (n1_i * (-0.16666666666666666e0)))) + (n0_i * (-0.5e0))))))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(n0_i * Float32(Float32(1.0) - u)) + Float32(Float32(u * n1_i) + Float32(Float32(normAngle * normAngle) * Float32(u * Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(Float32(n0_i * Float32(-0.16666666666666666)) + Float32(Float32(n0_i * Float32(0.5)) + Float32(u * Float32(Float32(u * Float32(Float32(n0_i * Float32(0.16666666666666666)) + Float32(n1_i * Float32(-0.16666666666666666)))) + Float32(n0_i * Float32(-0.5)))))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n0_i * (single(1.0) - u)) + ((u * n1_i) + ((normAngle * normAngle) * (u * ((n1_i * single(0.16666666666666666)) + ((n0_i * single(-0.16666666666666666)) + ((n0_i * single(0.5)) + (u * ((u * ((n0_i * single(0.16666666666666666)) + (n1_i * single(-0.16666666666666666)))) + (n0_i * single(-0.5))))))))));
end

Derivation

1. Initial program 97.0%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.6%

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

6. Taylor expanded in u around 0

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

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

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

   2. associate--l+ N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

   7. metadata-eval N/A

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

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

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

8. Simplified 98.6%

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

9. Final simplification 98.6%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(u \cdot n1\_i + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \left(n1\_i \cdot 0.16666666666666666 + \left(n0\_i \cdot -0.16666666666666666 + \left(n0\_i \cdot 0.5 + u \cdot \left(u \cdot \left(n0\_i \cdot 0.16666666666666666 + n1\_i \cdot -0.16666666666666666\right) + n0\_i \cdot -0.5\right)\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 3: 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 97.0%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.6%

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

6. Taylor expanded in u around 0

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

8. Simplified 98.9%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

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

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

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

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

    5. unpow2 N/A

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

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

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

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

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

    8. metadata-eval N/A

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

    9. +-commutative N/A

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

    10. +-commutative N/A

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

    11. associate-+r+ N/A

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

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

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

    13. distribute-rgt-out N/A

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

    14. metadata-eval N/A

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

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

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

    16. *-lowering-*.f32 98.6%

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

11. Simplified 98.6%

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

12. Final simplification 98.6%

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

Alternative 4: 69.8% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n0\_i \cdot \left(1 - u\right)\\ \mathbf{if}\;n0\_i \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 3.2000000759751296 \cdot 10^{-31}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* n0_i (- 1.0 u))))
   (if (<= n0_i -5.000000156871975e-23)
     t_0
     (if (<= n0_i 3.2000000759751296e-31) (* 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 <= -5.000000156871975e-23f) {
		tmp = t_0;
	} else if (n0_i <= 3.2000000759751296e-31f) {
		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 <= (-5.000000156871975e-23)) then
        tmp = t_0
    else if (n0_i <= 3.2000000759751296e-31) 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(-5.000000156871975e-23))
		tmp = t_0;
	elseif (n0_i <= Float32(3.2000000759751296e-31))
		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(-5.000000156871975e-23))
		tmp = t_0;
	elseif (n0_i <= single(3.2000000759751296e-31))
		tmp = u * n1_i;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -5.00000016e-23 or 3.20000008e-31 < n0_i

   1. Initial program 98.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l/ N/A

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

      7. +-commutative N/A

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

      8. *-lft-identity N/A

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

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

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

   3. Simplified 75.3%

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

   5. Taylor expanded in n0_i around inf

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

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

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

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

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

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

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

      4. --lowering--.f32 56.7%

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

   7. Simplified 56.7%

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

   8. Taylor expanded in normAngle around 0

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

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

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

      2. --lowering--.f32 73.0%

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

   10. Simplified 73.0%

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

   if -5.00000016e-23 < n0_i < 3.20000008e-31

   1. Initial program 94.2%

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

   3. Taylor expanded in normAngle around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

   5. Simplified 94.4%

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

   6. Taylor expanded in normAngle around 0

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

   8. Simplified 96.7%

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

   9. Taylor expanded in n0_i around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f32 67.0%

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

   11. Simplified 67.0%

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

Alternative 5: 98.7% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i + (u * ((n1_i - n0_i) + ((n1_i * 0.16666666666666666e0) * (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(n1_i * Float32(0.16666666666666666)) * Float32(normAngle * normAngle)))))
end

MATLAB

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

Derivation

1. Initial program 97.0%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.6%

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

6. Taylor expanded in u around 0

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

8. Simplified 98.9%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

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

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

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

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

    5. unpow2 N/A

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

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

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

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

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

    8. metadata-eval N/A

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

    9. +-commutative N/A

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

    10. +-commutative N/A

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

    11. associate-+r+ N/A

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

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

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

    13. distribute-rgt-out N/A

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

    14. metadata-eval N/A

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

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

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

    16. *-lowering-*.f32 98.6%

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

11. Simplified 98.6%

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

12. Taylor expanded in n0_i around 0

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

    1. *-commutative N/A

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

    2. *-lowering-*.f32 98.5%

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

14. Simplified 98.5%

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

15. Final simplification 98.5%

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

Alternative 6: 98.1% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(n1\_i + n0\_i \cdot \left(\left(normAngle \cdot normAngle\right) \cdot 0.3333333333333333 + -1\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+ n1_i (* n0_i (+ (* (* normAngle normAngle) 0.3333333333333333) -1.0))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * (n1_i + (n0_i * (((normAngle * normAngle) * 0.3333333333333333f) + -1.0f))));
}

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) * 0.3333333333333333e0) + (-1.0e0)))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(n1_i + Float32(n0_i * Float32(Float32(Float32(normAngle * normAngle) * Float32(0.3333333333333333)) + Float32(-1.0))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * (n1_i + (n0_i * (((normAngle * normAngle) * single(0.3333333333333333)) + single(-1.0)))));
end

Derivation

1. Initial program 97.0%

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

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

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

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

5. Simplified 97.1%

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

6. Taylor expanded in normAngle around 0

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

8. Simplified 98.1%

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

9. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

    5. sub-neg N/A

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

    6. metadata-eval N/A

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

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

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

    8. *-commutative N/A

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

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

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

    10. unpow2 N/A

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

    11. *-lowering-*.f32 98.0%

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

11. Simplified 98.0%

    \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + n0\_i \cdot \left(\left(normAngle \cdot normAngle\right) \cdot 0.3333333333333333 + -1\right)\right)} \]
12. Add Preprocessing

Alternative 7: 57.7% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 3.2000000759751296 \cdot 10^{-31}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -9.9999998245167e-15)
   n0_i
   (if (<= n0_i 3.2000000759751296e-31) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -9.9999998245167e-15f) {
		tmp = n0_i;
	} else if (n0_i <= 3.2000000759751296e-31f) {
		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 <= (-9.9999998245167e-15)) then
        tmp = n0_i
    else if (n0_i <= 3.2000000759751296e-31) 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(-9.9999998245167e-15))
		tmp = n0_i;
	elseif (n0_i <= Float32(3.2000000759751296e-31))
		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(-9.9999998245167e-15))
		tmp = n0_i;
	elseif (n0_i <= single(3.2000000759751296e-31))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -9.99999982e-15 or 3.20000008e-31 < n0_i

   1. Initial program 98.6%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 63.3%

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

   if -9.99999982e-15 < n0_i < 3.20000008e-31

   1. Initial program 95.1%

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

   3. Taylor expanded in normAngle around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

   5. Simplified 95.3%

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

   6. Taylor expanded in normAngle around 0

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

   8. Simplified 97.4%

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

   9. Taylor expanded in n0_i around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f32 58.8%

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

   11. Simplified 58.8%

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

Alternative 8: 98.0% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.0%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.6%

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

6. Taylor expanded in u around 0

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

8. Simplified 98.9%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

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

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

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

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

    5. unpow2 N/A

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

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

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

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

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

    8. metadata-eval N/A

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

    9. +-commutative N/A

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

    10. +-commutative N/A

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

    11. associate-+r+ N/A

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

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

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

    13. distribute-rgt-out N/A

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

    14. metadata-eval N/A

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

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

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

    16. *-lowering-*.f32 98.6%

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

11. Simplified 98.6%

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

12. Taylor expanded in normAngle around 0

    \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i - n0\_i\right)} \]
13. 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 98.0%

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

14. Simplified 98.0%

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

Alternative 9: 81.5% accurate, 84.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.0%

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

3. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

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

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

5. Simplified 97.1%

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

6. Taylor expanded in normAngle around 0

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

8. Simplified 98.1%

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

9. Taylor expanded in u around 0

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

11. Simplified 80.6%

    \[\leadsto \color{blue}{n0\_i} + u \cdot n1\_i \]
12. Add Preprocessing

Alternative 10: 46.9% 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 97.0%

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

3. Taylor expanded in u around 0

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

5. Simplified 47.6%

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

Reproduce

herbie shell --seed 2024163 
(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)))