Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 98.8%

Time: 15.2s

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: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 + u \cdot 0.16666666666666666\\ n0\_i \cdot \left(1 - u\right) + \left(u \cdot n1\_i + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot \left(t\_0 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right) + \left(\left(n1\_i \cdot -0.16666666666666666\right) \cdot \left(u \cdot \left(u \cdot u\right) - u\right) - n0\_i \cdot t\_0\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (+ -0.16666666666666666 (* u 0.16666666666666666))))
   (+
    (* n0_i (- 1.0 u))
    (+
     (* u n1_i)
     (*
      (* normAngle normAngle)
      (+
       (* n0_i (* t_0 (* (- 1.0 u) (- 1.0 u))))
       (-
        (* (* n1_i -0.16666666666666666) (- (* u (* u u)) u))
        (* n0_i t_0))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = -0.16666666666666666f + (u * 0.16666666666666666f);
	return (n0_i * (1.0f - u)) + ((u * n1_i) + ((normAngle * normAngle) * ((n0_i * (t_0 * ((1.0f - u) * (1.0f - u)))) + (((n1_i * -0.16666666666666666f) * ((u * (u * u)) - u)) - (n0_i * t_0)))));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: t_0
    t_0 = (-0.16666666666666666e0) + (u * 0.16666666666666666e0)
    code = (n0_i * (1.0e0 - u)) + ((u * n1_i) + ((normangle * normangle) * ((n0_i * (t_0 * ((1.0e0 - u) * (1.0e0 - u)))) + (((n1_i * (-0.16666666666666666e0)) * ((u * (u * u)) - u)) - (n0_i * t_0)))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(-0.16666666666666666) + Float32(u * Float32(0.16666666666666666)))
	return Float32(Float32(n0_i * Float32(Float32(1.0) - u)) + Float32(Float32(u * n1_i) + Float32(Float32(normAngle * normAngle) * Float32(Float32(n0_i * Float32(t_0 * Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)))) + Float32(Float32(Float32(n1_i * Float32(-0.16666666666666666)) * Float32(Float32(u * Float32(u * u)) - u)) - Float32(n0_i * t_0))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	t_0 = single(-0.16666666666666666) + (u * single(0.16666666666666666));
	tmp = (n0_i * (single(1.0) - u)) + ((u * n1_i) + ((normAngle * normAngle) * ((n0_i * (t_0 * ((single(1.0) - u) * (single(1.0) - u)))) + (((n1_i * single(-0.16666666666666666)) * ((u * (u * u)) - u)) - (n0_i * t_0)))));
end

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.8%

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

Alternative 2: 98.9% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. 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 98.8%

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

7. Applied egg-rr 98.7%

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

8. Final simplification 98.7%

   \[\leadsto \left(n0\_i + \left(u \cdot n1\_i + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left(n1\_i \cdot \left(u \cdot \left(u \cdot u + -1\right)\right)\right) + \left(n0\_i \cdot \left(-0.16666666666666666 + u \cdot 0.16666666666666666\right)\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right) + -1\right)\right)\right)\right) - n0\_i \cdot u \]
9. 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 97.4%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.8%

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

6. Taylor expanded in u around 0

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

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

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

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

   10. associate--l+ N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. unpow2 N/A

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

   18. *-lowering-*.f32 98.7%

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

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

Alternative 4: 71.1% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -3.999999954906409 \cdot 10^{-26}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{elif}\;n0\_i \leq 1.0000000195414814 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -3.999999954906409e-26)
   (* n0_i (- 1.0 u))
   (if (<= n0_i 1.0000000195414814e-24) (* u n1_i) (- n0_i (* n0_i u)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -3.999999954906409e-26f) {
		tmp = n0_i * (1.0f - u);
	} else if (n0_i <= 1.0000000195414814e-24f) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i - (n0_i * u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-3.999999954906409e-26)) then
        tmp = n0_i * (1.0e0 - u)
    else if (n0_i <= 1.0000000195414814e-24) then
        tmp = u * n1_i
    else
        tmp = n0_i - (n0_i * u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-3.999999954906409e-26))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	elseif (n0_i <= Float32(1.0000000195414814e-24))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(n0_i - Float32(n0_i * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-3.999999954906409e-26))
		tmp = n0_i * (single(1.0) - u);
	elseif (n0_i <= single(1.0000000195414814e-24))
		tmp = u * n1_i;
	else
		tmp = n0_i - (n0_i * u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if n0_i < -3.99999995e-26

   1. Initial program 97.7%

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

   3. Taylor expanded in n0_i around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. sin-lowering-sin.f32 76.0%

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

   5. Simplified 76.0%

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

   6. Taylor expanded in normAngle around 0

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

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

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

      2. --lowering--.f32 76.1%

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

   8. Simplified 76.1%

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

   if -3.99999995e-26 < n0_i < 1.00000002e-24

   1. Initial program 96.1%

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

   3. Taylor expanded in normAngle around 0

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

   5. Simplified 98.1%

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

   6. Taylor expanded in n0_i around -inf

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

   8. Simplified 97.8%

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

   9. Taylor expanded in normAngle around 0

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

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

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

       2. associate--r+ N/A

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

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

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f32 95.5%

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

   11. Simplified 95.5%

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

   12. Taylor expanded in n0_i around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f32 63.9%

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

   14. Simplified 63.9%

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

   if 1.00000002e-24 < n0_i

   1. Initial program 98.4%

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

   3. Taylor expanded in n0_i around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. sin-lowering-sin.f32 78.6%

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

   5. Simplified 78.6%

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

   6. Taylor expanded in normAngle around 0

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

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

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

      2. --lowering--.f32 78.3%

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

   8. Simplified 78.3%

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

      1. sub-neg N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. neg-mul-1 N/A

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

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

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

      7. unsub-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f32 78.3%

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

   10. Applied egg-rr 78.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -3.999999954906409 \cdot 10^{-26}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{elif}\;n0\_i \leq 1.0000000195414814 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.0% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n0\_i \cdot \left(1 - u\right)\\ \mathbf{if}\;n0\_i \leq -3.999999954906409 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 1.0000000195414814 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* n0_i (- 1.0 u))))
   (if (<= n0_i -3.999999954906409e-26)
     t_0
     (if (<= n0_i 1.0000000195414814e-24) (* 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 <= -3.999999954906409e-26f) {
		tmp = t_0;
	} else if (n0_i <= 1.0000000195414814e-24f) {
		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 <= (-3.999999954906409e-26)) then
        tmp = t_0
    else if (n0_i <= 1.0000000195414814e-24) 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(-3.999999954906409e-26))
		tmp = t_0;
	elseif (n0_i <= Float32(1.0000000195414814e-24))
		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(-3.999999954906409e-26))
		tmp = t_0;
	elseif (n0_i <= single(1.0000000195414814e-24))
		tmp = u * n1_i;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -3.99999995e-26 or 1.00000002e-24 < n0_i

   1. Initial program 98.0%

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

   3. Taylor expanded in n0_i around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. sin-lowering-sin.f32 77.2%

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

   5. Simplified 77.2%

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

   6. Taylor expanded in normAngle around 0

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

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

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

      2. --lowering--.f32 77.1%

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

   8. Simplified 77.1%

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

   if -3.99999995e-26 < n0_i < 1.00000002e-24

   1. Initial program 96.1%

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

   3. Taylor expanded in normAngle around 0

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

   5. Simplified 98.1%

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

   6. Taylor expanded in n0_i around -inf

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

   8. Simplified 97.8%

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

   9. Taylor expanded in normAngle around 0

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

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

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

       2. associate--r+ N/A

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

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

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f32 95.5%

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

   11. Simplified 95.5%

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

   12. Taylor expanded in n0_i around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f32 63.9%

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

   14. Simplified 63.9%

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

Alternative 6: 98.8% accurate, 28.1× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    (- n1_i n0_i)
    (* n1_i (* (* normAngle normAngle) 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 * ((normAngle * normAngle) * 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 * ((normangle * normangle) * 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(n1_i * Float32(Float32(normAngle * normAngle) * Float32(0.16666666666666666))))))
end

MATLAB

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.8%

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

6. Taylor expanded in u around 0

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

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

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

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

   10. associate--l+ N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. unpow2 N/A

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

   18. *-lowering-*.f32 98.7%

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

8. Simplified 98.7%

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

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. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

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

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

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

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

    6. unpow2 N/A

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

    7. *-lowering-*.f32 98.6%

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

11. Simplified 98.6%

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

12. Final simplification 98.6%

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

Alternative 7: 98.2% accurate, 28.1× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((n1_i - n0_i) + (n0_i * ((normAngle * normAngle) * 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) + (n0_i * ((normangle * normangle) * 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(n0_i * Float32(Float32(normAngle * normAngle) * Float32(0.3333333333333333))))))
end

MATLAB

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.8%

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

6. Taylor expanded in u around 0

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

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

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

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

   10. associate--l+ N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. unpow2 N/A

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

   18. *-lowering-*.f32 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in n1_i around 0

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

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

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

    5. *-commutative N/A

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

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

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

    7. unpow2 N/A

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

    8. *-lowering-*.f32 97.8%

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

11. Simplified 97.8%

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

Alternative 8: 60.7% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 1.0000000195414814 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.999999936531045e-20)
   n0_i
   (if (<= n0_i 1.0000000195414814e-24) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.999999936531045e-20f) {
		tmp = n0_i;
	} else if (n0_i <= 1.0000000195414814e-24f) {
		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 <= (-1.999999936531045e-20)) then
        tmp = n0_i
    else if (n0_i <= 1.0000000195414814e-24) 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(-1.999999936531045e-20))
		tmp = n0_i;
	elseif (n0_i <= Float32(1.0000000195414814e-24))
		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(-1.999999936531045e-20))
		tmp = n0_i;
	elseif (n0_i <= single(1.0000000195414814e-24))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.99999994e-20 or 1.00000002e-24 < 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 u around 0

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

   5. Simplified 61.1%

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

   if -1.99999994e-20 < n0_i < 1.00000002e-24

   1. Initial program 96.3%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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 98.2%

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

   6. Taylor expanded in n0_i around -inf

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

   8. Simplified 98.0%

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

   9. Taylor expanded in normAngle around 0

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

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

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

       2. associate--r+ N/A

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

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

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f32 95.8%

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

   11. Simplified 95.8%

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

   12. Taylor expanded in n0_i around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f32 60.7%

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

   14. Simplified 60.7%

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

Alternative 9: 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 97.4%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.8%

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

7. Applied egg-rr 98.7%

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

8. Taylor expanded in normAngle around 0

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

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

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

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

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. --lowering--.f32 97.7%

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

10. Simplified 97.7%

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

Alternative 10: 46.6% 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 97.4%

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

3. Taylor expanded in u around 0

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

5. Simplified 48.7%

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

Reproduce

herbie shell --seed 2024164 
(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)))