Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.3%

Time: 17.7s

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: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot \left(1 - u\right)\\ t_1 := u \cdot \left(-1 + u \cdot u\right)\\ t_2 := n0\_i \cdot \left(1 - u\right)\\ \mathsf{fma}\left(u, n1\_i, t\_2 + \left(normAngle \cdot normAngle\right) \cdot \left(\left(0.008333333333333333 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5} + n1\_i \cdot {u}^{5}\right) + \left(-0.027777777777777776 \cdot \left(n0\_i \cdot \left(\left(1 - u\right) \cdot \left(t\_0 + -1\right)\right) + n1\_i \cdot t\_1\right) + \left(t\_2 + u \cdot n1\_i\right) \cdot -0.008333333333333333\right)\right) \cdot \left(normAngle \cdot normAngle\right) + \left(n1\_i \cdot \left(t\_1 \cdot -0.16666666666666666\right) + \left(-0.16666666666666666 + u \cdot 0.16666666666666666\right) \cdot \left(n0\_i \cdot t\_0 - n0\_i\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) (- 1.0 u)))
        (t_1 (* u (+ -1.0 (* u u))))
        (t_2 (* n0_i (- 1.0 u))))
   (fma
    u
    n1_i
    (+
     t_2
     (*
      (* normAngle normAngle)
      (+
       (*
        (+
         (*
          0.008333333333333333
          (+ (* n0_i (pow (- 1.0 u) 5.0)) (* n1_i (pow u 5.0))))
         (+
          (*
           -0.027777777777777776
           (+ (* n0_i (* (- 1.0 u) (+ t_0 -1.0))) (* n1_i t_1)))
          (* (+ t_2 (* u n1_i)) -0.008333333333333333)))
        (* normAngle normAngle))
       (+
        (* n1_i (* t_1 -0.16666666666666666))
        (*
         (+ -0.16666666666666666 (* u 0.16666666666666666))
         (- (* n0_i t_0) n0_i)))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = (1.0f - u) * (1.0f - u);
	float t_1 = u * (-1.0f + (u * u));
	float t_2 = n0_i * (1.0f - u);
	return fmaf(u, n1_i, (t_2 + ((normAngle * normAngle) * ((((0.008333333333333333f * ((n0_i * powf((1.0f - u), 5.0f)) + (n1_i * powf(u, 5.0f)))) + ((-0.027777777777777776f * ((n0_i * ((1.0f - u) * (t_0 + -1.0f))) + (n1_i * t_1))) + ((t_2 + (u * n1_i)) * -0.008333333333333333f))) * (normAngle * normAngle)) + ((n1_i * (t_1 * -0.16666666666666666f)) + ((-0.16666666666666666f + (u * 0.16666666666666666f)) * ((n0_i * t_0) - n0_i)))))));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))
	t_1 = Float32(u * Float32(Float32(-1.0) + Float32(u * u)))
	t_2 = Float32(n0_i * Float32(Float32(1.0) - u))
	return fma(u, n1_i, Float32(t_2 + Float32(Float32(normAngle * normAngle) * Float32(Float32(Float32(Float32(Float32(0.008333333333333333) * Float32(Float32(n0_i * (Float32(Float32(1.0) - u) ^ Float32(5.0))) + Float32(n1_i * (u ^ Float32(5.0))))) + Float32(Float32(Float32(-0.027777777777777776) * Float32(Float32(n0_i * Float32(Float32(Float32(1.0) - u) * Float32(t_0 + Float32(-1.0)))) + Float32(n1_i * t_1))) + Float32(Float32(t_2 + Float32(u * n1_i)) * Float32(-0.008333333333333333)))) * Float32(normAngle * normAngle)) + Float32(Float32(n1_i * Float32(t_1 * Float32(-0.16666666666666666))) + Float32(Float32(Float32(-0.16666666666666666) + Float32(u * Float32(0.16666666666666666))) * Float32(Float32(n0_i * t_0) - n0_i)))))))
end

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 99.1%

   \[\leadsto \color{blue}{\left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) + normAngle \cdot \left(normAngle \cdot \left(n0\_i \cdot \left(\left(-0.16666666666666666 + u \cdot 0.16666666666666666\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right) + \left(normAngle \cdot \left(normAngle \cdot \left(0.008333333333333333 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5} + n1\_i \cdot {u}^{5}\right) - \left(0.027777777777777776 \cdot \left(n0\_i \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) - \left(1 - u\right)\right)\right) + \left(0.027777777777777776 \cdot \left(n1\_i \cdot \left(u \cdot \left(u \cdot u\right) - u\right)\right) + \left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) \cdot 0.008333333333333333\right)\right)\right)\right) + \left(\left(n1\_i \cdot -0.16666666666666666\right) \cdot \left(u \cdot \left(u \cdot u\right) - u\right) - n0\_i \cdot \left(-0.16666666666666666 + u \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

7. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1\_i, n0\_i \cdot \left(1 - u\right) + \left(\left(0.008333333333333333 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5} + n1\_i \cdot {u}^{5}\right) + \left(-0.027777777777777776 \cdot \left(n0\_i \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right) + -1\right)\right) + n1\_i \cdot \left(u \cdot \left(u \cdot u + -1\right)\right)\right) + \left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) \cdot -0.008333333333333333\right)\right) \cdot \left(normAngle \cdot normAngle\right) + \left(n1\_i \cdot \left(-0.16666666666666666 \cdot \left(u \cdot \left(u \cdot u + -1\right)\right)\right) - \left(-0.16666666666666666 + u \cdot 0.16666666666666666\right) \cdot \left(n0\_i - n0\_i \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right)\right)\right) \cdot \left(normAngle \cdot normAngle\right)\right)} \]

8. Final simplification 99.2%

   \[\leadsto \mathsf{fma}\left(u, n1\_i, n0\_i \cdot \left(1 - u\right) + \left(normAngle \cdot normAngle\right) \cdot \left(\left(0.008333333333333333 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5} + n1\_i \cdot {u}^{5}\right) + \left(-0.027777777777777776 \cdot \left(n0\_i \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right) + -1\right)\right) + n1\_i \cdot \left(u \cdot \left(-1 + u \cdot u\right)\right)\right) + \left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) \cdot -0.008333333333333333\right)\right) \cdot \left(normAngle \cdot normAngle\right) + \left(n1\_i \cdot \left(\left(u \cdot \left(-1 + u \cdot u\right)\right) \cdot -0.16666666666666666\right) + \left(-0.16666666666666666 + u \cdot 0.16666666666666666\right) \cdot \left(n0\_i \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) - n0\_i\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 2: 99.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) + normAngle \cdot \left(normAngle \cdot \left(u \cdot \left(\left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.5\right) + \left(\left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.041666666666666664 + \left(0.008333333333333333 \cdot \left(n0\_i - n1\_i\right) - \left(n0\_i \cdot -0.05555555555555555 + n1\_i \cdot -0.027777777777777776\right)\right)\right) + u \cdot \left(n0\_i \cdot -0.5 + u \cdot \left(n1\_i \cdot -0.16666666666666666 + \left(\left(\left(n0\_i \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right)\right) \cdot 0.041666666666666664 + n0\_i \cdot 0.16666666666666666\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot -0.027777777777777776 + \left(n0\_i \cdot -0.08333333333333333 + n0\_i \cdot 0.027777777777777776\right)\right)\right)\right)\right)\right)\right) + n0\_i \cdot -0.16666666666666666\right)\right)\right) \end{array} \]

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 0.5))
       (+
        (*
         (* normAngle normAngle)
         (+
          (* n0_i -0.041666666666666664)
          (-
           (* 0.008333333333333333 (- n0_i n1_i))
           (+ (* n0_i -0.05555555555555555) (* n1_i -0.027777777777777776)))))
        (*
         u
         (+
          (* n0_i -0.5)
          (*
           u
           (+
            (* n1_i -0.16666666666666666)
            (+
             (+
              (* (* n0_i (* u (* normAngle normAngle))) 0.041666666666666664)
              (* n0_i 0.16666666666666666))
             (*
              (* normAngle normAngle)
              (+
               (* n1_i -0.027777777777777776)
               (+
                (* n0_i -0.08333333333333333)
                (* n0_i 0.027777777777777776)))))))))))
      (* n0_i -0.16666666666666666)))))))

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 * 0.5f)) + (((normAngle * normAngle) * ((n0_i * -0.041666666666666664f) + ((0.008333333333333333f * (n0_i - n1_i)) - ((n0_i * -0.05555555555555555f) + (n1_i * -0.027777777777777776f))))) + (u * ((n0_i * -0.5f) + (u * ((n1_i * -0.16666666666666666f) + ((((n0_i * (u * (normAngle * normAngle))) * 0.041666666666666664f) + (n0_i * 0.16666666666666666f)) + ((normAngle * normAngle) * ((n1_i * -0.027777777777777776f) + ((n0_i * -0.08333333333333333f) + (n0_i * 0.027777777777777776f))))))))))) + (n0_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 * (1.0e0 - u)) + (u * n1_i)) + (normangle * (normangle * (u * ((((n1_i * 0.16666666666666666e0) + (n0_i * 0.5e0)) + (((normangle * normangle) * ((n0_i * (-0.041666666666666664e0)) + ((0.008333333333333333e0 * (n0_i - n1_i)) - ((n0_i * (-0.05555555555555555e0)) + (n1_i * (-0.027777777777777776e0)))))) + (u * ((n0_i * (-0.5e0)) + (u * ((n1_i * (-0.16666666666666666e0)) + ((((n0_i * (u * (normangle * normangle))) * 0.041666666666666664e0) + (n0_i * 0.16666666666666666e0)) + ((normangle * normangle) * ((n1_i * (-0.027777777777777776e0)) + ((n0_i * (-0.08333333333333333e0)) + (n0_i * 0.027777777777777776e0))))))))))) + (n0_i * (-0.16666666666666666e0))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(Float32(n0_i * Float32(Float32(1.0) - u)) + Float32(u * n1_i)) + Float32(normAngle * Float32(normAngle * Float32(u * Float32(Float32(Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.5))) + Float32(Float32(Float32(normAngle * normAngle) * Float32(Float32(n0_i * Float32(-0.041666666666666664)) + Float32(Float32(Float32(0.008333333333333333) * Float32(n0_i - n1_i)) - Float32(Float32(n0_i * Float32(-0.05555555555555555)) + Float32(n1_i * Float32(-0.027777777777777776)))))) + Float32(u * Float32(Float32(n0_i * Float32(-0.5)) + Float32(u * Float32(Float32(n1_i * Float32(-0.16666666666666666)) + Float32(Float32(Float32(Float32(n0_i * Float32(u * Float32(normAngle * normAngle))) * Float32(0.041666666666666664)) + Float32(n0_i * Float32(0.16666666666666666))) + Float32(Float32(normAngle * normAngle) * Float32(Float32(n1_i * Float32(-0.027777777777777776)) + Float32(Float32(n0_i * Float32(-0.08333333333333333)) + Float32(n0_i * Float32(0.027777777777777776)))))))))))) + Float32(n0_i * Float32(-0.16666666666666666)))))))
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(0.5))) + (((normAngle * normAngle) * ((n0_i * single(-0.041666666666666664)) + ((single(0.008333333333333333) * (n0_i - n1_i)) - ((n0_i * single(-0.05555555555555555)) + (n1_i * single(-0.027777777777777776)))))) + (u * ((n0_i * single(-0.5)) + (u * ((n1_i * single(-0.16666666666666666)) + ((((n0_i * (u * (normAngle * normAngle))) * single(0.041666666666666664)) + (n0_i * single(0.16666666666666666))) + ((normAngle * normAngle) * ((n1_i * single(-0.027777777777777776)) + ((n0_i * single(-0.08333333333333333)) + (n0_i * single(0.027777777777777776)))))))))))) + (n0_i * single(-0.16666666666666666))))));
end

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 99.1%

   \[\leadsto \color{blue}{\left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) + normAngle \cdot \left(normAngle \cdot \left(n0\_i \cdot \left(\left(-0.16666666666666666 + u \cdot 0.16666666666666666\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right) + \left(normAngle \cdot \left(normAngle \cdot \left(0.008333333333333333 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5} + n1\_i \cdot {u}^{5}\right) - \left(0.027777777777777776 \cdot \left(n0\_i \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) - \left(1 - u\right)\right)\right) + \left(0.027777777777777776 \cdot \left(n1\_i \cdot \left(u \cdot \left(u \cdot u\right) - u\right)\right) + \left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) \cdot 0.008333333333333333\right)\right)\right)\right) + \left(\left(n1\_i \cdot -0.16666666666666666\right) \cdot \left(u \cdot \left(u \cdot u\right) - u\right) - n0\_i \cdot \left(-0.16666666666666666 + u \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

6. Taylor expanded in u around 0

   \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(1, u\right)\right), \mathsf{*.f32}\left(u, n1\_i\right)\right), \mathsf{*.f32}\left(normAngle, \mathsf{*.f32}\left(normAngle, \color{blue}{\left(u \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \left(\frac{1}{2} \cdot n0\_i + \left(u \cdot \left(\frac{-1}{2} \cdot n0\_i + u \cdot \left(\frac{-1}{6} \cdot n1\_i + \left(\frac{1}{24} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \left(\frac{1}{6} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{-1}{12} \cdot n0\_i - \left(\frac{-1}{36} \cdot n0\_i + \frac{1}{36} \cdot n1\_i\right)\right)\right)\right)\right)\right) + {normAngle}^{2} \cdot \left(\frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{18} \cdot n0\_i + \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right)\right)\right)\right) - \frac{1}{6} \cdot n0\_i\right)\right)}\right)\right)\right) \]

8. Simplified 99.1%

   \[\leadsto \left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) + normAngle \cdot \left(normAngle \cdot \color{blue}{\left(u \cdot \left(\left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.5\right) + \left(\left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.041666666666666664 - \left(\left(n0\_i \cdot -0.05555555555555555 + n1\_i \cdot -0.027777777777777776\right) + 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right) + u \cdot \left(n0\_i \cdot -0.5 + u \cdot \left(n1\_i \cdot -0.16666666666666666 + \left(\left(\left(n0\_i \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right)\right) \cdot 0.041666666666666664 + n0\_i \cdot 0.16666666666666666\right) + \left(normAngle \cdot normAngle\right) \cdot \left(\left(n0\_i \cdot -0.08333333333333333 + 0.027777777777777776 \cdot n0\_i\right) + n1\_i \cdot -0.027777777777777776\right)\right)\right)\right)\right)\right) + -0.16666666666666666 \cdot n0\_i\right)\right)}\right) \]

9. Final simplification 99.1%

   \[\leadsto \left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) + normAngle \cdot \left(normAngle \cdot \left(u \cdot \left(\left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.5\right) + \left(\left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.041666666666666664 + \left(0.008333333333333333 \cdot \left(n0\_i - n1\_i\right) - \left(n0\_i \cdot -0.05555555555555555 + n1\_i \cdot -0.027777777777777776\right)\right)\right) + u \cdot \left(n0\_i \cdot -0.5 + u \cdot \left(n1\_i \cdot -0.16666666666666666 + \left(\left(\left(n0\_i \cdot \left(u \cdot \left(normAngle \cdot normAngle\right)\right)\right) \cdot 0.041666666666666664 + n0\_i \cdot 0.16666666666666666\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot -0.027777777777777776 + \left(n0\_i \cdot -0.08333333333333333 + n0\_i \cdot 0.027777777777777776\right)\right)\right)\right)\right)\right)\right) + n0\_i \cdot -0.16666666666666666\right)\right)\right) \]
10. Add Preprocessing

Alternative 3: 99.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.16666666666666666 + \left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.5\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.041666666666666664 + \left(0.008333333333333333 \cdot \left(n0\_i - n1\_i\right) - \left(n0\_i \cdot -0.05555555555555555 + n1\_i \cdot -0.027777777777777776\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    (- n1_i n0_i)
    (*
     (* normAngle normAngle)
     (+
      (* n0_i -0.16666666666666666)
      (+
       (+ (* n1_i 0.16666666666666666) (* n0_i 0.5))
       (*
        (* normAngle normAngle)
        (+
         (* n0_i -0.041666666666666664)
         (-
          (* 0.008333333333333333 (- n0_i n1_i))
          (+
           (* n0_i -0.05555555555555555)
           (* n1_i -0.027777777777777776))))))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((n1_i - n0_i) + ((normAngle * normAngle) * ((n0_i * -0.16666666666666666f) + (((n1_i * 0.16666666666666666f) + (n0_i * 0.5f)) + ((normAngle * normAngle) * ((n0_i * -0.041666666666666664f) + ((0.008333333333333333f * (n0_i - n1_i)) - ((n0_i * -0.05555555555555555f) + (n1_i * -0.027777777777777776f))))))))));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i + (u * ((n1_i - n0_i) + ((normangle * normangle) * ((n0_i * (-0.16666666666666666e0)) + (((n1_i * 0.16666666666666666e0) + (n0_i * 0.5e0)) + ((normangle * normangle) * ((n0_i * (-0.041666666666666664e0)) + ((0.008333333333333333e0 * (n0_i - n1_i)) - ((n0_i * (-0.05555555555555555e0)) + (n1_i * (-0.027777777777777776e0)))))))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(Float32(n1_i - n0_i) + Float32(Float32(normAngle * normAngle) * Float32(Float32(n0_i * Float32(-0.16666666666666666)) + Float32(Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.5))) + Float32(Float32(normAngle * normAngle) * Float32(Float32(n0_i * Float32(-0.041666666666666664)) + Float32(Float32(Float32(0.008333333333333333) * Float32(n0_i - n1_i)) - Float32(Float32(n0_i * Float32(-0.05555555555555555)) + Float32(n1_i * Float32(-0.027777777777777776))))))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * ((n1_i - n0_i) + ((normAngle * normAngle) * ((n0_i * single(-0.16666666666666666)) + (((n1_i * single(0.16666666666666666)) + (n0_i * single(0.5))) + ((normAngle * normAngle) * ((n0_i * single(-0.041666666666666664)) + ((single(0.008333333333333333) * (n0_i - n1_i)) - ((n0_i * single(-0.05555555555555555)) + (n1_i * single(-0.027777777777777776)))))))))));
end

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 99.1%

   \[\leadsto \color{blue}{\left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) + normAngle \cdot \left(normAngle \cdot \left(n0\_i \cdot \left(\left(-0.16666666666666666 + u \cdot 0.16666666666666666\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right) + \left(normAngle \cdot \left(normAngle \cdot \left(0.008333333333333333 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5} + n1\_i \cdot {u}^{5}\right) - \left(0.027777777777777776 \cdot \left(n0\_i \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) - \left(1 - u\right)\right)\right) + \left(0.027777777777777776 \cdot \left(n1\_i \cdot \left(u \cdot \left(u \cdot u\right) - u\right)\right) + \left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) \cdot 0.008333333333333333\right)\right)\right)\right) + \left(\left(n1\_i \cdot -0.16666666666666666\right) \cdot \left(u \cdot \left(u \cdot u\right) - u\right) - n0\_i \cdot \left(-0.16666666666666666 + u \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \left(\frac{1}{2} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{18} \cdot n0\_i + \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right)\right)\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)} \]

8. Simplified 99.1%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(\left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.5\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.041666666666666664 - \left(\left(n0\_i \cdot -0.05555555555555555 + n1\_i \cdot -0.027777777777777776\right) + 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right) + -0.16666666666666666 \cdot n0\_i\right)\right)} \]

9. Final simplification 99.1%

   \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.16666666666666666 + \left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.5\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.041666666666666664 + \left(0.008333333333333333 \cdot \left(n0\_i - n1\_i\right) - \left(n0\_i \cdot -0.05555555555555555 + n1\_i \cdot -0.027777777777777776\right)\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 4: 99.0% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.16666666666666666 + \left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.5\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.041666666666666664 - \left(\left(n0\_i \cdot -0.05555555555555555 + n1\_i \cdot -0.027777777777777776\right) + n0\_i \cdot -0.008333333333333333\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    (- n1_i n0_i)
    (*
     (* normAngle normAngle)
     (+
      (* n0_i -0.16666666666666666)
      (+
       (+ (* n1_i 0.16666666666666666) (* n0_i 0.5))
       (*
        (* normAngle normAngle)
        (-
         (* n0_i -0.041666666666666664)
         (+
          (+ (* n0_i -0.05555555555555555) (* n1_i -0.027777777777777776))
          (* n0_i -0.008333333333333333)))))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((n1_i - n0_i) + ((normAngle * normAngle) * ((n0_i * -0.16666666666666666f) + (((n1_i * 0.16666666666666666f) + (n0_i * 0.5f)) + ((normAngle * normAngle) * ((n0_i * -0.041666666666666664f) - (((n0_i * -0.05555555555555555f) + (n1_i * -0.027777777777777776f)) + (n0_i * -0.008333333333333333f)))))))));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i + (u * ((n1_i - n0_i) + ((normangle * normangle) * ((n0_i * (-0.16666666666666666e0)) + (((n1_i * 0.16666666666666666e0) + (n0_i * 0.5e0)) + ((normangle * normangle) * ((n0_i * (-0.041666666666666664e0)) - (((n0_i * (-0.05555555555555555e0)) + (n1_i * (-0.027777777777777776e0))) + (n0_i * (-0.008333333333333333e0))))))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(Float32(n1_i - n0_i) + Float32(Float32(normAngle * normAngle) * Float32(Float32(n0_i * Float32(-0.16666666666666666)) + Float32(Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.5))) + Float32(Float32(normAngle * normAngle) * Float32(Float32(n0_i * Float32(-0.041666666666666664)) - Float32(Float32(Float32(n0_i * Float32(-0.05555555555555555)) + Float32(n1_i * Float32(-0.027777777777777776))) + Float32(n0_i * Float32(-0.008333333333333333)))))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * ((n1_i - n0_i) + ((normAngle * normAngle) * ((n0_i * single(-0.16666666666666666)) + (((n1_i * single(0.16666666666666666)) + (n0_i * single(0.5))) + ((normAngle * normAngle) * ((n0_i * single(-0.041666666666666664)) - (((n0_i * single(-0.05555555555555555)) + (n1_i * single(-0.027777777777777776))) + (n0_i * single(-0.008333333333333333))))))))));
end

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 99.1%

   \[\leadsto \color{blue}{\left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) + normAngle \cdot \left(normAngle \cdot \left(n0\_i \cdot \left(\left(-0.16666666666666666 + u \cdot 0.16666666666666666\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right) + \left(normAngle \cdot \left(normAngle \cdot \left(0.008333333333333333 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5} + n1\_i \cdot {u}^{5}\right) - \left(0.027777777777777776 \cdot \left(n0\_i \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) - \left(1 - u\right)\right)\right) + \left(0.027777777777777776 \cdot \left(n1\_i \cdot \left(u \cdot \left(u \cdot u\right) - u\right)\right) + \left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) \cdot 0.008333333333333333\right)\right)\right)\right) + \left(\left(n1\_i \cdot -0.16666666666666666\right) \cdot \left(u \cdot \left(u \cdot u\right) - u\right) - n0\_i \cdot \left(-0.16666666666666666 + u \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \left(\frac{1}{2} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{18} \cdot n0\_i + \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right)\right)\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)} \]

8. Simplified 99.1%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(\left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.5\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.041666666666666664 - \left(\left(n0\_i \cdot -0.05555555555555555 + n1\_i \cdot -0.027777777777777776\right) + 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right) + -0.16666666666666666 \cdot n0\_i\right)\right)} \]

9. Taylor expanded in n1_i around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{*.f32}\left(n0\_i, \frac{1}{2}\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{-1}{24}\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{-1}{18}\right), \mathsf{*.f32}\left(n1\_i, \frac{-1}{36}\right)\right), \color{blue}{\left(\frac{-1}{120} \cdot n0\_i\right)}\right)\right)\right)\right), \mathsf{*.f32}\left(\frac{-1}{6}, n0\_i\right)\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{*.f32}\left(n0\_i, \frac{1}{2}\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{-1}{24}\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{-1}{18}\right), \mathsf{*.f32}\left(n1\_i, \frac{-1}{36}\right)\right), \left(n0\_i \cdot \frac{-1}{120}\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\frac{-1}{6}, n0\_i\right)\right)\right)\right)\right)\right) \]

    2. *-lowering-*.f32 99.0%

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{*.f32}\left(n0\_i, \frac{1}{2}\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{-1}{24}\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{-1}{18}\right), \mathsf{*.f32}\left(n1\_i, \frac{-1}{36}\right)\right), \mathsf{*.f32}\left(n0\_i, \frac{-1}{120}\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\frac{-1}{6}, n0\_i\right)\right)\right)\right)\right)\right) \]

11. Simplified 99.0%

    \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(\left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.5\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.041666666666666664 - \left(\left(n0\_i \cdot -0.05555555555555555 + n1\_i \cdot -0.027777777777777776\right) + \color{blue}{n0\_i \cdot -0.008333333333333333}\right)\right)\right) + -0.16666666666666666 \cdot n0\_i\right)\right) \]

12. Final simplification 99.0%

    \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.16666666666666666 + \left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.5\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.041666666666666664 - \left(\left(n0\_i \cdot -0.05555555555555555 + n1\_i \cdot -0.027777777777777776\right) + n0\_i \cdot -0.008333333333333333\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 5: 98.9% accurate, 20.0× speedup

Math

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

FPCore

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

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 + ((normAngle * normAngle) * 0.019444444444444445f))))));
}

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 + ((normangle * normangle) * 0.019444444444444445e0))))))
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(n1_i * Float32(Float32(0.16666666666666666) + Float32(Float32(normAngle * normAngle) * Float32(0.019444444444444445))))))))
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) + ((normAngle * normAngle) * single(0.019444444444444445)))))));
end

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 99.1%

   \[\leadsto \color{blue}{\left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) + normAngle \cdot \left(normAngle \cdot \left(n0\_i \cdot \left(\left(-0.16666666666666666 + u \cdot 0.16666666666666666\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right) + \left(normAngle \cdot \left(normAngle \cdot \left(0.008333333333333333 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5} + n1\_i \cdot {u}^{5}\right) - \left(0.027777777777777776 \cdot \left(n0\_i \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) - \left(1 - u\right)\right)\right) + \left(0.027777777777777776 \cdot \left(n1\_i \cdot \left(u \cdot \left(u \cdot u\right) - u\right)\right) + \left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) \cdot 0.008333333333333333\right)\right)\right)\right) + \left(\left(n1\_i \cdot -0.16666666666666666\right) \cdot \left(u \cdot \left(u \cdot u\right) - u\right) - n0\_i \cdot \left(-0.16666666666666666 + u \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \left(\frac{1}{2} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{18} \cdot n0\_i + \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right)\right)\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)} \]

8. Simplified 99.1%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(\left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.5\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.041666666666666664 - \left(\left(n0\_i \cdot -0.05555555555555555 + n1\_i \cdot -0.027777777777777776\right) + 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right) + -0.16666666666666666 \cdot n0\_i\right)\right)} \]

9. Taylor expanded in n1_i around inf

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

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

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

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

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

    3. *-commutative N/A

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

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

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

    5. unpow2 N/A

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

    6. *-lowering-*.f32 99.0%

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

11. Simplified 99.0%

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

Alternative 6: 69.8% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n0\_i - u \cdot n0\_i\\ \mathbf{if}\;n0\_i \leq -2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 5.000000015855384 \cdot 10^{-31}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (- n0_i (* u n0_i))))
   (if (<= n0_i -2.0000000390829628e-24)
     t_0
     (if (<= n0_i 5.000000015855384e-31) (* u n1_i) t_0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = n0_i - (u * n0_i);
	float tmp;
	if (n0_i <= -2.0000000390829628e-24f) {
		tmp = t_0;
	} else if (n0_i <= 5.000000015855384e-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 - (u * n0_i)
    if (n0_i <= (-2.0000000390829628e-24)) then
        tmp = t_0
    else if (n0_i <= 5.000000015855384e-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(u * n0_i))
	tmp = Float32(0.0)
	if (n0_i <= Float32(-2.0000000390829628e-24))
		tmp = t_0;
	elseif (n0_i <= Float32(5.000000015855384e-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 - (u * n0_i);
	tmp = single(0.0);
	if (n0_i <= single(-2.0000000390829628e-24))
		tmp = t_0;
	elseif (n0_i <= single(5.000000015855384e-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 < -2.00000004e-24 or 5e-31 < n0_i

   1. Initial program 97.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

   3. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i}{\sin normAngle}} \]
   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 61.7%

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

   7. Simplified 61.7%

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

   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 76.7%

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

   10. Simplified 76.7%

       \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
   11. 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. distribute-rgt-in N/A

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

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

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

       4. *-commutative N/A

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

       5. unsub-neg N/A

          \[\leadsto 1 \cdot n0\_i - \color{blue}{n0\_i \cdot u} \]

       6. *-lft-identity N/A

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f32 76.8%

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

   12. Applied egg-rr 76.8%

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

   if -2.00000004e-24 < n0_i < 5e-31

   1. Initial program 95.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

   3. Simplified 61.0%

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

   5. Taylor expanded in n0_i around 0

      \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(n1\_i \cdot \sin \left(normAngle \cdot u\right)\right)}, \mathsf{sin.f32}\left(normAngle\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f32 53.3%

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

   7. Simplified 53.3%

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

   8. Taylor expanded in normAngle around 0

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

      1. *-lowering-*.f32 70.5%

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

   10. Simplified 70.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \mathbf{elif}\;n0\_i \leq 5.000000015855384 \cdot 10^{-31}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.7% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n0\_i \cdot \left(1 - u\right)\\ \mathbf{if}\;n0\_i \leq -2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 5.000000015855384 \cdot 10^{-31}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* n0_i (- 1.0 u))))
   (if (<= n0_i -2.0000000390829628e-24)
     t_0
     (if (<= n0_i 5.000000015855384e-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 <= -2.0000000390829628e-24f) {
		tmp = t_0;
	} else if (n0_i <= 5.000000015855384e-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 <= (-2.0000000390829628e-24)) then
        tmp = t_0
    else if (n0_i <= 5.000000015855384e-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(-2.0000000390829628e-24))
		tmp = t_0;
	elseif (n0_i <= Float32(5.000000015855384e-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(-2.0000000390829628e-24))
		tmp = t_0;
	elseif (n0_i <= single(5.000000015855384e-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 < -2.00000004e-24 or 5e-31 < n0_i

   1. Initial program 97.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

   3. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i}{\sin normAngle}} \]
   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 61.7%

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

   7. Simplified 61.7%

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

   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 76.7%

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

   10. Simplified 76.7%

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

   if -2.00000004e-24 < n0_i < 5e-31

   1. Initial program 95.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

   3. Simplified 61.0%

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

   5. Taylor expanded in n0_i around 0

      \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(n1\_i \cdot \sin \left(normAngle \cdot u\right)\right)}, \mathsf{sin.f32}\left(normAngle\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f32 53.3%

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

   7. Simplified 53.3%

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

   8. Taylor expanded in normAngle around 0

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

      1. *-lowering-*.f32 70.5%

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

   10. Simplified 70.5%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{elif}\;n0\_i \leq 5.000000015855384 \cdot 10^{-31}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.2% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 1.4999999800084155 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -2.0000000390829628e-24)
   n0_i
   (if (<= n0_i 1.4999999800084155e-24) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -2.0000000390829628e-24f) {
		tmp = n0_i;
	} else if (n0_i <= 1.4999999800084155e-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 <= (-2.0000000390829628e-24)) then
        tmp = n0_i
    else if (n0_i <= 1.4999999800084155e-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(-2.0000000390829628e-24))
		tmp = n0_i;
	elseif (n0_i <= Float32(1.4999999800084155e-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(-2.0000000390829628e-24))
		tmp = n0_i;
	elseif (n0_i <= single(1.4999999800084155e-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 < -2.00000004e-24 or 1.49999998e-24 < n0_i

   1. Initial program 98.5%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 63.0%

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

   if -2.00000004e-24 < n0_i < 1.49999998e-24

   1. Initial program 95.5%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

   3. Simplified 54.7%

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

   5. Taylor expanded in n0_i around 0

      \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(n1\_i \cdot \sin \left(normAngle \cdot u\right)\right)}, \mathsf{sin.f32}\left(normAngle\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f32 46.7%

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

   7. Simplified 46.7%

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

   8. Taylor expanded in normAngle around 0

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

      1. *-lowering-*.f32 64.5%

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

   10. Simplified 64.5%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 1.4999999800084155 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. 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 97.3%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 99.1%

   \[\leadsto \color{blue}{\left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) + normAngle \cdot \left(normAngle \cdot \left(n0\_i \cdot \left(\left(-0.16666666666666666 + u \cdot 0.16666666666666666\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right) + \left(normAngle \cdot \left(normAngle \cdot \left(0.008333333333333333 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5} + n1\_i \cdot {u}^{5}\right) - \left(0.027777777777777776 \cdot \left(n0\_i \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) - \left(1 - u\right)\right)\right) + \left(0.027777777777777776 \cdot \left(n1\_i \cdot \left(u \cdot \left(u \cdot u\right) - u\right)\right) + \left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) \cdot 0.008333333333333333\right)\right)\right)\right) + \left(\left(n1\_i \cdot -0.16666666666666666\right) \cdot \left(u \cdot \left(u \cdot u\right) - u\right) - n0\_i \cdot \left(-0.16666666666666666 + u \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \left(\frac{1}{2} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{18} \cdot n0\_i + \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right)\right)\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)} \]

8. Simplified 99.1%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(\left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.5\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.041666666666666664 - \left(\left(n0\_i \cdot -0.05555555555555555 + n1\_i \cdot -0.027777777777777776\right) + 0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right) + -0.16666666666666666 \cdot n0\_i\right)\right)} \]

9. Taylor expanded in normAngle around 0

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

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

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

    2. sub-neg N/A

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

    3. mul-1-neg N/A

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

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

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

    5. mul-1-neg N/A

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

    6. sub-neg N/A

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

    7. --lowering--.f32 98.4%

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

11. Simplified 98.4%

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

Alternative 10: 46.7% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in u around 0

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

5. Simplified 48.5%

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

Reproduce

herbie shell --seed 2024159 
(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)))