Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.1%

Time: 15.9s

Alternatives: 10

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot -0.16666666666666666 - \left(n0\_i \cdot -0.5 + n1\_i \cdot -0.16666666666666666\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.16666666666666666 \cdot \left(n0\_i \cdot -0.5 - n0\_i \cdot -0.16666666666666666\right) + n0\_i \cdot 0.008333333333333333\right) - \left(n1\_i \cdot -0.027777777777777776 + \left(n1\_i \cdot 0.008333333333333333 + n0\_i \cdot 0.041666666666666664\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (+
   (* u (- n1_i n0_i))
   (*
    (pow normAngle 2.0)
    (+
     (*
      u
      (-
       (* n0_i -0.16666666666666666)
       (+ (* n0_i -0.5) (* n1_i -0.16666666666666666))))
     (*
      (pow normAngle 2.0)
      (*
       u
       (-
        (+
         (*
          -0.16666666666666666
          (- (* n0_i -0.5) (* n0_i -0.16666666666666666)))
         (* n0_i 0.008333333333333333))
        (+
         (* n1_i -0.027777777777777776)
         (+
          (* n1_i 0.008333333333333333)
          (* n0_i 0.041666666666666664)))))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + ((u * (n1_i - n0_i)) + (powf(normAngle, 2.0f) * ((u * ((n0_i * -0.16666666666666666f) - ((n0_i * -0.5f) + (n1_i * -0.16666666666666666f)))) + (powf(normAngle, 2.0f) * (u * (((-0.16666666666666666f * ((n0_i * -0.5f) - (n0_i * -0.16666666666666666f))) + (n0_i * 0.008333333333333333f)) - ((n1_i * -0.027777777777777776f) + ((n1_i * 0.008333333333333333f) + (n0_i * 0.041666666666666664f)))))))));
}

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 ** 2.0e0) * ((u * ((n0_i * (-0.16666666666666666e0)) - ((n0_i * (-0.5e0)) + (n1_i * (-0.16666666666666666e0))))) + ((normangle ** 2.0e0) * (u * ((((-0.16666666666666666e0) * ((n0_i * (-0.5e0)) - (n0_i * (-0.16666666666666666e0)))) + (n0_i * 0.008333333333333333e0)) - ((n1_i * (-0.027777777777777776e0)) + ((n1_i * 0.008333333333333333e0) + (n0_i * 0.041666666666666664e0)))))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(Float32(u * Float32(n1_i - n0_i)) + Float32((normAngle ^ Float32(2.0)) * Float32(Float32(u * Float32(Float32(n0_i * Float32(-0.16666666666666666)) - Float32(Float32(n0_i * Float32(-0.5)) + Float32(n1_i * Float32(-0.16666666666666666))))) + Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(Float32(Float32(Float32(-0.16666666666666666) * Float32(Float32(n0_i * Float32(-0.5)) - Float32(n0_i * Float32(-0.16666666666666666)))) + Float32(n0_i * Float32(0.008333333333333333))) - Float32(Float32(n1_i * Float32(-0.027777777777777776)) + Float32(Float32(n1_i * Float32(0.008333333333333333)) + Float32(n0_i * Float32(0.041666666666666664)))))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + ((u * (n1_i - n0_i)) + ((normAngle ^ single(2.0)) * ((u * ((n0_i * single(-0.16666666666666666)) - ((n0_i * single(-0.5)) + (n1_i * single(-0.16666666666666666))))) + ((normAngle ^ single(2.0)) * (u * (((single(-0.16666666666666666) * ((n0_i * single(-0.5)) - (n0_i * single(-0.16666666666666666)))) + (n0_i * single(0.008333333333333333))) - ((n1_i * single(-0.027777777777777776)) + ((n1_i * single(0.008333333333333333)) + (n0_i * single(0.041666666666666664))))))))));
end

Derivation

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. fma-define 98.0%

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

   2. associate-*r/ 98.3%

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

   3. *-rgt-identity 98.3%

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

   4. associate-*r/ 98.4%

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

   5. *-rgt-identity 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in u around 0 86.2%

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

   1. +-commutative 86.2%

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

   2. mul-1-neg 86.2%

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

   3. unsub-neg 86.2%

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

   4. *-commutative 86.2%

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

   5. associate-*r* 86.2%

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

7. Simplified 86.2%

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

8. Taylor expanded in normAngle around 0 98.6%

   \[\leadsto n0\_i + \color{blue}{\left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.16666666666666666 \cdot \left(-0.5 \cdot n0\_i - -0.16666666666666666 \cdot n0\_i\right) + 0.008333333333333333 \cdot n0\_i\right) - \left(-0.027777777777777776 \cdot n1\_i + \left(0.008333333333333333 \cdot n1\_i + 0.041666666666666664 \cdot n0\_i\right)\right)\right)\right)\right)\right)} \]

9. Final simplification 98.6%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot -0.16666666666666666 - \left(n0\_i \cdot -0.5 + n1\_i \cdot -0.16666666666666666\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.16666666666666666 \cdot \left(n0\_i \cdot -0.5 - n0\_i \cdot -0.16666666666666666\right) + n0\_i \cdot 0.008333333333333333\right) - \left(n1\_i \cdot -0.027777777777777776 + \left(n1\_i \cdot 0.008333333333333333 + n0\_i \cdot 0.041666666666666664\right)\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 2: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left({normAngle}^{2} \cdot \left(u \cdot \left(n1\_i \cdot 0.019444444444444445\right)\right) - u \cdot \left(\left(n0\_i \cdot -0.5 + n1\_i \cdot -0.16666666666666666\right) - n0\_i \cdot -0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (+
   (* u (- n1_i n0_i))
   (*
    (pow normAngle 2.0)
    (-
     (* (pow normAngle 2.0) (* u (* n1_i 0.019444444444444445)))
     (*
      u
      (-
       (+ (* n0_i -0.5) (* n1_i -0.16666666666666666))
       (* n0_i -0.16666666666666666))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + ((u * (n1_i - n0_i)) + (powf(normAngle, 2.0f) * ((powf(normAngle, 2.0f) * (u * (n1_i * 0.019444444444444445f))) - (u * (((n0_i * -0.5f) + (n1_i * -0.16666666666666666f)) - (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 + ((u * (n1_i - n0_i)) + ((normangle ** 2.0e0) * (((normangle ** 2.0e0) * (u * (n1_i * 0.019444444444444445e0))) - (u * (((n0_i * (-0.5e0)) + (n1_i * (-0.16666666666666666e0))) - (n0_i * (-0.16666666666666666e0)))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(Float32(u * Float32(n1_i - n0_i)) + Float32((normAngle ^ Float32(2.0)) * Float32(Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(n1_i * Float32(0.019444444444444445)))) - Float32(u * Float32(Float32(Float32(n0_i * Float32(-0.5)) + Float32(n1_i * Float32(-0.16666666666666666))) - Float32(n0_i * Float32(-0.16666666666666666))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + ((u * (n1_i - n0_i)) + ((normAngle ^ single(2.0)) * (((normAngle ^ single(2.0)) * (u * (n1_i * single(0.019444444444444445)))) - (u * (((n0_i * single(-0.5)) + (n1_i * single(-0.16666666666666666))) - (n0_i * single(-0.16666666666666666)))))));
end

Derivation

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. fma-define 98.0%

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

   2. associate-*r/ 98.3%

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

   3. *-rgt-identity 98.3%

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

   4. associate-*r/ 98.4%

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

   5. *-rgt-identity 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in u around 0 86.2%

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

   1. +-commutative 86.2%

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

   2. mul-1-neg 86.2%

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

   3. unsub-neg 86.2%

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

   4. *-commutative 86.2%

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

   5. associate-*r* 86.2%

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

7. Simplified 86.2%

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

8. Taylor expanded in normAngle around 0 98.6%

   \[\leadsto n0\_i + \color{blue}{\left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.16666666666666666 \cdot \left(-0.5 \cdot n0\_i - -0.16666666666666666 \cdot n0\_i\right) + 0.008333333333333333 \cdot n0\_i\right) - \left(-0.027777777777777776 \cdot n1\_i + \left(0.008333333333333333 \cdot n1\_i + 0.041666666666666664 \cdot n0\_i\right)\right)\right)\right)\right)\right)} \]

9. Taylor expanded in n0_i around 0 98.6%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \color{blue}{\left(-1 \cdot \left(-0.027777777777777776 \cdot n1\_i + 0.008333333333333333 \cdot n1\_i\right)\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. mul-1-neg 98.6%

       \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \color{blue}{\left(-\left(-0.027777777777777776 \cdot n1\_i + 0.008333333333333333 \cdot n1\_i\right)\right)}\right)\right)\right) \]

    2. distribute-rgt-out 98.6%

       \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-\color{blue}{n1\_i \cdot \left(-0.027777777777777776 + 0.008333333333333333\right)}\right)\right)\right)\right) \]

    3. metadata-eval 98.6%

       \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-n1\_i \cdot \color{blue}{-0.019444444444444445}\right)\right)\right)\right) \]

    4. metadata-eval 98.6%

       \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-n1\_i \cdot \color{blue}{\left(-0.019444444444444445\right)}\right)\right)\right)\right) \]

    5. distribute-rgt-neg-in 98.6%

       \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \color{blue}{\left(n1\_i \cdot \left(-\left(-0.019444444444444445\right)\right)\right)}\right)\right)\right) \]

    6. metadata-eval 98.6%

       \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n1\_i \cdot \left(-\color{blue}{-0.019444444444444445}\right)\right)\right)\right)\right) \]

    7. metadata-eval 98.6%

       \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n1\_i \cdot \color{blue}{0.019444444444444445}\right)\right)\right)\right) \]

11. Simplified 98.6%

    \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \color{blue}{\left(n1\_i \cdot 0.019444444444444445\right)}\right)\right)\right) \]

12. Final simplification 98.6%

    \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left({normAngle}^{2} \cdot \left(u \cdot \left(n1\_i \cdot 0.019444444444444445\right)\right) - u \cdot \left(\left(n0\_i \cdot -0.5 + n1\_i \cdot -0.16666666666666666\right) - n0\_i \cdot -0.16666666666666666\right)\right)\right) \]
13. Add Preprocessing

Alternative 3: 98.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ n0\_i \cdot \left(1 - u\right) + \left(u \cdot n1\_i - {normAngle}^{2} \cdot \left(u \cdot \left(\left(n1\_i - n0\_i\right) \cdot -0.16666666666666666 - \left(n0\_i \cdot 0.5 + u \cdot \left(n0\_i \cdot -0.5 + u \cdot \left(n1\_i \cdot -0.16666666666666666 + n0\_i \cdot 0.16666666666666666\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* n0_i (- 1.0 u))
  (-
   (* u n1_i)
   (*
    (pow normAngle 2.0)
    (*
     u
     (-
      (* (- n1_i n0_i) -0.16666666666666666)
      (+
       (* n0_i 0.5)
       (*
        u
        (+
         (* n0_i -0.5)
         (*
          u
          (+
           (* n1_i -0.16666666666666666)
           (* 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) - (powf(normAngle, 2.0f) * (u * (((n1_i - n0_i) * -0.16666666666666666f) - ((n0_i * 0.5f) + (u * ((n0_i * -0.5f) + (u * ((n1_i * -0.16666666666666666f) + (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 ** 2.0e0) * (u * (((n1_i - n0_i) * (-0.16666666666666666e0)) - ((n0_i * 0.5e0) + (u * ((n0_i * (-0.5e0)) + (u * ((n1_i * (-0.16666666666666666e0)) + (n0_i * 0.16666666666666666e0))))))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(n0_i * Float32(Float32(1.0) - u)) + Float32(Float32(u * n1_i) - Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(Float32(Float32(n1_i - n0_i) * Float32(-0.16666666666666666)) - Float32(Float32(n0_i * Float32(0.5)) + Float32(u * Float32(Float32(n0_i * Float32(-0.5)) + Float32(u * Float32(Float32(n1_i * Float32(-0.16666666666666666)) + 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 ^ single(2.0)) * (u * (((n1_i - n0_i) * single(-0.16666666666666666)) - ((n0_i * single(0.5)) + (u * ((n0_i * single(-0.5)) + (u * ((n1_i * single(-0.16666666666666666)) + (n0_i * single(0.16666666666666666)))))))))));
end

Derivation

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 97.9%

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

   2. associate-*l* 79.0%

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

   3. *-commutative 79.0%

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

   4. associate-*l* 72.1%

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

   5. distribute-lft-out 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in normAngle around 0 98.3%

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right)\right)\right)} \]

6. Taylor expanded in u around 0 98.3%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(\left(0.5 \cdot n0\_i + u \cdot \left(-0.5 \cdot n0\_i + u \cdot \left(-0.16666666666666666 \cdot n1\_i + 0.16666666666666666 \cdot n0\_i\right)\right)\right) - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)}\right) \]

7. Final simplification 98.3%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(u \cdot n1\_i - {normAngle}^{2} \cdot \left(u \cdot \left(\left(n1\_i - n0\_i\right) \cdot -0.16666666666666666 - \left(n0\_i \cdot 0.5 + u \cdot \left(n0\_i \cdot -0.5 + u \cdot \left(n1\_i \cdot -0.16666666666666666 + n0\_i \cdot 0.16666666666666666\right)\right)\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 4: 98.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ n0\_i - u \cdot \left(n0\_i - \left(n1\_i - {normAngle}^{2} \cdot \left(\left(n0\_i \cdot -0.5 + n1\_i \cdot -0.16666666666666666\right) - n0\_i \cdot -0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (-
  n0_i
  (*
   u
   (-
    n0_i
    (-
     n1_i
     (*
      (pow normAngle 2.0)
      (-
       (+ (* n0_i -0.5) (* n1_i -0.16666666666666666))
       (* n0_i -0.16666666666666666))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i - (u * (n0_i - (n1_i - (powf(normAngle, 2.0f) * (((n0_i * -0.5f) + (n1_i * -0.16666666666666666f)) - (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 - (u * (n0_i - (n1_i - ((normangle ** 2.0e0) * (((n0_i * (-0.5e0)) + (n1_i * (-0.16666666666666666e0))) - (n0_i * (-0.16666666666666666e0)))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i - Float32(u * Float32(n0_i - Float32(n1_i - Float32((normAngle ^ Float32(2.0)) * Float32(Float32(Float32(n0_i * Float32(-0.5)) + Float32(n1_i * Float32(-0.16666666666666666))) - Float32(n0_i * Float32(-0.16666666666666666))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i - (u * (n0_i - (n1_i - ((normAngle ^ single(2.0)) * (((n0_i * single(-0.5)) + (n1_i * single(-0.16666666666666666))) - (n0_i * single(-0.16666666666666666)))))));
end

Derivation

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. fma-define 98.0%

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

   2. associate-*r/ 98.3%

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

   3. *-rgt-identity 98.3%

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

   4. associate-*r/ 98.4%

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

   5. *-rgt-identity 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in u around 0 86.2%

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

   1. +-commutative 86.2%

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

   2. mul-1-neg 86.2%

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

   3. unsub-neg 86.2%

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

   4. *-commutative 86.2%

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

   5. associate-*r* 86.2%

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

7. Simplified 86.2%

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

8. Taylor expanded in normAngle around 0 98.3%

   \[\leadsto n0\_i + u \cdot \color{blue}{\left(\left(n1\_i + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right)\right) - n0\_i\right)} \]

9. Final simplification 98.3%

   \[\leadsto n0\_i - u \cdot \left(n0\_i - \left(n1\_i - {normAngle}^{2} \cdot \left(\left(n0\_i \cdot -0.5 + n1\_i \cdot -0.16666666666666666\right) - n0\_i \cdot -0.16666666666666666\right)\right)\right) \]
10. Add Preprocessing

Alternative 5: 98.7% accurate, 16.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 97.9%

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

   2. associate-*l* 79.0%

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

   3. *-commutative 79.0%

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

   4. associate-*l* 72.1%

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

   5. distribute-lft-out 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in normAngle around 0 98.3%

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right)\right)\right)} \]

6. Taylor expanded in u around 0 98.0%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. cancel-sign-sub-inv 98.0%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \color{blue}{\left(0.5 \cdot n0\_i + \left(--0.16666666666666666\right) \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)}\right)\right) \]

   2. *-commutative 98.0%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\color{blue}{n0\_i \cdot 0.5} + \left(--0.16666666666666666\right) \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

   3. metadata-eval 98.0%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot 0.5 + \color{blue}{0.16666666666666666} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

   4. neg-mul-1 98.0%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1\_i + \color{blue}{\left(-n0\_i\right)}\right)\right)\right)\right) \]

   5. sub-neg 98.0%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot 0.5 + 0.16666666666666666 \cdot \color{blue}{\left(n1\_i - n0\_i\right)}\right)\right)\right) \]

8. Simplified 98.0%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(n0\_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1\_i - n0\_i\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. unpow2 98.0%

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

10. Applied egg-rr 98.0%

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

11. Final simplification 98.0%

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

Alternative 6: 70.5% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -5.99999989471002 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 6.999999919513338 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -5.99999989471002e-14) (not (<= n1_i 6.999999919513338e-16)))
   (* u n1_i)
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -5.99999989471002e-14f) || !(n1_i <= 6.999999919513338e-16f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-5.99999989471002e-14)) .or. (.not. (n1_i <= 6.999999919513338e-16))) then
        tmp = u * n1_i
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-5.99999989471002e-14)) || !(n1_i <= Float32(6.999999919513338e-16)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-5.99999989471002e-14)) || ~((n1_i <= single(6.999999919513338e-16))))
		tmp = u * n1_i;
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -5.99999989e-14 or 6.99999992e-16 < n1_i

   1. Initial program 97.2%

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

      1. fma-define 97.2%

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

      2. associate-*r/ 97.4%

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

      3. *-rgt-identity 97.4%

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

      4. associate-*r/ 97.6%

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

      5. *-rgt-identity 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in u around 0 91.8%

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

      1. +-commutative 91.8%

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

      2. mul-1-neg 91.8%

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

      3. unsub-neg 91.8%

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

      4. *-commutative 91.8%

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

      5. associate-*r* 91.8%

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

   7. Simplified 91.8%

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

   8. Taylor expanded in normAngle around 0 96.2%

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

   9. Taylor expanded in n0_i around inf 95.8%

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

       1. +-commutative 95.8%

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

       2. +-commutative 95.8%

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

       3. associate-+l+ 95.8%

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

       4. +-commutative 95.8%

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

       5. mul-1-neg 95.8%

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

       6. sub-neg 95.8%

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

       7. *-commutative 95.8%

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

       8. associate-/l* 95.5%

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

   11. Simplified 95.5%

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

   12. Taylor expanded in n0_i around 0 63.2%

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

   if -5.99999989e-14 < n1_i < 6.99999992e-16

   1. Initial program 98.3%

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

      1. fma-define 98.4%

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

      2. associate-*r/ 98.7%

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

      3. *-rgt-identity 98.7%

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

      4. associate-*r/ 98.8%

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

      5. *-rgt-identity 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in u around 0 83.2%

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

      1. +-commutative 83.2%

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

      2. mul-1-neg 83.2%

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

      3. unsub-neg 83.2%

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

      4. *-commutative 83.2%

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

      5. associate-*r* 83.2%

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

   7. Simplified 83.2%

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

   8. Taylor expanded in normAngle around 0 97.6%

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

   9. Taylor expanded in n0_i around inf 75.6%

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

       1. mul-1-neg 75.6%

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

       2. sub-neg 75.6%

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

   11. Simplified 75.6%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -5.99999989471002 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 6.999999919513338 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.2% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -5.99999989471002 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 6.999999919513338 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -5.99999989471002e-14) (not (<= n1_i 6.999999919513338e-16)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -5.99999989471002e-14f) || !(n1_i <= 6.999999919513338e-16f)) {
		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 ((n1_i <= (-5.99999989471002e-14)) .or. (.not. (n1_i <= 6.999999919513338e-16))) 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 ((n1_i <= Float32(-5.99999989471002e-14)) || !(n1_i <= Float32(6.999999919513338e-16)))
		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 ((n1_i <= single(-5.99999989471002e-14)) || ~((n1_i <= single(6.999999919513338e-16))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -5.99999989e-14 or 6.99999992e-16 < n1_i

   1. Initial program 97.2%

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

      1. fma-define 97.2%

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

      2. associate-*r/ 97.4%

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

      3. *-rgt-identity 97.4%

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

      4. associate-*r/ 97.6%

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

      5. *-rgt-identity 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in u around 0 91.8%

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

      1. +-commutative 91.8%

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

      2. mul-1-neg 91.8%

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

      3. unsub-neg 91.8%

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

      4. *-commutative 91.8%

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

      5. associate-*r* 91.8%

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

   7. Simplified 91.8%

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

   8. Taylor expanded in normAngle around 0 96.2%

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

   9. Taylor expanded in n0_i around inf 95.8%

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

       1. +-commutative 95.8%

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

       2. +-commutative 95.8%

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

       3. associate-+l+ 95.8%

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

       4. +-commutative 95.8%

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

       5. mul-1-neg 95.8%

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

       6. sub-neg 95.8%

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

       7. *-commutative 95.8%

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

       8. associate-/l* 95.5%

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

   11. Simplified 95.5%

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

   12. Taylor expanded in n0_i around 0 63.2%

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

   if -5.99999989e-14 < n1_i < 6.99999992e-16

   1. Initial program 98.3%

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

      1. fma-define 98.4%

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

      2. associate-*r/ 98.7%

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

      3. *-rgt-identity 98.7%

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

      4. associate-*r/ 98.8%

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

      5. *-rgt-identity 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in u around 0 61.0%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -5.99999989471002 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 6.999999919513338 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.1% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. fma-define 98.0%

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

   2. associate-*r/ 98.3%

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

   3. *-rgt-identity 98.3%

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

   4. associate-*r/ 98.4%

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

   5. *-rgt-identity 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in u around 0 86.2%

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

   1. +-commutative 86.2%

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

   2. mul-1-neg 86.2%

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

   3. unsub-neg 86.2%

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

   4. *-commutative 86.2%

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

   5. associate-*r* 86.2%

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

7. Simplified 86.2%

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

8. Taylor expanded in normAngle around 0 97.1%

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

9. Final simplification 97.1%

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

Alternative 9: 81.6% accurate, 84.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. fma-define 98.0%

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

   2. associate-*r/ 98.3%

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

   3. *-rgt-identity 98.3%

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

   4. associate-*r/ 98.4%

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

   5. *-rgt-identity 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in u around 0 86.2%

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

   1. +-commutative 86.2%

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

   2. mul-1-neg 86.2%

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

   3. unsub-neg 86.2%

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

   4. *-commutative 86.2%

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

   5. associate-*r* 86.2%

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

7. Simplified 86.2%

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

8. Taylor expanded in normAngle around 0 97.1%

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

9. Taylor expanded in n1_i around inf 82.0%

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

    1. *-commutative 82.0%

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

11. Simplified 82.0%

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

Alternative 10: 47.0% 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 97.9%

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

   1. fma-define 98.0%

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

   2. associate-*r/ 98.3%

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

   3. *-rgt-identity 98.3%

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

   4. associate-*r/ 98.4%

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

   5. *-rgt-identity 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in u around 0 48.6%

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

Reproduce

herbie shell --seed 2024145 
(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)))