Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.2%

Time: 16.6s

Alternatives: 9

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.2% 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: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n0\_i \cdot -0.16666666666666666 - n0\_i \cdot -0.5\\ n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \left({normAngle}^{2} \cdot \left(u \cdot \left(t\_0 - n1\_i \cdot -0.16666666666666666\right)\right) - {normAngle}^{4} \cdot \left(u \cdot \left(\left(n0\_i \cdot 0.041666666666666664 + \left(-0.16666666666666666 \cdot t\_0 - n0\_i \cdot 0.008333333333333333\right)\right) + \left(n1\_i \cdot -0.027777777777777776 + n1\_i \cdot 0.008333333333333333\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (- (* n0_i -0.16666666666666666) (* n0_i -0.5))))
   (+
    n0_i
    (+
     (* u (- n1_i n0_i))
     (-
      (* (pow normAngle 2.0) (* u (- t_0 (* n1_i -0.16666666666666666))))
      (*
       (pow normAngle 4.0)
       (*
        u
        (+
         (+
          (* n0_i 0.041666666666666664)
          (- (* -0.16666666666666666 t_0) (* n0_i 0.008333333333333333)))
         (+
          (* n1_i -0.027777777777777776)
          (* n1_i 0.008333333333333333))))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = (n0_i * -0.16666666666666666f) - (n0_i * -0.5f);
	return n0_i + ((u * (n1_i - n0_i)) + ((powf(normAngle, 2.0f) * (u * (t_0 - (n1_i * -0.16666666666666666f)))) - (powf(normAngle, 4.0f) * (u * (((n0_i * 0.041666666666666664f) + ((-0.16666666666666666f * t_0) - (n0_i * 0.008333333333333333f))) + ((n1_i * -0.027777777777777776f) + (n1_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
    real(4) :: t_0
    t_0 = (n0_i * (-0.16666666666666666e0)) - (n0_i * (-0.5e0))
    code = n0_i + ((u * (n1_i - n0_i)) + (((normangle ** 2.0e0) * (u * (t_0 - (n1_i * (-0.16666666666666666e0))))) - ((normangle ** 4.0e0) * (u * (((n0_i * 0.041666666666666664e0) + (((-0.16666666666666666e0) * t_0) - (n0_i * 0.008333333333333333e0))) + ((n1_i * (-0.027777777777777776e0)) + (n1_i * 0.008333333333333333e0)))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(n0_i * Float32(-0.16666666666666666)) - Float32(n0_i * Float32(-0.5)))
	return Float32(n0_i + Float32(Float32(u * Float32(n1_i - n0_i)) + Float32(Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(t_0 - Float32(n1_i * Float32(-0.16666666666666666))))) - Float32((normAngle ^ Float32(4.0)) * Float32(u * Float32(Float32(Float32(n0_i * Float32(0.041666666666666664)) + Float32(Float32(Float32(-0.16666666666666666) * t_0) - Float32(n0_i * Float32(0.008333333333333333)))) + Float32(Float32(n1_i * Float32(-0.027777777777777776)) + Float32(n1_i * Float32(0.008333333333333333)))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	t_0 = (n0_i * single(-0.16666666666666666)) - (n0_i * single(-0.5));
	tmp = n0_i + ((u * (n1_i - n0_i)) + (((normAngle ^ single(2.0)) * (u * (t_0 - (n1_i * single(-0.16666666666666666))))) - ((normAngle ^ single(4.0)) * (u * (((n0_i * single(0.041666666666666664)) + ((single(-0.16666666666666666) * t_0) - (n0_i * single(0.008333333333333333)))) + ((n1_i * single(-0.027777777777777776)) + (n1_i * single(0.008333333333333333))))))));
end

Derivation

1. Initial program 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*l* 82.7%

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

   3. *-commutative 82.7%

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

   4. associate-*l* 76.3%

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

   5. distribute-lft-out 76.3%

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

3. Simplified 76.3%

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

5. Taylor expanded in u around 0 87.0%

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

6. Taylor expanded in normAngle around 0 99.3%

   \[\leadsto n0\_i + \color{blue}{\left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + \left({normAngle}^{2} \cdot \left(u \cdot \left(-1 \cdot \left(-0.5 \cdot n0\_i - -0.16666666666666666 \cdot n0\_i\right) - -0.16666666666666666 \cdot n1\_i\right)\right) + {normAngle}^{4} \cdot \left(u \cdot \left(-1 \cdot \left(0.041666666666666664 \cdot n0\_i - \left(-0.16666666666666666 \cdot \left(-0.5 \cdot n0\_i - -0.16666666666666666 \cdot n0\_i\right) + 0.008333333333333333 \cdot n0\_i\right)\right) - \left(-0.027777777777777776 \cdot n1\_i + 0.008333333333333333 \cdot n1\_i\right)\right)\right)\right)\right)} \]

7. Final simplification 99.3%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \left({normAngle}^{2} \cdot \left(u \cdot \left(\left(n0\_i \cdot -0.16666666666666666 - n0\_i \cdot -0.5\right) - n1\_i \cdot -0.16666666666666666\right)\right) - {normAngle}^{4} \cdot \left(u \cdot \left(\left(n0\_i \cdot 0.041666666666666664 + \left(-0.16666666666666666 \cdot \left(n0\_i \cdot -0.16666666666666666 - n0\_i \cdot -0.5\right) - n0\_i \cdot 0.008333333333333333\right)\right) + \left(n1\_i \cdot -0.027777777777777776 + n1\_i \cdot 0.008333333333333333\right)\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 2: 99.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot 0.3333333333333333 + n1\_i \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (+
   (* u (- n1_i n0_i))
   (*
    (pow normAngle 2.0)
    (* u (+ (* n0_i 0.3333333333333333) (* n1_i 0.16666666666666666)))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + ((u * (n1_i - n0_i)) + (powf(normAngle, 2.0f) * (u * ((n0_i * 0.3333333333333333f) + (n1_i * 0.16666666666666666f)))));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i + ((u * (n1_i - n0_i)) + ((normangle ** 2.0e0) * (u * ((n0_i * 0.3333333333333333e0) + (n1_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(u * Float32(Float32(n0_i * Float32(0.3333333333333333)) + Float32(n1_i * Float32(0.16666666666666666)))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + ((u * (n1_i - n0_i)) + ((normAngle ^ single(2.0)) * (u * ((n0_i * single(0.3333333333333333)) + (n1_i * single(0.16666666666666666))))));
end

Derivation

1. Initial program 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*l* 82.7%

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

   3. *-commutative 82.7%

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

   4. associate-*l* 76.3%

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

   5. distribute-lft-out 76.3%

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

3. Simplified 76.3%

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

5. Taylor expanded in u around 0 87.0%

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

6. Taylor expanded in normAngle around 0 99.1%

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

7. Taylor expanded in n0_i around 0 99.1%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(n1\_i \cdot u\right) + 0.3333333333333333 \cdot \left(n0\_i \cdot u\right)\right)}\right) \]
8. Step-by-step derivation

   1. +-commutative 99.1%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(n0\_i \cdot u\right) + 0.16666666666666666 \cdot \left(n1\_i \cdot u\right)\right)}\right) \]

   2. *-commutative 99.1%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(0.3333333333333333 \cdot \left(n0\_i \cdot u\right) + \color{blue}{\left(n1\_i \cdot u\right) \cdot 0.16666666666666666}\right)\right) \]

   3. *-commutative 99.1%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(0.3333333333333333 \cdot \left(n0\_i \cdot u\right) + \color{blue}{\left(u \cdot n1\_i\right)} \cdot 0.16666666666666666\right)\right) \]

   4. associate-*r* 99.1%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(0.3333333333333333 \cdot \left(n0\_i \cdot u\right) + \color{blue}{u \cdot \left(n1\_i \cdot 0.16666666666666666\right)}\right)\right) \]

   5. associate-*r* 99.1%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(\color{blue}{\left(0.3333333333333333 \cdot n0\_i\right) \cdot u} + u \cdot \left(n1\_i \cdot 0.16666666666666666\right)\right)\right) \]

   6. *-commutative 99.1%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(\left(0.3333333333333333 \cdot n0\_i\right) \cdot u + \color{blue}{\left(n1\_i \cdot 0.16666666666666666\right) \cdot u}\right)\right) \]

   7. distribute-rgt-out 99.1%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(0.3333333333333333 \cdot n0\_i + n1\_i \cdot 0.16666666666666666\right)\right)}\right) \]

   8. *-commutative 99.1%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(0.3333333333333333 \cdot n0\_i + \color{blue}{0.16666666666666666 \cdot n1\_i}\right)\right)\right) \]

9. Simplified 99.1%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(0.3333333333333333 \cdot n0\_i + 0.16666666666666666 \cdot n1\_i\right)\right)}\right) \]

10. Final simplification 99.1%

    \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot 0.3333333333333333 + n1\_i \cdot 0.16666666666666666\right)\right)\right) \]
11. Add Preprocessing

Alternative 3: 98.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(n0\_i \cdot \left(u \cdot 0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (+
   (* u (- n1_i n0_i))
   (* (pow normAngle 2.0) (* n0_i (* u 0.3333333333333333))))))

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) * (n0_i * (u * 0.3333333333333333f))));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i + ((u * (n1_i - n0_i)) + ((normangle ** 2.0e0) * (n0_i * (u * 0.3333333333333333e0))))
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(n0_i * Float32(u * Float32(0.3333333333333333))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + ((u * (n1_i - n0_i)) + ((normAngle ^ single(2.0)) * (n0_i * (u * single(0.3333333333333333)))));
end

Derivation

1. Initial program 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*l* 82.7%

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

   3. *-commutative 82.7%

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

   4. associate-*l* 76.3%

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

   5. distribute-lft-out 76.3%

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

3. Simplified 76.3%

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

5. Taylor expanded in u around 0 87.0%

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

6. Taylor expanded in normAngle around 0 99.1%

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

7. Taylor expanded in n0_i around inf 98.3%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(n0\_i \cdot u\right)\right)}\right) \]
8. Step-by-step derivation

   1. *-commutative 98.3%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(\left(n0\_i \cdot u\right) \cdot 0.3333333333333333\right)}\right) \]

   2. associate-*l* 98.3%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(n0\_i \cdot \left(u \cdot 0.3333333333333333\right)\right)}\right) \]

9. Simplified 98.3%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(n0\_i \cdot \left(u \cdot 0.3333333333333333\right)\right)}\right) \]

10. Final simplification 98.3%

    \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(n0\_i \cdot \left(u \cdot 0.3333333333333333\right)\right)\right) \]
11. Add Preprocessing

Alternative 4: 98.8% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (+
   (* u (- n1_i n0_i))
   (* (pow normAngle 2.0) (* n1_i (* u 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) * (n1_i * (u * 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) * (n1_i * (u * 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(n1_i * Float32(u * 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)) * (n1_i * (u * single(0.16666666666666666)))));
end

Derivation

1. Initial program 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*l* 82.7%

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

   3. *-commutative 82.7%

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

   4. associate-*l* 76.3%

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

   5. distribute-lft-out 76.3%

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

3. Simplified 76.3%

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

5. Taylor expanded in u around 0 87.0%

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

6. Taylor expanded in normAngle around 0 99.1%

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

7. Taylor expanded in n0_i around 0 98.9%

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

   1. associate-*r* 98.9%

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

   2. *-commutative 98.9%

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

   3. associate-*l* 98.9%

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

9. Simplified 98.9%

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

10. Final simplification 98.9%

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

Alternative 5: 69.6% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -5.000000229068525 \cdot 10^{-19} \lor \neg \left(n0\_i \leq 4.0000000126843074 \cdot 10^{-28}\right):\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -5.000000229068525e-19)
         (not (<= n0_i 4.0000000126843074e-28)))
   (* n0_i (- 1.0 u))
   (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -5.000000229068525e-19f) || !(n0_i <= 4.0000000126843074e-28f)) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = u * n1_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 <= (-5.000000229068525e-19)) .or. (.not. (n0_i <= 4.0000000126843074e-28))) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-5.000000229068525e-19)) || !(n0_i <= Float32(4.0000000126843074e-28)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-5.000000229068525e-19)) || ~((n0_i <= single(4.0000000126843074e-28))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -5.00000023e-19 or 4.00000001e-28 < n0_i

   1. Initial program 98.1%

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

      1. *-commutative 98.1%

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

      2. associate-*l* 85.7%

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

      3. *-commutative 85.7%

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

      4. associate-*l* 84.2%

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

      5. distribute-lft-out 84.2%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in normAngle around 0 97.8%

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

      1. *-commutative 97.8%

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

      2. fma-def 97.8%

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

      3. *-commutative 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in n0_i around inf 71.9%

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

   if -5.00000023e-19 < n0_i < 4.00000001e-28

   1. Initial program 96.6%

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

      1. *-commutative 96.6%

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

      2. associate-*l* 76.9%

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

      3. *-commutative 76.9%

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

      4. associate-*l* 61.4%

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

      5. distribute-lft-out 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in normAngle around 0 98.0%

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

   6. Taylor expanded in n0_i around 0 67.7%

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

      1. *-commutative 67.7%

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

   8. Simplified 67.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -5.000000229068525 \cdot 10^{-19} \lor \neg \left(n0\_i \leq 4.0000000126843074 \cdot 10^{-28}\right):\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.5% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -3.0000000340435383 \cdot 10^{-18}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 2.1999999301841496 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -3.0000000340435383e-18)
   n0_i
   (if (<= n0_i 2.1999999301841496e-20) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -3.0000000340435383e-18f) {
		tmp = n0_i;
	} else if (n0_i <= 2.1999999301841496e-20f) {
		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 <= (-3.0000000340435383e-18)) then
        tmp = n0_i
    else if (n0_i <= 2.1999999301841496e-20) 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(-3.0000000340435383e-18))
		tmp = n0_i;
	elseif (n0_i <= Float32(2.1999999301841496e-20))
		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(-3.0000000340435383e-18))
		tmp = n0_i;
	elseif (n0_i <= single(2.1999999301841496e-20))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -3.00000003e-18 or 2.19999993e-20 < n0_i

   1. Initial program 98.7%

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

   3. Taylor expanded in u around 0 77.6%

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

   4. Taylor expanded in u around 0 77.2%

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

   5. Taylor expanded in n0_i around inf 62.6%

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

   if -3.00000003e-18 < n0_i < 2.19999993e-20

   1. Initial program 96.5%

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

      1. *-commutative 96.5%

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

      2. associate-*l* 75.0%

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

      3. *-commutative 75.0%

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

      4. associate-*l* 62.0%

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

      5. distribute-lft-out 62.0%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in normAngle around 0 98.1%

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

   6. Taylor expanded in n0_i around 0 61.7%

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

      1. *-commutative 61.7%

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

   8. Simplified 61.7%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -3.0000000340435383 \cdot 10^{-18}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 2.1999999301841496 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*l* 82.7%

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

   3. *-commutative 82.7%

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

   4. associate-*l* 76.3%

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

   5. distribute-lft-out 76.3%

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

3. Simplified 76.3%

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

5. Taylor expanded in normAngle around 0 97.9%

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

6. Taylor expanded in u around -inf 98.1%

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

   1. mul-1-neg 98.1%

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

   2. unsub-neg 98.1%

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

   3. mul-1-neg 98.1%

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

   4. unsub-neg 98.1%

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

8. Simplified 98.1%

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

9. Final simplification 98.1%

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

Alternative 8: 81.2% 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 97.6%

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

3. Taylor expanded in u around 0 80.6%

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

4. Taylor expanded in normAngle around 0 81.1%

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

   1. *-commutative 81.1%

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

6. Simplified 81.1%

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

7. Final simplification 81.1%

   \[\leadsto n0\_i + u \cdot n1\_i \]
8. Add Preprocessing

Alternative 9: 47.3% 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 97.6%

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

3. Taylor expanded in u around 0 80.6%

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

4. Taylor expanded in u around 0 71.7%

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

5. Taylor expanded in n0_i around inf 45.4%

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

6. Final simplification 45.4%

   \[\leadsto n0\_i \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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)))