Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.3%

Time: 17.8s

Alternatives: 6

Speedup: 22.2×

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: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, normAngle \cdot \left(\frac{n1\_i}{\sin normAngle} - \frac{n0\_i}{\tan normAngle}\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (* normAngle (- (/ n1_i (sin normAngle)) (/ n0_i (tan normAngle))))
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, (normAngle * ((n1_i / sinf(normAngle)) - (n0_i / tanf(normAngle)))), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, Float32(normAngle * Float32(Float32(n1_i / sin(normAngle)) - Float32(n0_i / tan(normAngle)))), n0_i)
end

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in u around 0 86.0%

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

   1. +-commutative 86.0%

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

   2. mul-1-neg 86.0%

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

   3. unsub-neg 86.0%

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

   4. associate-/l* 94.4%

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

   5. associate-/l* 99.4%

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

   6. associate-/l* 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in normAngle around inf 99.1%

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

   1. associate-*r* 97.6%

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

   2. *-commutative 97.6%

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

   3. associate-/l* 97.6%

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

8. Simplified 97.6%

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

   1. *-un-lft-identity 97.6%

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

   2. +-commutative 97.6%

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

   3. associate-*l* 99.2%

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

   4. fma-define 99.5%

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

   5. clear-num 99.5%

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

   6. tan-quot 99.5%

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

10. Applied egg-rr 99.5%

    \[\leadsto \color{blue}{1 \cdot \mathsf{fma}\left(u, normAngle \cdot \left(\frac{n1\_i}{\sin normAngle} - n0\_i \cdot \frac{1}{\tan normAngle}\right), n0\_i\right)} \]
11. Step-by-step derivation

    1. *-lft-identity 99.5%

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

    2. associate-*r/ 99.5%

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

    3. *-rgt-identity 99.5%

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

12. Simplified 99.5%

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

Alternative 2: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in u around 0 86.0%

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

   1. +-commutative 86.0%

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

   2. mul-1-neg 86.0%

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

   3. unsub-neg 86.0%

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

   4. associate-/l* 94.4%

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

   5. associate-/l* 99.4%

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

   6. associate-/l* 99.4%

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

5. Simplified 99.4%

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

   1. pow1 99.4%

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

   2. clear-num 99.4%

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

   3. quot-tan 99.4%

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

7. Applied egg-rr 99.4%

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

   1. unpow1 99.4%

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

   2. associate-*r/ 99.4%

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

   3. *-rgt-identity 99.4%

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

9. Simplified 99.4%

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

    1. clear-num 99.4%

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

    2. inv-pow 99.4%

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

11. Applied egg-rr 99.4%

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

    1. unpow-1 99.4%

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

13. Simplified 99.4%

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

14. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in u around 0 86.0%

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

   1. +-commutative 86.0%

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

   2. mul-1-neg 86.0%

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

   3. unsub-neg 86.0%

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

   4. associate-/l* 94.4%

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

   5. associate-/l* 99.4%

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

   6. associate-/l* 99.4%

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

5. Simplified 99.4%

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

   1. pow1 99.4%

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

   2. clear-num 99.4%

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

   3. quot-tan 99.4%

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

7. Applied egg-rr 99.4%

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

   1. unpow1 99.4%

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

   2. associate-*r/ 99.4%

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

   3. *-rgt-identity 99.4%

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

9. Simplified 99.4%

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

Alternative 4: 99.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in u around 0 86.0%

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

   1. +-commutative 86.0%

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

   2. mul-1-neg 86.0%

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

   3. unsub-neg 86.0%

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

   4. associate-/l* 94.4%

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

   5. associate-/l* 99.4%

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

   6. associate-/l* 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in normAngle around inf 99.1%

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

   1. associate-*r* 97.6%

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

   2. *-commutative 97.6%

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

   3. associate-/l* 97.6%

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

8. Simplified 97.6%

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

   1. *-un-lft-identity 97.6%

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

   2. +-commutative 97.6%

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

   3. associate-*l* 99.2%

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

   4. fma-define 99.5%

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

   5. clear-num 99.5%

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

   6. tan-quot 99.5%

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

10. Applied egg-rr 99.5%

    \[\leadsto \color{blue}{1 \cdot \mathsf{fma}\left(u, normAngle \cdot \left(\frac{n1\_i}{\sin normAngle} - n0\_i \cdot \frac{1}{\tan normAngle}\right), n0\_i\right)} \]
11. Step-by-step derivation

    1. *-lft-identity 99.5%

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

    2. associate-*r/ 99.5%

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

    3. *-rgt-identity 99.5%

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

12. Simplified 99.5%

    \[\leadsto \color{blue}{\mathsf{fma}\left(u, normAngle \cdot \left(\frac{n1\_i}{\sin normAngle} - \frac{n0\_i}{\tan normAngle}\right), n0\_i\right)} \]
13. Step-by-step derivation

    1. fma-undefine 99.3%

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

14. Applied egg-rr 99.3%

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

15. Final simplification 99.3%

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

Alternative 5: 98.8% accurate, 16.8× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (((u * n1_i) + ((normAngle * normAngle) * (((u * n0_i) * 0.3333333333333333f) - ((u * n1_i) * -0.16666666666666666f)))) - (u * 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) + ((normangle * normangle) * (((u * n0_i) * 0.3333333333333333e0) - ((u * n1_i) * (-0.16666666666666666e0))))) - (u * n0_i))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(Float32(Float32(u * n1_i) + Float32(Float32(normAngle * normAngle) * Float32(Float32(Float32(u * n0_i) * Float32(0.3333333333333333)) - Float32(Float32(u * n1_i) * Float32(-0.16666666666666666))))) - Float32(u * n0_i)))
end

MATLAB

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in u around 0 86.0%

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

   1. +-commutative 86.0%

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

   2. mul-1-neg 86.0%

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

   3. unsub-neg 86.0%

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

   4. associate-/l* 94.4%

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

   5. associate-/l* 99.4%

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

   6. associate-/l* 99.4%

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

5. Simplified 99.4%

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

   1. sub-neg 99.4%

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

   2. associate-*r/ 88.7%

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

   3. associate-*r/ 88.8%

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

   4. associate-/l* 86.0%

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

   5. mul-1-neg 86.0%

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

   6. +-commutative 86.0%

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

   7. distribute-rgt-in 86.0%

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

7. Applied egg-rr 94.4%

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

8. Taylor expanded in normAngle around 0 99.0%

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

   1. unpow2 99.0%

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

10. Applied egg-rr 99.0%

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

11. Final simplification 99.0%

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

Alternative 6: 98.8% accurate, 22.2× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    (- n1_i n0_i)
    (*
     (* normAngle normAngle)
     (+ (* n0_i 0.3333333333333333) (* 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) + ((normAngle * normAngle) * ((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 * normangle) * ((n0_i * 0.3333333333333333e0) + (n1_i * 0.16666666666666666e0)))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(Float32(n1_i - n0_i) + Float32(Float32(normAngle * normAngle) * 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 * normAngle) * ((n0_i * single(0.3333333333333333)) + (n1_i * single(0.16666666666666666))))));
end

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in u around 0 86.0%

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

4. Taylor expanded in normAngle around 0 99.0%

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

   1. associate-+r+ 99.0%

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

   2. neg-mul-1 99.0%

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

   3. sub-neg 99.0%

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

   4. cancel-sign-sub-inv 99.0%

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

   5. mul-1-neg 99.0%

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

   6. distribute-rgt-out-- 99.0%

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

   7. metadata-eval 99.0%

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

   8. metadata-eval 99.0%

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

   9. *-commutative 99.0%

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

6. Simplified 99.0%

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

   1. unpow2 99.0%

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

8. Applied egg-rr 99.0%

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

9. Taylor expanded in n0_i around 0 99.0%

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

    1. *-commutative 99.0%

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

11. Simplified 99.0%

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

Reproduce

herbie shell --seed 2024179 
(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)))