Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 99.4%

Time: 22.2s

Alternatives: 8

Speedup: 60.1×

Specification

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t_0\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot t_0\right) \cdot n1_i \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t_0\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot t_0\right) \cdot n1_i \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \frac{n0_i}{\frac{\sin normAngle}{normAngle \cdot \cos normAngle}}, n0_i\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. fma-def 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

   2. associate-*r/ 97.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   3. *-rgt-identity 97.7%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   4. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

   5. *-rgt-identity 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

4. Taylor expanded in u around 0 88.3%

   \[\leadsto \color{blue}{n0_i + u \cdot \left(-1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}\right)} \]
5. Step-by-step derivation

   1. +-commutative 88.3%

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}\right) + n0_i} \]

   2. fma-def 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, -1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}, n0_i\right)} \]

   3. +-commutative 88.4%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\frac{n1_i \cdot normAngle}{\sin normAngle} + -1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}, n0_i\right) \]

   4. mul-1-neg 88.4%

      \[\leadsto \mathsf{fma}\left(u, \frac{n1_i \cdot normAngle}{\sin normAngle} + \color{blue}{\left(-\frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)}, n0_i\right) \]

   5. unsub-neg 88.4%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\frac{n1_i \cdot normAngle}{\sin normAngle} - \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}, n0_i\right) \]

   6. associate-/l* 93.5%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\frac{n1_i}{\frac{\sin normAngle}{normAngle}}} - \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}, n0_i\right) \]

   7. associate-/l* 99.6%

      \[\leadsto \mathsf{fma}\left(u, \frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \color{blue}{\frac{n0_i}{\frac{\sin normAngle}{normAngle \cdot \cos normAngle}}}, n0_i\right) \]

6. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \frac{n0_i}{\frac{\sin normAngle}{normAngle \cdot \cos normAngle}}, n0_i\right)} \]

7. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(u, \frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \frac{n0_i}{\frac{\sin normAngle}{normAngle \cdot \cos normAngle}}, n0_i\right) \]

Alternative 2: 99.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(u \cdot \left(normAngle \cdot normAngle\right)\right) \cdot \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right) + \mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. fma-def 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

   2. associate-*r/ 97.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   3. *-rgt-identity 97.7%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   4. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

   5. *-rgt-identity 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

4. Taylor expanded in u around 0 88.3%

   \[\leadsto \color{blue}{n0_i + u \cdot \left(-1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}\right)} \]
5. Step-by-step derivation

   1. +-commutative 88.3%

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}\right) + n0_i} \]

   2. fma-def 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, -1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}, n0_i\right)} \]

   3. +-commutative 88.4%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\frac{n1_i \cdot normAngle}{\sin normAngle} + -1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}, n0_i\right) \]

   4. mul-1-neg 88.4%

      \[\leadsto \mathsf{fma}\left(u, \frac{n1_i \cdot normAngle}{\sin normAngle} + \color{blue}{\left(-\frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)}, n0_i\right) \]

   5. unsub-neg 88.4%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\frac{n1_i \cdot normAngle}{\sin normAngle} - \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}, n0_i\right) \]

   6. associate-/l* 93.5%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\frac{n1_i}{\frac{\sin normAngle}{normAngle}}} - \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}, n0_i\right) \]

   7. associate-/l* 99.6%

      \[\leadsto \mathsf{fma}\left(u, \frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \color{blue}{\frac{n0_i}{\frac{\sin normAngle}{normAngle \cdot \cos normAngle}}}, n0_i\right) \]

6. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \frac{n0_i}{\frac{\sin normAngle}{normAngle \cdot \cos normAngle}}, n0_i\right)} \]

7. Taylor expanded in normAngle around 0 99.1%

   \[\leadsto \color{blue}{n0_i + \left(u \cdot \left(n1_i - n0_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0_i - \left(-0.5 \cdot n0_i + -0.16666666666666666 \cdot n1_i\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative 99.1%

      \[\leadsto \color{blue}{\left(u \cdot \left(n1_i - n0_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0_i - \left(-0.5 \cdot n0_i + -0.16666666666666666 \cdot n1_i\right)\right)\right)\right) + n0_i} \]

   2. +-commutative 99.1%

      \[\leadsto \color{blue}{\left({normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0_i - \left(-0.5 \cdot n0_i + -0.16666666666666666 \cdot n1_i\right)\right)\right) + u \cdot \left(n1_i - n0_i\right)\right)} + n0_i \]

   3. associate-+l+ 99.1%

      \[\leadsto \color{blue}{{normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0_i - \left(-0.5 \cdot n0_i + -0.16666666666666666 \cdot n1_i\right)\right)\right) + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right)} \]

   4. associate-*r* 99.1%

      \[\leadsto \color{blue}{\left({normAngle}^{2} \cdot u\right) \cdot \left(-0.16666666666666666 \cdot n0_i - \left(-0.5 \cdot n0_i + -0.16666666666666666 \cdot n1_i\right)\right)} + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   5. unpow2 99.1%

      \[\leadsto \left(\color{blue}{\left(normAngle \cdot normAngle\right)} \cdot u\right) \cdot \left(-0.16666666666666666 \cdot n0_i - \left(-0.5 \cdot n0_i + -0.16666666666666666 \cdot n1_i\right)\right) + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   6. associate--r+ 99.1%

      \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot n0_i - -0.5 \cdot n0_i\right) - -0.16666666666666666 \cdot n1_i\right)} + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   7. cancel-sign-sub-inv 99.1%

      \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot n0_i - -0.5 \cdot n0_i\right) + \left(--0.16666666666666666\right) \cdot n1_i\right)} + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   8. distribute-rgt-out-- 99.1%

      \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(\color{blue}{n0_i \cdot \left(-0.16666666666666666 - -0.5\right)} + \left(--0.16666666666666666\right) \cdot n1_i\right) + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   9. metadata-eval 99.1%

      \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(n0_i \cdot \color{blue}{0.3333333333333333} + \left(--0.16666666666666666\right) \cdot n1_i\right) + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   10. metadata-eval 99.1%

       \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(n0_i \cdot 0.3333333333333333 + \color{blue}{0.16666666666666666} \cdot n1_i\right) + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   11. fma-def 99.3%

       \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right) + \color{blue}{\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right)} \]

9. Simplified 99.3%

   \[\leadsto \color{blue}{\left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right) + \mathsf{fma}\left(u, n1_i - n0_i, n0_i\right)} \]

10. Final simplification 99.3%

    \[\leadsto \left(u \cdot \left(normAngle \cdot normAngle\right)\right) \cdot \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right) + \mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \]

Alternative 3: 98.8% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ \left(u \cdot \left(normAngle \cdot normAngle\right)\right) \cdot \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right) + \left(n0_i - u \cdot \left(n0_i - n1_i\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. fma-def 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

   2. associate-*r/ 97.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   3. *-rgt-identity 97.7%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   4. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

   5. *-rgt-identity 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

4. Taylor expanded in u around 0 88.3%

   \[\leadsto \color{blue}{n0_i + u \cdot \left(-1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}\right)} \]
5. Step-by-step derivation

   1. +-commutative 88.3%

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}\right) + n0_i} \]

   2. fma-def 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, -1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}, n0_i\right)} \]

   3. +-commutative 88.4%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\frac{n1_i \cdot normAngle}{\sin normAngle} + -1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}, n0_i\right) \]

   4. mul-1-neg 88.4%

      \[\leadsto \mathsf{fma}\left(u, \frac{n1_i \cdot normAngle}{\sin normAngle} + \color{blue}{\left(-\frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)}, n0_i\right) \]

   5. unsub-neg 88.4%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\frac{n1_i \cdot normAngle}{\sin normAngle} - \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}, n0_i\right) \]

   6. associate-/l* 93.5%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\frac{n1_i}{\frac{\sin normAngle}{normAngle}}} - \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}, n0_i\right) \]

   7. associate-/l* 99.6%

      \[\leadsto \mathsf{fma}\left(u, \frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \color{blue}{\frac{n0_i}{\frac{\sin normAngle}{normAngle \cdot \cos normAngle}}}, n0_i\right) \]

6. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \frac{n0_i}{\frac{\sin normAngle}{normAngle \cdot \cos normAngle}}, n0_i\right)} \]

7. Taylor expanded in normAngle around 0 99.1%

   \[\leadsto \color{blue}{n0_i + \left(u \cdot \left(n1_i - n0_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0_i - \left(-0.5 \cdot n0_i + -0.16666666666666666 \cdot n1_i\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative 99.1%

      \[\leadsto \color{blue}{\left(u \cdot \left(n1_i - n0_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0_i - \left(-0.5 \cdot n0_i + -0.16666666666666666 \cdot n1_i\right)\right)\right)\right) + n0_i} \]

   2. +-commutative 99.1%

      \[\leadsto \color{blue}{\left({normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0_i - \left(-0.5 \cdot n0_i + -0.16666666666666666 \cdot n1_i\right)\right)\right) + u \cdot \left(n1_i - n0_i\right)\right)} + n0_i \]

   3. associate-+l+ 99.1%

      \[\leadsto \color{blue}{{normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0_i - \left(-0.5 \cdot n0_i + -0.16666666666666666 \cdot n1_i\right)\right)\right) + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right)} \]

   4. associate-*r* 99.1%

      \[\leadsto \color{blue}{\left({normAngle}^{2} \cdot u\right) \cdot \left(-0.16666666666666666 \cdot n0_i - \left(-0.5 \cdot n0_i + -0.16666666666666666 \cdot n1_i\right)\right)} + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   5. unpow2 99.1%

      \[\leadsto \left(\color{blue}{\left(normAngle \cdot normAngle\right)} \cdot u\right) \cdot \left(-0.16666666666666666 \cdot n0_i - \left(-0.5 \cdot n0_i + -0.16666666666666666 \cdot n1_i\right)\right) + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   6. associate--r+ 99.1%

      \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot n0_i - -0.5 \cdot n0_i\right) - -0.16666666666666666 \cdot n1_i\right)} + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   7. cancel-sign-sub-inv 99.1%

      \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot n0_i - -0.5 \cdot n0_i\right) + \left(--0.16666666666666666\right) \cdot n1_i\right)} + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   8. distribute-rgt-out-- 99.1%

      \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(\color{blue}{n0_i \cdot \left(-0.16666666666666666 - -0.5\right)} + \left(--0.16666666666666666\right) \cdot n1_i\right) + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   9. metadata-eval 99.1%

      \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(n0_i \cdot \color{blue}{0.3333333333333333} + \left(--0.16666666666666666\right) \cdot n1_i\right) + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   10. metadata-eval 99.1%

       \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(n0_i \cdot 0.3333333333333333 + \color{blue}{0.16666666666666666} \cdot n1_i\right) + \left(u \cdot \left(n1_i - n0_i\right) + n0_i\right) \]

   11. fma-def 99.3%

       \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right) + \color{blue}{\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right)} \]

9. Simplified 99.3%

   \[\leadsto \color{blue}{\left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right) + \mathsf{fma}\left(u, n1_i - n0_i, n0_i\right)} \]
10. Step-by-step derivation

    1. fma-udef 99.1%

       \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right) + \color{blue}{\left(u \cdot \left(n1_i - n0_i\right) + n0_i\right)} \]

11. Applied egg-rr 99.1%

    \[\leadsto \left(\left(normAngle \cdot normAngle\right) \cdot u\right) \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right) + \color{blue}{\left(u \cdot \left(n1_i - n0_i\right) + n0_i\right)} \]

12. Final simplification 99.1%

    \[\leadsto \left(u \cdot \left(normAngle \cdot normAngle\right)\right) \cdot \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right) + \left(n0_i - u \cdot \left(n0_i - n1_i\right)\right) \]

Alternative 4: 69.4% accurate, 45.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -2.4999999292951713 \cdot 10^{-10}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -2.4999999292951713e-10)
   (* u n1_i)
   (if (<= n1_i 4.999999969612645e-9) (* n0_i (- 1.0 u)) (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -2.4999999292951713e-10f) {
		tmp = u * n1_i;
	} else if (n1_i <= 4.999999969612645e-9f) {
		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 (n1_i <= (-2.4999999292951713e-10)) then
        tmp = u * n1_i
    else if (n1_i <= 4.999999969612645e-9) 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 (n1_i <= Float32(-2.4999999292951713e-10))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(4.999999969612645e-9))
		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 (n1_i <= single(-2.4999999292951713e-10))
		tmp = u * n1_i;
	elseif (n1_i <= single(4.999999969612645e-9))
		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 n1_i < -2.49999993e-10 or 4.99999997e-9 < n1_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. fma-def 98.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

      2. associate-*r/ 98.1%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      3. *-rgt-identity 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      4. associate-*r/ 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

      5. *-rgt-identity 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 97.1%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

   5. Taylor expanded in n0_i around 0 71.2%

      \[\leadsto \color{blue}{n1_i \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 71.2%

         \[\leadsto \color{blue}{u \cdot n1_i} \]

   7. Simplified 71.2%

      \[\leadsto \color{blue}{u \cdot n1_i} \]

   if -2.49999993e-10 < n1_i < 4.99999997e-9

   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. Step-by-step derivation

      1. fma-def 97.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

      2. associate-*r/ 97.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      3. *-rgt-identity 97.5%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      4. associate-*r/ 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

      5. *-rgt-identity 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 98.5%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

   5. Taylor expanded in n0_i around inf 70.9%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -2.4999999292951713 \cdot 10^{-10}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 5: 69.6% accurate, 45.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -2.4999999292951713 \cdot 10^{-10}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -2.4999999292951713e-10)
   (* u n1_i)
   (if (<= n1_i 4.999999969612645e-9) (- n0_i (* u n0_i)) (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -2.4999999292951713e-10f) {
		tmp = u * n1_i;
	} else if (n1_i <= 4.999999969612645e-9f) {
		tmp = n0_i - (u * n0_i);
	} 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 (n1_i <= (-2.4999999292951713e-10)) then
        tmp = u * n1_i
    else if (n1_i <= 4.999999969612645e-9) then
        tmp = n0_i - (u * n0_i)
    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 (n1_i <= Float32(-2.4999999292951713e-10))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(4.999999969612645e-9))
		tmp = Float32(n0_i - Float32(u * n0_i));
	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 (n1_i <= single(-2.4999999292951713e-10))
		tmp = u * n1_i;
	elseif (n1_i <= single(4.999999969612645e-9))
		tmp = n0_i - (u * n0_i);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -2.49999993e-10 or 4.99999997e-9 < n1_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. fma-def 98.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

      2. associate-*r/ 98.1%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      3. *-rgt-identity 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      4. associate-*r/ 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

      5. *-rgt-identity 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 97.1%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

   5. Taylor expanded in n0_i around 0 71.2%

      \[\leadsto \color{blue}{n1_i \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 71.2%

         \[\leadsto \color{blue}{u \cdot n1_i} \]

   7. Simplified 71.2%

      \[\leadsto \color{blue}{u \cdot n1_i} \]

   if -2.49999993e-10 < n1_i < 4.99999997e-9

   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. Step-by-step derivation

      1. fma-def 97.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

      2. associate-*r/ 97.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      3. *-rgt-identity 97.5%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      4. associate-*r/ 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

      5. *-rgt-identity 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

   4. Taylor expanded in u around 0 85.5%

      \[\leadsto \color{blue}{n0_i + u \cdot \left(-1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 85.5%

         \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}\right) + n0_i} \]

      2. fma-def 85.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(u, -1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}, n0_i\right)} \]

      3. +-commutative 85.6%

         \[\leadsto \mathsf{fma}\left(u, \color{blue}{\frac{n1_i \cdot normAngle}{\sin normAngle} + -1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}, n0_i\right) \]

      4. mul-1-neg 85.6%

         \[\leadsto \mathsf{fma}\left(u, \frac{n1_i \cdot normAngle}{\sin normAngle} + \color{blue}{\left(-\frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}\right)}, n0_i\right) \]

      5. unsub-neg 85.6%

         \[\leadsto \mathsf{fma}\left(u, \color{blue}{\frac{n1_i \cdot normAngle}{\sin normAngle} - \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}}, n0_i\right) \]

      6. associate-/l* 91.6%

         \[\leadsto \mathsf{fma}\left(u, \color{blue}{\frac{n1_i}{\frac{\sin normAngle}{normAngle}}} - \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle}, n0_i\right) \]

      7. associate-/l* 99.6%

         \[\leadsto \mathsf{fma}\left(u, \frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \color{blue}{\frac{n0_i}{\frac{\sin normAngle}{normAngle \cdot \cos normAngle}}}, n0_i\right) \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, \frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \frac{n0_i}{\frac{\sin normAngle}{normAngle \cdot \cos normAngle}}, n0_i\right)} \]

   7. Taylor expanded in normAngle around 0 98.8%

      \[\leadsto \color{blue}{n0_i + u \cdot \left(n1_i - n0_i\right)} \]
   8. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \color{blue}{u \cdot \left(n1_i - n0_i\right) + n0_i} \]

      2. fma-def 99.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right)} \]

   9. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right)} \]

   10. Taylor expanded in n1_i around 0 71.1%

       \[\leadsto \color{blue}{n0_i + -1 \cdot \left(n0_i \cdot u\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 71.1%

          \[\leadsto n0_i + \color{blue}{\left(-n0_i \cdot u\right)} \]

       2. sub-neg 71.1%

          \[\leadsto \color{blue}{n0_i - n0_i \cdot u} \]

   12. Simplified 71.1%

       \[\leadsto \color{blue}{n0_i - n0_i \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -2.4999999292951713 \cdot 10^{-10}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 6: 60.3% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -7.99999974612418 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -7.99999974612418e-20)
   (* u n1_i)
   (if (<= n1_i 3.999999935100636e-17) n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -7.99999974612418e-20f) {
		tmp = u * n1_i;
	} else if (n1_i <= 3.999999935100636e-17f) {
		tmp = n0_i;
	} 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 (n1_i <= (-7.99999974612418e-20)) then
        tmp = u * n1_i
    else if (n1_i <= 3.999999935100636e-17) then
        tmp = n0_i
    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 (n1_i <= Float32(-7.99999974612418e-20))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(3.999999935100636e-17))
		tmp = n0_i;
	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 (n1_i <= single(-7.99999974612418e-20))
		tmp = u * n1_i;
	elseif (n1_i <= single(3.999999935100636e-17))
		tmp = n0_i;
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -7.99999975e-20 or 3.99999994e-17 < n1_i

   1. Initial program 96.9%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. fma-def 97.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

      2. associate-*r/ 97.1%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      3. *-rgt-identity 97.1%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      4. associate-*r/ 97.2%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

      5. *-rgt-identity 97.2%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

   3. Simplified 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 97.1%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

   5. Taylor expanded in n0_i around 0 58.2%

      \[\leadsto \color{blue}{n1_i \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 58.2%

         \[\leadsto \color{blue}{u \cdot n1_i} \]

   7. Simplified 58.2%

      \[\leadsto \color{blue}{u \cdot n1_i} \]

   if -7.99999975e-20 < n1_i < 3.99999994e-17

   1. Initial program 98.0%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. fma-def 98.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

      2. associate-*r/ 98.2%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      3. *-rgt-identity 98.2%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      4. associate-*r/ 98.9%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

      5. *-rgt-identity 98.9%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

   4. Taylor expanded in u around 0 63.7%

      \[\leadsto \color{blue}{n0_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -7.99999974612418 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 7: 98.0% 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 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. fma-def 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

   2. associate-*r/ 97.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   3. *-rgt-identity 97.7%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   4. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

   5. *-rgt-identity 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

4. Taylor expanded in normAngle around 0 98.2%

   \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

5. Taylor expanded in u around -inf 98.4%

   \[\leadsto \color{blue}{n0_i + -1 \cdot \left(u \cdot \left(n0_i + -1 \cdot n1_i\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 98.4%

      \[\leadsto n0_i + \color{blue}{\left(-u \cdot \left(n0_i + -1 \cdot n1_i\right)\right)} \]

   2. unsub-neg 98.4%

      \[\leadsto \color{blue}{n0_i - u \cdot \left(n0_i + -1 \cdot n1_i\right)} \]

   3. mul-1-neg 98.4%

      \[\leadsto n0_i - u \cdot \left(n0_i + \color{blue}{\left(-n1_i\right)}\right) \]

   4. unsub-neg 98.4%

      \[\leadsto n0_i - u \cdot \color{blue}{\left(n0_i - n1_i\right)} \]

7. Simplified 98.4%

   \[\leadsto \color{blue}{n0_i - u \cdot \left(n0_i - n1_i\right)} \]

8. Final simplification 98.4%

   \[\leadsto n0_i + u \cdot \left(n1_i - n0_i\right) \]

Alternative 8: 47.1% 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 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. fma-def 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

   2. associate-*r/ 97.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   3. *-rgt-identity 97.7%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   4. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

   5. *-rgt-identity 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

4. Taylor expanded in u around 0 47.4%

   \[\leadsto \color{blue}{n0_i} \]

5. Final simplification 47.4%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023279 
(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)))