Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.5%

Time: 14.8s

Alternatives: 11

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% 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 97.1%

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

   1. *-commutative 97.1%

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

   2. associate-*l* 85.4%

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

   3. *-commutative 85.4%

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

   4. associate-*l* 73.9%

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

   5. distribute-lft-out 73.9%

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

   6. associate-*l/ 74.0%

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

   7. *-lft-identity 74.0%

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

   8. fma-def 74.0%

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

   9. *-commutative 74.0%

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

   10. distribute-rgt-out-- 74.0%

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

   11. *-lft-identity 74.0%

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

3. Simplified 74.0%

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

4. Taylor expanded in u around 0 87.4%

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

   1. +-commutative 87.4%

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

   2. fma-def 87.4%

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

   3. +-commutative 87.4%

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

   4. mul-1-neg 87.4%

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

   5. unsub-neg 87.4%

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

   6. associate-/l* 94.5%

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

   7. associate-/l* 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.1%

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

   1. *-commutative 97.1%

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

   2. associate-*l* 85.4%

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

   3. *-commutative 85.4%

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

   4. associate-*l* 73.9%

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

   5. distribute-lft-out 73.9%

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

   6. associate-*l/ 74.0%

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

   7. *-lft-identity 74.0%

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

   8. fma-def 74.0%

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

   9. *-commutative 74.0%

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

   10. distribute-rgt-out-- 74.0%

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

   11. *-lft-identity 74.0%

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

3. Simplified 74.0%

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

4. Taylor expanded in u around 0 87.4%

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

   1. +-commutative 87.4%

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

   2. fma-def 87.4%

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

   3. +-commutative 87.4%

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

   4. mul-1-neg 87.4%

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

   5. unsub-neg 87.4%

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

   6. associate-/l* 94.5%

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

   7. associate-/l* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in normAngle around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-*r* 99.4%

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

   6. unpow2 99.4%

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

   7. associate--r+ 99.4%

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

   8. distribute-rgt-out-- 99.4%

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

   9. metadata-eval 99.4%

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

   10. fma-def 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 3: 98.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.1%

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

   1. *-commutative 97.1%

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

   2. associate-*l* 85.4%

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

   3. *-commutative 85.4%

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

   4. associate-*l* 73.9%

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

   5. distribute-lft-out 73.9%

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

   6. associate-*l/ 74.0%

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

   7. *-lft-identity 74.0%

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

   8. fma-def 74.0%

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

   9. *-commutative 74.0%

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

   10. distribute-rgt-out-- 74.0%

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

   11. *-lft-identity 74.0%

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

3. Simplified 74.0%

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

4. Taylor expanded in u around 0 87.4%

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

   1. +-commutative 87.4%

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

   2. fma-def 87.4%

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

   3. +-commutative 87.4%

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

   4. mul-1-neg 87.4%

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

   5. unsub-neg 87.4%

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

   6. associate-/l* 94.5%

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

   7. associate-/l* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in normAngle around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-*r* 99.4%

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

   6. unpow2 99.4%

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

   7. associate--r+ 99.4%

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

   8. distribute-rgt-out-- 99.4%

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

   9. metadata-eval 99.4%

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

   10. fma-def 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in n0_i around 0 99.2%

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

    1. associate-*r* 99.2%

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

    2. *-commutative 99.2%

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

    3. unpow2 99.2%

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

    4. *-commutative 99.2%

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

12. Simplified 99.2%

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

13. Final simplification 99.2%

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

Alternative 4: 98.8% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.1%

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

   1. *-commutative 97.1%

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

   2. associate-*l* 85.4%

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

   3. *-commutative 85.4%

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

   4. associate-*l* 73.9%

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

   5. distribute-lft-out 73.9%

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

   6. associate-*l/ 74.0%

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

   7. *-lft-identity 74.0%

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

   8. fma-def 74.0%

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

   9. *-commutative 74.0%

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

   10. distribute-rgt-out-- 74.0%

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

   11. *-lft-identity 74.0%

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

3. Simplified 74.0%

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

4. Taylor expanded in u around 0 87.4%

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

   1. +-commutative 87.4%

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

   2. fma-def 87.4%

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

   3. +-commutative 87.4%

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

   4. mul-1-neg 87.4%

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

   5. unsub-neg 87.4%

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

   6. associate-/l* 94.5%

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

   7. associate-/l* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in normAngle around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-*r* 99.4%

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

   6. unpow2 99.4%

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

   7. associate--r+ 99.4%

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

   8. distribute-rgt-out-- 99.4%

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

   9. metadata-eval 99.4%

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

   10. fma-def 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in n0_i around 0 99.2%

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

    1. associate-*r* 99.2%

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

    2. *-commutative 99.2%

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

    3. unpow2 99.2%

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

    4. *-commutative 99.2%

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

12. Simplified 99.2%

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

    1. fma-udef 99.1%

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

14. Applied egg-rr 99.1%

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

15. Final simplification 99.1%

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

Alternative 5: 98.3% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.1%

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

   1. *-commutative 97.1%

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

   2. associate-*l* 85.4%

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

   3. *-commutative 85.4%

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

   4. associate-*l* 73.9%

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

   5. distribute-lft-out 73.9%

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

   6. associate-*l/ 74.0%

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

   7. *-lft-identity 74.0%

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

   8. fma-def 74.0%

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

   9. *-commutative 74.0%

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

   10. distribute-rgt-out-- 74.0%

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

   11. *-lft-identity 74.0%

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

3. Simplified 74.0%

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

4. Taylor expanded in u around 0 87.4%

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

   1. +-commutative 87.4%

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

   2. fma-def 87.4%

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

   3. +-commutative 87.4%

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

   4. mul-1-neg 87.4%

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

   5. unsub-neg 87.4%

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

   6. associate-/l* 94.5%

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

   7. associate-/l* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in normAngle around 0 99.4%

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

8. Taylor expanded in n0_i around inf 98.8%

   \[\leadsto n0_i + \left(u \cdot \left(n1_i - n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(n0_i \cdot u\right)\right)}\right) \]

9. Taylor expanded in u around -inf 98.8%

   \[\leadsto \color{blue}{n0_i + -1 \cdot \left(u \cdot \left(-1 \cdot \left(n1_i - n0_i\right) + -0.3333333333333333 \cdot \left(n0_i \cdot {normAngle}^{2}\right)\right)\right)} \]
10. Step-by-step derivation

    1. mul-1-neg 98.8%

       \[\leadsto n0_i + \color{blue}{\left(-u \cdot \left(-1 \cdot \left(n1_i - n0_i\right) + -0.3333333333333333 \cdot \left(n0_i \cdot {normAngle}^{2}\right)\right)\right)} \]

    2. unsub-neg 98.8%

       \[\leadsto \color{blue}{n0_i - u \cdot \left(-1 \cdot \left(n1_i - n0_i\right) + -0.3333333333333333 \cdot \left(n0_i \cdot {normAngle}^{2}\right)\right)} \]

    3. +-commutative 98.8%

       \[\leadsto n0_i - u \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(n0_i \cdot {normAngle}^{2}\right) + -1 \cdot \left(n1_i - n0_i\right)\right)} \]

    4. mul-1-neg 98.8%

       \[\leadsto n0_i - u \cdot \left(-0.3333333333333333 \cdot \left(n0_i \cdot {normAngle}^{2}\right) + \color{blue}{\left(-\left(n1_i - n0_i\right)\right)}\right) \]

    5. unsub-neg 98.8%

       \[\leadsto n0_i - u \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(n0_i \cdot {normAngle}^{2}\right) - \left(n1_i - n0_i\right)\right)} \]

    6. *-commutative 98.8%

       \[\leadsto n0_i - u \cdot \left(\color{blue}{\left(n0_i \cdot {normAngle}^{2}\right) \cdot -0.3333333333333333} - \left(n1_i - n0_i\right)\right) \]

    7. unpow2 98.8%

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

    8. associate-*l* 98.8%

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

    9. *-commutative 98.8%

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

11. Simplified 98.8%

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

12. Final simplification 98.8%

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

Alternative 6: 70.1% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -1.999999936531045 \cdot 10^{-21} \lor \neg \left(n0_i \leq 1.999999936531045 \cdot 10^{-21}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -1.999999936531045e-21)
         (not (<= n0_i 1.999999936531045e-21)))
   (* n0_i (- 1.0 u))
   (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -1.999999936531045e-21f) || !(n0_i <= 1.999999936531045e-21f)) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = u * n1_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n0_i <= (-1.999999936531045e-21)) .or. (.not. (n0_i <= 1.999999936531045e-21))) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-1.999999936531045e-21)) || !(n0_i <= Float32(1.999999936531045e-21)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-1.999999936531045e-21)) || ~((n0_i <= single(1.999999936531045e-21))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.9999999e-21 or 1.9999999e-21 < n0_i

   1. Initial program 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-*l* 96.6%

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

      3. *-commutative 96.6%

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

      4. associate-*l* 85.6%

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

      5. distribute-lft-out 85.6%

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

      6. associate-*l/ 85.8%

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

      7. *-lft-identity 85.8%

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

      8. fma-def 85.7%

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

      9. *-commutative 85.7%

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

      10. distribute-rgt-out-- 85.8%

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

      11. *-lft-identity 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in normAngle around 0 99.0%

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

   5. Taylor expanded in n0_i around inf 84.6%

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

   if -1.9999999e-21 < n0_i < 1.9999999e-21

   1. Initial program 96.2%

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

      1. *-commutative 96.2%

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

      2. associate-*l* 70.8%

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

      3. *-commutative 70.8%

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

      4. associate-*l* 58.5%

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

      5. distribute-lft-out 58.5%

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

      6. associate-*l/ 58.7%

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

      7. *-lft-identity 58.7%

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

      8. fma-def 58.7%

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

      9. *-commutative 58.7%

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

      10. distribute-rgt-out-- 58.7%

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

      11. *-lft-identity 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in normAngle around 0 97.2%

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

   5. Taylor expanded in n0_i around 0 60.4%

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

      1. *-commutative 60.4%

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

   7. Simplified 60.4%

      \[\leadsto \color{blue}{u \cdot n1_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -1.999999936531045 \cdot 10^{-21} \lor \neg \left(n0_i \leq 1.999999936531045 \cdot 10^{-21}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 7: 85.1% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -1.99999996490334 \cdot 10^{-13} \lor \neg \left(n0_i \leq 3.99999987306209 \cdot 10^{-20}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -1.99999996490334e-13) (not (<= n0_i 3.99999987306209e-20)))
   (* n0_i (- 1.0 u))
   (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -1.99999996490334e-13f) || !(n0_i <= 3.99999987306209e-20f)) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = n0_i + (u * n1_i);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n0_i <= (-1.99999996490334e-13)) .or. (.not. (n0_i <= 3.99999987306209e-20))) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = n0_i + (u * n1_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-1.99999996490334e-13)) || !(n0_i <= Float32(3.99999987306209e-20)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(n0_i + Float32(u * n1_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-1.99999996490334e-13)) || ~((n0_i <= single(3.99999987306209e-20))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.99999996e-13 or 3.99999987e-20 < n0_i

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*l* 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*l* 89.3%

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

      5. distribute-lft-out 89.3%

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

      6. associate-*l/ 89.5%

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

      7. *-lft-identity 89.5%

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

      8. fma-def 89.4%

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

      9. *-commutative 89.4%

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

      10. distribute-rgt-out-- 89.4%

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

      11. *-lft-identity 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in normAngle around 0 99.4%

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

   5. Taylor expanded in n0_i around inf 89.6%

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

   if -1.99999996e-13 < n0_i < 3.99999987e-20

   1. Initial program 95.9%

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

   2. Taylor expanded in normAngle around 0 97.6%

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

   3. Taylor expanded in u around 0 85.5%

      \[\leadsto \color{blue}{n0_i} + u \cdot n1_i \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -1.99999996490334 \cdot 10^{-13} \lor \neg \left(n0_i \leq 3.99999987306209 \cdot 10^{-20}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \]

Alternative 8: 85.2% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -1.99999996490334 \cdot 10^{-13} \lor \neg \left(n0_i \leq 3.99999987306209 \cdot 10^{-20}\right):\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -1.99999996490334e-13) (not (<= n0_i 3.99999987306209e-20)))
   (- n0_i (* u n0_i))
   (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -1.99999996490334e-13f) || !(n0_i <= 3.99999987306209e-20f)) {
		tmp = n0_i - (u * n0_i);
	} else {
		tmp = n0_i + (u * n1_i);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n0_i <= (-1.99999996490334e-13)) .or. (.not. (n0_i <= 3.99999987306209e-20))) then
        tmp = n0_i - (u * n0_i)
    else
        tmp = n0_i + (u * n1_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-1.99999996490334e-13)) || !(n0_i <= Float32(3.99999987306209e-20)))
		tmp = Float32(n0_i - Float32(u * n0_i));
	else
		tmp = Float32(n0_i + Float32(u * n1_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-1.99999996490334e-13)) || ~((n0_i <= single(3.99999987306209e-20))))
		tmp = n0_i - (u * n0_i);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.99999996e-13 or 3.99999987e-20 < n0_i

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*l* 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*l* 89.3%

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

      5. distribute-lft-out 89.3%

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

      6. associate-*l/ 89.5%

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

      7. *-lft-identity 89.5%

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

      8. fma-def 89.4%

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

      9. *-commutative 89.4%

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

      10. distribute-rgt-out-- 89.4%

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

      11. *-lft-identity 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in normAngle around 0 99.4%

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

   5. Taylor expanded in n0_i around inf 89.6%

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

   6. Taylor expanded in u around 0 89.7%

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

      1. mul-1-neg 89.7%

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

      2. unsub-neg 89.7%

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

   8. Simplified 89.7%

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

   if -1.99999996e-13 < n0_i < 3.99999987e-20

   1. Initial program 95.9%

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

   2. Taylor expanded in normAngle around 0 97.6%

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

   3. Taylor expanded in u around 0 85.5%

      \[\leadsto \color{blue}{n0_i} + u \cdot n1_i \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -1.99999996490334 \cdot 10^{-13} \lor \neg \left(n0_i \leq 3.99999987306209 \cdot 10^{-20}\right):\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \]

Alternative 9: 60.9% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.999999936531045e-21)
   n0_i
   (if (<= n0_i 1.999999936531045e-21) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.999999936531045e-21f) {
		tmp = n0_i;
	} else if (n0_i <= 1.999999936531045e-21f) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-1.999999936531045e-21)) then
        tmp = n0_i
    else if (n0_i <= 1.999999936531045e-21) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-1.999999936531045e-21))
		tmp = n0_i;
	elseif (n0_i <= Float32(1.999999936531045e-21))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-1.999999936531045e-21))
		tmp = n0_i;
	elseif (n0_i <= single(1.999999936531045e-21))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.9999999e-21 or 1.9999999e-21 < n0_i

   1. Initial program 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-*l* 96.6%

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

      3. *-commutative 96.6%

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

      4. associate-*l* 85.6%

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

      5. distribute-lft-out 85.6%

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

      6. associate-*l/ 85.8%

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

      7. *-lft-identity 85.8%

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

      8. fma-def 85.7%

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

      9. *-commutative 85.7%

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

      10. distribute-rgt-out-- 85.8%

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

      11. *-lft-identity 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in normAngle around 0 99.0%

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

      1. flip-+ 97.6%

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

      2. *-commutative 97.6%

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

      3. *-commutative 97.6%

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

      4. *-commutative 97.6%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in u around 0 62.6%

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

   if -1.9999999e-21 < n0_i < 1.9999999e-21

   1. Initial program 96.2%

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

      1. *-commutative 96.2%

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

      2. associate-*l* 70.8%

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

      3. *-commutative 70.8%

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

      4. associate-*l* 58.5%

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

      5. distribute-lft-out 58.5%

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

      6. associate-*l/ 58.7%

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

      7. *-lft-identity 58.7%

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

      8. fma-def 58.7%

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

      9. *-commutative 58.7%

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

      10. distribute-rgt-out-- 58.7%

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

      11. *-lft-identity 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in normAngle around 0 97.2%

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

   5. Taylor expanded in n0_i around 0 60.4%

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

      1. *-commutative 60.4%

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

   7. Simplified 60.4%

      \[\leadsto \color{blue}{u \cdot n1_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 10: 98.2% 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 97.1%

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

   1. *-commutative 97.1%

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

   2. associate-*l* 85.4%

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

   3. *-commutative 85.4%

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

   4. associate-*l* 73.9%

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

   5. distribute-lft-out 73.9%

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

   6. associate-*l/ 74.0%

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

   7. *-lft-identity 74.0%

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

   8. fma-def 74.0%

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

   9. *-commutative 74.0%

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

   10. distribute-rgt-out-- 74.0%

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

   11. *-lft-identity 74.0%

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

3. Simplified 74.0%

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

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 98.5%

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

   1. mul-1-neg 98.5%

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

   2. unsub-neg 98.5%

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

   3. mul-1-neg 98.5%

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

   4. unsub-neg 98.5%

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

7. Simplified 98.5%

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

8. Final simplification 98.5%

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

Alternative 11: 47.0% accurate, 421.0× speedup

Math

\[\begin{array}{l} \\ n0_i \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.1%

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

   1. *-commutative 97.1%

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

   2. associate-*l* 85.4%

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

   3. *-commutative 85.4%

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

   4. associate-*l* 73.9%

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

   5. distribute-lft-out 73.9%

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

   6. associate-*l/ 74.0%

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

   7. *-lft-identity 74.0%

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

   8. fma-def 74.0%

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

   9. *-commutative 74.0%

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

   10. distribute-rgt-out-- 74.0%

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

   11. *-lft-identity 74.0%

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

3. Simplified 74.0%

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

4. Taylor expanded in normAngle around 0 98.2%

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

   1. flip-+ 78.8%

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

   2. *-commutative 78.8%

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

   3. *-commutative 78.8%

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

   4. *-commutative 78.8%

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

6. Applied egg-rr 78.8%

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

7. Taylor expanded in u around 0 47.3%

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

8. Final simplification 47.3%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023278 
(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)))