Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.4%

Time: 12.0s

Alternatives: 8

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.2% 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: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.8%

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

   1. associate-*l* 96.7%

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

   2. cancel-sign-sub 96.7%

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

   3. *-commutative 96.7%

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

   4. associate-*r* 82.3%

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

   5. associate-*r/ 82.5%

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

   6. *-rgt-identity 82.5%

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

   7. sin-neg 82.5%

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

   8. distribute-lft-neg-out 82.5%

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

   9. associate-*l* 82.5%

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

   10. *-commutative 82.5%

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

   11. distribute-lft-neg-out 82.5%

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

   12. distribute-rgt-neg-out 82.5%

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

   13. associate-*r/ 82.6%

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

3. Simplified 74.2%

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

5. Taylor expanded in u around 0 87.6%

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

   1. +-commutative 87.6%

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

   2. fma-define 87.7%

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

   3. +-commutative 87.7%

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

   4. mul-1-neg 87.7%

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

   5. unsub-neg 87.7%

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

   6. *-commutative 87.7%

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

   7. associate-/l* 96.0%

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

   8. fma-neg 96.0%

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

   9. associate-/l* 99.7%

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

   10. distribute-rgt-neg-in 99.7%

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

   11. distribute-neg-frac2 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 98.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.8%

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

   1. associate-*l* 96.7%

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

   2. cancel-sign-sub 96.7%

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

   3. *-commutative 96.7%

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

   4. associate-*r* 82.3%

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

   5. associate-*r/ 82.5%

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

   6. *-rgt-identity 82.5%

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

   7. sin-neg 82.5%

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

   8. distribute-lft-neg-out 82.5%

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

   9. associate-*l* 82.5%

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

   10. *-commutative 82.5%

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

   11. distribute-lft-neg-out 82.5%

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

   12. distribute-rgt-neg-out 82.5%

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

   13. associate-*r/ 82.6%

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

3. Simplified 74.2%

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

5. Taylor expanded in u around 0 87.6%

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

   1. +-commutative 87.6%

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

   2. fma-define 87.7%

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

   3. +-commutative 87.7%

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

   4. mul-1-neg 87.7%

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

   5. unsub-neg 87.7%

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

   6. *-commutative 87.7%

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

   7. associate-/l* 96.0%

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

   8. fma-neg 96.0%

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

   9. associate-/l* 99.7%

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

   10. distribute-rgt-neg-in 99.7%

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

   11. distribute-neg-frac2 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in normAngle around 0 98.2%

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

   1. mul-1-neg 98.2%

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

   2. unsub-neg 98.2%

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

10. Simplified 98.2%

    \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i - n0\_i}, n0\_i\right) \]
11. Add Preprocessing

Alternative 3: 86.0% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.000000031374395 \cdot 10^{-22} \lor \neg \left(n1\_i \leq 6.000000240508405 \cdot 10^{-26}\right):\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.000000031374395e-22)
         (not (<= n1_i 6.000000240508405e-26)))
   (+ n0_i (* u n1_i))
   (- n0_i (* u n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.000000031374395e-22f) || !(n1_i <= 6.000000240508405e-26f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i - (u * n0_i);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-1.000000031374395e-22)) .or. (.not. (n1_i <= 6.000000240508405e-26))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i - (u * n0_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-1.000000031374395e-22)) || !(n1_i <= Float32(6.000000240508405e-26)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i - Float32(u * n0_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-1.000000031374395e-22)) || ~((n1_i <= single(6.000000240508405e-26))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i - (u * n0_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.00000003e-22 or 6.00000024e-26 < n1_i

   1. Initial program 96.1%

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

   3. Taylor expanded in u around 0 86.9%

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

   4. Taylor expanded in normAngle around 0 87.3%

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

   if -1.00000003e-22 < n1_i < 6.00000024e-26

   1. Initial program 97.9%

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

      1. associate-*l* 97.8%

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

      2. cancel-sign-sub 97.8%

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

      3. *-commutative 97.8%

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

      4. associate-*r* 73.2%

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

      5. associate-*r/ 73.4%

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

      6. *-rgt-identity 73.4%

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

      7. sin-neg 73.4%

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

      8. distribute-lft-neg-out 73.4%

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

      9. associate-*l* 73.4%

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

      10. *-commutative 73.4%

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

      11. distribute-lft-neg-out 73.4%

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

      12. distribute-rgt-neg-out 73.4%

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

      13. associate-*r/ 73.4%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in u around 0 88.5%

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

      1. +-commutative 88.5%

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

      2. fma-define 88.6%

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

      3. +-commutative 88.6%

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

      4. mul-1-neg 88.6%

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

      5. unsub-neg 88.6%

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

      6. *-commutative 88.6%

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

      7. associate-/l* 92.5%

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

      8. fma-neg 92.6%

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

      9. associate-/l* 99.9%

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

      10. distribute-rgt-neg-in 99.9%

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

      11. distribute-neg-frac2 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in normAngle around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. unsub-neg 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in n1_i around 0 91.2%

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

       1. mul-1-neg 91.2%

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

       2. distribute-rgt-neg-in 91.2%

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

   13. Simplified 91.2%

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

       1. distribute-rgt-neg-out 91.2%

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

       2. unsub-neg 91.2%

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

       3. *-commutative 91.2%

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

   15. Applied egg-rr 91.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.000000031374395 \cdot 10^{-22} \lor \neg \left(n1\_i \leq 6.000000240508405 \cdot 10^{-26}\right):\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.000000031374395 \cdot 10^{-22} \lor \neg \left(n1\_i \leq 6.000000240508405 \cdot 10^{-26}\right):\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.000000031374395e-22)
         (not (<= n1_i 6.000000240508405e-26)))
   (+ n0_i (* u n1_i))
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.000000031374395e-22f) || !(n1_i <= 6.000000240508405e-26f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-1.000000031374395e-22)) .or. (.not. (n1_i <= 6.000000240508405e-26))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-1.000000031374395e-22)) || !(n1_i <= Float32(6.000000240508405e-26)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-1.000000031374395e-22)) || ~((n1_i <= single(6.000000240508405e-26))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.00000003e-22 or 6.00000024e-26 < n1_i

   1. Initial program 96.1%

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

   3. Taylor expanded in u around 0 86.9%

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

   4. Taylor expanded in normAngle around 0 87.3%

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

   if -1.00000003e-22 < n1_i < 6.00000024e-26

   1. Initial program 97.9%

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

      1. associate-*l* 97.8%

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

      2. cancel-sign-sub 97.8%

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

      3. *-commutative 97.8%

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

      4. associate-*r* 73.2%

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

      5. associate-*r/ 73.4%

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

      6. *-rgt-identity 73.4%

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

      7. sin-neg 73.4%

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

      8. distribute-lft-neg-out 73.4%

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

      9. associate-*l* 73.4%

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

      10. *-commutative 73.4%

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

      11. distribute-lft-neg-out 73.4%

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

      12. distribute-rgt-neg-out 73.4%

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

      13. associate-*r/ 73.4%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in u around 0 88.5%

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

      1. +-commutative 88.5%

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

      2. fma-define 88.6%

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

      3. +-commutative 88.6%

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

      4. mul-1-neg 88.6%

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

      5. unsub-neg 88.6%

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

      6. *-commutative 88.6%

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

      7. associate-/l* 92.5%

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

      8. fma-neg 92.6%

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

      9. associate-/l* 99.9%

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

      10. distribute-rgt-neg-in 99.9%

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

      11. distribute-neg-frac2 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in normAngle around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in n1_i around 0 91.2%

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

       1. *-rgt-identity 91.2%

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

       2. mul-1-neg 91.2%

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

       3. distribute-rgt-neg-in 91.2%

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

       4. mul-1-neg 91.2%

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

       5. distribute-lft-in 91.0%

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

       6. mul-1-neg 91.0%

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

       7. unsub-neg 91.0%

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

   13. Simplified 91.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.000000031374395 \cdot 10^{-22} \lor \neg \left(n1\_i \leq 6.000000240508405 \cdot 10^{-26}\right):\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -2.4999999206638063 \cdot 10^{-21} \lor \neg \left(n1\_i \leq 1.500000013088254 \cdot 10^{-9}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -2.4999999206638063e-21)
         (not (<= n1_i 1.500000013088254e-9)))
   (* u n1_i)
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -2.4999999206638063e-21f) || !(n1_i <= 1.500000013088254e-9f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-2.4999999206638063e-21)) .or. (.not. (n1_i <= 1.500000013088254e-9))) then
        tmp = u * n1_i
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-2.4999999206638063e-21)) || !(n1_i <= Float32(1.500000013088254e-9)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-2.4999999206638063e-21)) || ~((n1_i <= single(1.500000013088254e-9))))
		tmp = u * n1_i;
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -2.49999992e-21 or 1.50000001e-9 < n1_i

   1. Initial program 95.6%

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

      1. associate-*l* 95.6%

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

      2. cancel-sign-sub 95.6%

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

      3. *-commutative 95.6%

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

      4. associate-*r* 89.6%

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

      5. associate-*r/ 89.7%

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

      6. *-rgt-identity 89.7%

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

      7. sin-neg 89.7%

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

      8. distribute-lft-neg-out 89.7%

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

      9. associate-*l* 89.8%

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

      10. *-commutative 89.8%

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

      11. distribute-lft-neg-out 89.8%

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

      12. distribute-rgt-neg-out 89.8%

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

      13. associate-*r/ 90.1%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in normAngle around 0 97.2%

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

   6. Taylor expanded in n0_i around 0 67.3%

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

   if -2.49999992e-21 < n1_i < 1.50000001e-9

   1. Initial program 97.4%

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

      1. associate-*l* 97.4%

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

      2. cancel-sign-sub 97.4%

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

      3. *-commutative 97.4%

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

      4. associate-*r* 78.2%

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

      5. associate-*r/ 78.3%

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

      6. *-rgt-identity 78.3%

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

      7. sin-neg 78.3%

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

      8. distribute-lft-neg-out 78.3%

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

      9. associate-*l* 78.3%

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

      10. *-commutative 78.3%

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

      11. distribute-lft-neg-out 78.3%

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

      12. distribute-rgt-neg-out 78.3%

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

      13. associate-*r/ 78.4%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in u around 0 88.5%

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

      1. +-commutative 88.5%

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

      2. fma-define 88.6%

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

      3. +-commutative 88.6%

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

      4. mul-1-neg 88.6%

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

      5. unsub-neg 88.6%

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

      6. *-commutative 88.6%

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

      7. associate-/l* 94.6%

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

      8. fma-neg 94.6%

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

      9. associate-/l* 99.8%

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

      10. distribute-rgt-neg-in 99.8%

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

      11. distribute-neg-frac2 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in normAngle around 0 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in n1_i around 0 79.3%

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

       1. *-rgt-identity 79.3%

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

       2. mul-1-neg 79.3%

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

       3. distribute-rgt-neg-in 79.3%

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

       4. mul-1-neg 79.3%

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

       5. distribute-lft-in 79.2%

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

       6. mul-1-neg 79.2%

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

       7. unsub-neg 79.2%

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

   13. Simplified 79.2%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -2.4999999206638063 \cdot 10^{-21} \lor \neg \left(n1\_i \leq 1.500000013088254 \cdot 10^{-9}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.8% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.999999682655225 \cdot 10^{-22} \lor \neg \left(n1\_i \leq 9.000000350284799 \cdot 10^{-15}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -9.999999682655225e-22)
         (not (<= n1_i 9.000000350284799e-15)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -9.999999682655225e-22f) || !(n1_i <= 9.000000350284799e-15f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-9.999999682655225e-22)) .or. (.not. (n1_i <= 9.000000350284799e-15))) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-9.999999682655225e-22)) || !(n1_i <= Float32(9.000000350284799e-15)))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-9.999999682655225e-22)) || ~((n1_i <= single(9.000000350284799e-15))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -9.9999997e-22 or 9.00000035e-15 < n1_i

   1. Initial program 95.1%

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

      1. associate-*l* 95.0%

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

      2. cancel-sign-sub 95.0%

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

      3. *-commutative 95.0%

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

      4. associate-*r* 88.2%

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

      5. associate-*r/ 88.3%

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

      6. *-rgt-identity 88.3%

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

      7. sin-neg 88.3%

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

      8. distribute-lft-neg-out 88.3%

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

      9. associate-*l* 88.4%

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

      10. *-commutative 88.4%

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

      11. distribute-lft-neg-out 88.4%

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

      12. distribute-rgt-neg-out 88.4%

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

      13. associate-*r/ 88.7%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in normAngle around 0 97.1%

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

   6. Taylor expanded in n0_i around 0 63.5%

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

   if -9.9999997e-22 < n1_i < 9.00000035e-15

   1. Initial program 98.1%

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

      1. associate-*l* 98.0%

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

      2. cancel-sign-sub 98.0%

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

      3. *-commutative 98.0%

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

      4. associate-*r* 77.8%

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

      5. associate-*r/ 78.0%

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

      6. *-rgt-identity 78.0%

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

      7. sin-neg 78.0%

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

      8. distribute-lft-neg-out 78.0%

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

      9. associate-*l* 77.9%

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

      10. *-commutative 77.9%

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

      11. distribute-lft-neg-out 77.9%

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

      12. distribute-rgt-neg-out 77.9%

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

      13. associate-*r/ 78.0%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in u around 0 89.3%

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

      1. +-commutative 89.3%

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

      2. fma-define 89.4%

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

      3. +-commutative 89.4%

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

      4. mul-1-neg 89.4%

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

      5. unsub-neg 89.4%

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

      6. *-commutative 89.4%

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

      7. associate-/l* 94.3%

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

      8. fma-neg 94.3%

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

      9. associate-/l* 99.8%

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

      10. distribute-rgt-neg-in 99.8%

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

      11. distribute-neg-frac2 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in u around 0 61.7%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.999999682655225 \cdot 10^{-22} \lor \neg \left(n1\_i \leq 9.000000350284799 \cdot 10^{-15}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.8%

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

   1. associate-*l* 96.7%

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

   2. cancel-sign-sub 96.7%

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

   3. *-commutative 96.7%

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

   4. associate-*r* 82.3%

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

   5. associate-*r/ 82.5%

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

   6. *-rgt-identity 82.5%

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

   7. sin-neg 82.5%

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

   8. distribute-lft-neg-out 82.5%

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

   9. associate-*l* 82.5%

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

   10. *-commutative 82.5%

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

   11. distribute-lft-neg-out 82.5%

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

   12. distribute-rgt-neg-out 82.5%

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

   13. associate-*r/ 82.6%

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

3. Simplified 74.2%

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

5. Taylor expanded in u around 0 87.6%

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

   1. +-commutative 87.6%

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

   2. fma-define 87.7%

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

   3. +-commutative 87.7%

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

   4. mul-1-neg 87.7%

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

   5. unsub-neg 87.7%

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

   6. *-commutative 87.7%

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

   7. associate-/l* 96.0%

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

   8. fma-neg 96.0%

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

   9. associate-/l* 99.7%

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

   10. distribute-rgt-neg-in 99.7%

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

   11. distribute-neg-frac2 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in normAngle around 0 98.0%

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

   1. *-commutative 98.0%

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

   2. mul-1-neg 98.0%

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

   3. unsub-neg 98.0%

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

10. Simplified 98.0%

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

11. Final simplification 98.0%

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

Alternative 8: 46.7% 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 96.8%

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

   1. associate-*l* 96.7%

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

   2. cancel-sign-sub 96.7%

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

   3. *-commutative 96.7%

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

   4. associate-*r* 82.3%

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

   5. associate-*r/ 82.5%

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

   6. *-rgt-identity 82.5%

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

   7. sin-neg 82.5%

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

   8. distribute-lft-neg-out 82.5%

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

   9. associate-*l* 82.5%

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

   10. *-commutative 82.5%

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

   11. distribute-lft-neg-out 82.5%

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

   12. distribute-rgt-neg-out 82.5%

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

   13. associate-*r/ 82.6%

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

3. Simplified 74.2%

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

5. Taylor expanded in u around 0 87.6%

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

   1. +-commutative 87.6%

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

   2. fma-define 87.7%

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

   3. +-commutative 87.7%

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

   4. mul-1-neg 87.7%

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

   5. unsub-neg 87.7%

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

   6. *-commutative 87.7%

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

   7. associate-/l* 96.0%

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

   8. fma-neg 96.0%

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

   9. associate-/l* 99.7%

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

   10. distribute-rgt-neg-in 99.7%

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

   11. distribute-neg-frac2 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in u around 0 46.5%

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

Reproduce

herbie shell --seed 2024095 
(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)))