Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.4% → 99.5%

Time: 15.4s

Alternatives: 7

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: 97.4% 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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.8%

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

   1. fma-define 95.9%

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

   2. associate-*r/ 96.1%

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

   3. *-rgt-identity 96.1%

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

   4. associate-*r/ 96.3%

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

   5. *-rgt-identity 96.3%

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

3. Simplified 96.3%

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

5. Taylor expanded in u around 0 86.4%

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

   1. +-commutative 86.4%

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

   2. mul-1-neg 86.4%

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

   3. unsub-neg 86.4%

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

   4. associate-/l* 94.9%

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

   5. associate-/l* 98.7%

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

   6. associate-/l* 98.7%

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

7. Simplified 98.7%

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

   1. *-un-lft-identity 98.7%

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

   2. +-commutative 98.7%

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

   3. fma-define 98.9%

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

   4. associate-*r* 95.0%

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

   5. clear-num 95.0%

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

   6. quot-tan 95.0%

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

9. Applied egg-rr 95.0%

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

    1. *-lft-identity 95.0%

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

    2. associate-*l* 98.9%

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

    3. associate-*r/ 98.9%

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

    4. *-rgt-identity 98.9%

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

11. Simplified 98.9%

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

Alternative 2: 99.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.8%

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

   1. fma-define 95.9%

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

   2. associate-*r/ 96.1%

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

   3. *-rgt-identity 96.1%

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

   4. associate-*r/ 96.3%

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

   5. *-rgt-identity 96.3%

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

3. Simplified 96.3%

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

5. Taylor expanded in u around 0 86.4%

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

   1. +-commutative 86.4%

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

   2. mul-1-neg 86.4%

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

   3. unsub-neg 86.4%

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

   4. associate-/l* 94.9%

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

   5. associate-/l* 98.7%

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

   6. associate-/l* 98.7%

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

7. Simplified 98.7%

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

8. Taylor expanded in normAngle around 0 98.7%

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

9. Final simplification 98.7%

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

Alternative 3: 99.0% accurate, 22.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.8%

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

   1. fma-define 95.9%

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

   2. associate-*r/ 96.1%

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

   3. *-rgt-identity 96.1%

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

   4. associate-*r/ 96.3%

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

   5. *-rgt-identity 96.3%

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

3. Simplified 96.3%

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

5. Taylor expanded in u around 0 86.4%

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

   1. +-commutative 86.4%

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

   2. mul-1-neg 86.4%

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

   3. unsub-neg 86.4%

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

   4. associate-/l* 94.9%

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

   5. associate-/l* 98.7%

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

   6. associate-/l* 98.7%

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

7. Simplified 98.7%

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

8. Taylor expanded in normAngle around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate--r+ 98.4%

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

   3. sub-neg 98.4%

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

   4. distribute-rgt-out-- 98.4%

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

   5. metadata-eval 98.4%

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

   6. *-commutative 98.4%

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

   7. distribute-rgt-neg-in 98.4%

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

   8. metadata-eval 98.4%

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

10. Simplified 98.4%

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

    1. unpow2 98.4%

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

12. Applied egg-rr 98.4%

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

Alternative 4: 61.1% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -2.00000009162741e-18)
   n0_i
   (if (<= n0_i 1.999999936531045e-20) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -2.00000009162741e-18f) {
		tmp = n0_i;
	} else if (n0_i <= 1.999999936531045e-20f) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-2.00000009162741e-18)) then
        tmp = n0_i
    else if (n0_i <= 1.999999936531045e-20) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-2.00000009162741e-18))
		tmp = n0_i;
	elseif (n0_i <= Float32(1.999999936531045e-20))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-2.00000009162741e-18))
		tmp = n0_i;
	elseif (n0_i <= single(1.999999936531045e-20))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -2.00000009e-18 or 1.99999994e-20 < n0_i

   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. fma-define 98.1%

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

      2. associate-*r/ 98.4%

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

      3. *-rgt-identity 98.4%

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

      4. associate-*r/ 98.4%

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

      5. *-rgt-identity 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in u around 0 65.5%

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

   if -2.00000009e-18 < n0_i < 1.99999994e-20

   1. Initial program 93.8%

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

      1. fma-define 93.9%

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

      2. associate-*r/ 93.9%

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

      3. *-rgt-identity 93.9%

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

      4. associate-*r/ 94.3%

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

      5. *-rgt-identity 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in n0_i around 0 49.9%

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

   6. Taylor expanded in normAngle around 0 66.6%

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

      1. *-commutative 66.6%

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

   8. Simplified 66.6%

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

Alternative 5: 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 95.8%

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

   1. fma-define 95.9%

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

   2. associate-*r/ 96.1%

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

   3. *-rgt-identity 96.1%

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

   4. associate-*r/ 96.3%

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

   5. *-rgt-identity 96.3%

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

3. Simplified 96.3%

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

5. Taylor expanded in u around 0 86.4%

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

   1. +-commutative 86.4%

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

   2. mul-1-neg 86.4%

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

   3. unsub-neg 86.4%

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

   4. associate-/l* 94.9%

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

   5. associate-/l* 98.7%

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

   6. associate-/l* 98.7%

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

7. Simplified 98.7%

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

8. Taylor expanded in normAngle around 0 97.5%

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

Alternative 6: 82.0% accurate, 84.2× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot n1\_i \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.8%

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

   1. fma-define 95.9%

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

   2. associate-*r/ 96.1%

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

   3. *-rgt-identity 96.1%

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

   4. associate-*r/ 96.3%

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

   5. *-rgt-identity 96.3%

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

3. Simplified 96.3%

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

5. Taylor expanded in u around 0 86.4%

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

   1. +-commutative 86.4%

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

   2. mul-1-neg 86.4%

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

   3. unsub-neg 86.4%

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

   4. associate-/l* 94.9%

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

   5. associate-/l* 98.7%

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

   6. associate-/l* 98.7%

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

7. Simplified 98.7%

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

8. Taylor expanded in n1_i around inf 70.7%

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

   1. associate-/l* 80.6%

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

   2. associate-/l* 83.5%

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

10. Simplified 83.5%

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

11. Taylor expanded in normAngle around 0 82.4%

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

    1. *-commutative 82.4%

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

13. Simplified 82.4%

    \[\leadsto \color{blue}{n0\_i + u \cdot n1\_i} \]
14. Add Preprocessing

Alternative 7: 47.1% accurate, 421.0× speedup

Math

\[\begin{array}{l} \\ n0\_i \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.8%

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

   1. fma-define 95.9%

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

   2. associate-*r/ 96.1%

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

   3. *-rgt-identity 96.1%

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

   4. associate-*r/ 96.3%

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

   5. *-rgt-identity 96.3%

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

3. Simplified 96.3%

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

5. Taylor expanded in u around 0 45.7%

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

Reproduce

herbie shell --seed 2024135 
(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)))