Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.4%

Time: 14.7s

Alternatives: 9

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: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*l* 82.6%

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

   3. *-commutative 82.6%

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

   4. associate-*l* 74.3%

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

   5. distribute-lft-out 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in u around 0 88.9%

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

   1. +-commutative 88.9%

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

   2. fma-define 89.1%

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

   3. +-commutative 89.1%

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

   4. mul-1-neg 89.1%

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

   5. unsub-neg 89.1%

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

   6. associate-/l* 96.7%

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

   7. associate-/l* 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 98.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

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 = (n1_i * (u / (sin(normangle) / normangle))) + (n0_i * (1.0e0 - u))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. fma-define 97.6%

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

   2. associate-*l* 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in normAngle around 0 97.4%

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

6. Taylor expanded in u around 0 90.1%

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

7. Taylor expanded in n0_i around -inf 85.8%

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

   1. +-commutative 85.8%

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

   2. *-commutative 85.8%

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

   3. associate-*l/ 97.3%

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

   4. mul-1-neg 97.3%

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

   5. unsub-neg 97.3%

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

   6. associate-*l/ 85.8%

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

   7. associate-*r/ 97.5%

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

   8. associate-/l* 98.9%

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

   9. sub-neg 98.9%

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

   10. metadata-eval 98.9%

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

9. Simplified 98.9%

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

10. Final simplification 98.9%

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

Alternative 3: 98.1% 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 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*l* 82.6%

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

   3. *-commutative 82.6%

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

   4. associate-*l* 74.3%

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

   5. distribute-lft-out 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in u around 0 88.9%

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

   1. +-commutative 88.9%

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

   2. fma-define 89.1%

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

   3. +-commutative 89.1%

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

   4. mul-1-neg 89.1%

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

   5. unsub-neg 89.1%

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

   6. associate-/l* 96.7%

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

   7. associate-/l* 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in normAngle around 0 97.6%

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

   1. +-commutative 97.6%

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

   2. fma-define 97.7%

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

10. Simplified 97.7%

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

11. Final simplification 97.7%

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

Alternative 4: 70.8% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -8.999999840607448 \cdot 10^{-23} \lor \neg \left(n0\_i \leq 1.9999999996399175 \cdot 10^{-23}\right):\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -8.999999840607448e-23)
         (not (<= n0_i 1.9999999996399175e-23)))
   (* n0_i (- 1.0 u))
   (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -8.999999840607448e-23f) || !(n0_i <= 1.9999999996399175e-23f)) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = u * n1_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n0_i <= (-8.999999840607448e-23)) .or. (.not. (n0_i <= 1.9999999996399175e-23))) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-8.999999840607448e-23)) || !(n0_i <= Float32(1.9999999996399175e-23)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-8.999999840607448e-23)) || ~((n0_i <= single(1.9999999996399175e-23))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -8.99999984e-23 or 2e-23 < n0_i

   1. Initial program 98.3%

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

      1. fma-define 98.4%

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

      2. associate-*l* 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in normAngle around 0 98.3%

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

   6. Taylor expanded in u around 0 97.8%

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

   7. Taylor expanded in n0_i around inf 78.3%

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

      1. mul-1-neg 78.3%

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

      2. sub-neg 78.3%

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

   9. Simplified 78.3%

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

   if -8.99999984e-23 < n0_i < 2e-23

   1. Initial program 96.5%

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

      1. *-commutative 96.5%

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

      2. associate-*l* 78.8%

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

      3. *-commutative 78.8%

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

      4. associate-*l* 59.8%

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

      5. distribute-lft-out 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in normAngle around 0 96.3%

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

   6. Taylor expanded in n0_i around 0 71.0%

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

      1. *-commutative 71.0%

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

   8. Simplified 71.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -8.999999840607448 \cdot 10^{-23} \lor \neg \left(n0\_i \leq 1.9999999996399175 \cdot 10^{-23}\right):\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.0% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -8.999999840607448 \cdot 10^{-23} \lor \neg \left(n0\_i \leq 1.9999999996399175 \cdot 10^{-23}\right):\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -8.999999840607448e-23)
         (not (<= n0_i 1.9999999996399175e-23)))
   (- n0_i (* u n0_i))
   (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -8.999999840607448e-23f) || !(n0_i <= 1.9999999996399175e-23f)) {
		tmp = n0_i - (u * n0_i);
	} else {
		tmp = u * n1_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n0_i <= (-8.999999840607448e-23)) .or. (.not. (n0_i <= 1.9999999996399175e-23))) then
        tmp = n0_i - (u * n0_i)
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-8.999999840607448e-23)) || !(n0_i <= Float32(1.9999999996399175e-23)))
		tmp = Float32(n0_i - Float32(u * n0_i));
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-8.999999840607448e-23)) || ~((n0_i <= single(1.9999999996399175e-23))))
		tmp = n0_i - (u * n0_i);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -8.99999984e-23 or 2e-23 < n0_i

   1. Initial program 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 85.3%

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

      3. *-commutative 85.3%

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

      4. associate-*l* 84.5%

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

      5. distribute-lft-out 84.5%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in normAngle around 0 97.9%

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

   6. Taylor expanded in u around -inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unsub-neg 98.4%

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

      3. mul-1-neg 98.4%

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

      4. unsub-neg 98.4%

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

   8. Simplified 98.4%

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

   9. Taylor expanded in n1_i around 0 78.5%

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

   if -8.99999984e-23 < n0_i < 2e-23

   1. Initial program 96.5%

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

      1. *-commutative 96.5%

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

      2. associate-*l* 78.8%

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

      3. *-commutative 78.8%

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

      4. associate-*l* 59.8%

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

      5. distribute-lft-out 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in normAngle around 0 96.3%

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

   6. Taylor expanded in n0_i around 0 71.0%

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

      1. *-commutative 71.0%

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

   8. Simplified 71.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -8.999999840607448 \cdot 10^{-23} \lor \neg \left(n0\_i \leq 1.9999999996399175 \cdot 10^{-23}\right):\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.8% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -8.999999840607448 \cdot 10^{-23}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -8.999999840607448e-23)
   n0_i
   (if (<= n0_i 1.9999999996399175e-23) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -8.999999840607448e-23f) {
		tmp = n0_i;
	} else if (n0_i <= 1.9999999996399175e-23f) {
		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 <= (-8.999999840607448e-23)) then
        tmp = n0_i
    else if (n0_i <= 1.9999999996399175e-23) 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(-8.999999840607448e-23))
		tmp = n0_i;
	elseif (n0_i <= Float32(1.9999999996399175e-23))
		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(-8.999999840607448e-23))
		tmp = n0_i;
	elseif (n0_i <= single(1.9999999996399175e-23))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -8.99999984e-23 or 2e-23 < n0_i

   1. Initial program 98.3%

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

      1. fma-define 98.4%

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

      2. associate-*l* 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in normAngle around 0 98.3%

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

   6. Taylor expanded in u around 0 97.8%

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

   7. Taylor expanded in u around 0 60.7%

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

   if -8.99999984e-23 < n0_i < 2e-23

   1. Initial program 96.5%

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

      1. *-commutative 96.5%

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

      2. associate-*l* 78.8%

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

      3. *-commutative 78.8%

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

      4. associate-*l* 59.8%

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

      5. distribute-lft-out 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in normAngle around 0 96.3%

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

   6. Taylor expanded in n0_i around 0 71.0%

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

      1. *-commutative 71.0%

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

   8. Simplified 71.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -8.999999840607448 \cdot 10^{-23}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.0% accurate, 42.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i 2.0000000072549875e-15) (+ 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 (n0_i <= 2.0000000072549875e-15f) {
		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 (n0_i <= 2.0000000072549875e-15) 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 (n0_i <= Float32(2.0000000072549875e-15))
		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 (n0_i <= single(2.0000000072549875e-15))
		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 n0_i < 2.00000001e-15

   1. Initial program 97.3%

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

      1. *-commutative 97.3%

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

      2. associate-*l* 80.5%

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

      3. *-commutative 80.5%

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

      4. associate-*l* 70.0%

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

      5. distribute-lft-out 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in normAngle around 0 96.8%

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

   6. Taylor expanded in u around -inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. unsub-neg 97.1%

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

      3. mul-1-neg 97.1%

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

      4. unsub-neg 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in n0_i around 0 85.1%

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

       1. neg-mul-1 85.1%

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

       2. distribute-rgt-neg-in 85.1%

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

   11. Simplified 85.1%

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

   if 2.00000001e-15 < n0_i

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*l* 90.0%

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

      3. *-commutative 90.0%

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

      4. associate-*l* 90.0%

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

      5. distribute-lft-out 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in normAngle around 0 99.0%

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

   6. Taylor expanded in u around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. unsub-neg 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in n1_i around 0 90.6%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*l* 82.6%

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

   3. *-commutative 82.6%

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

   4. associate-*l* 74.3%

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

   5. distribute-lft-out 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in normAngle around 0 97.3%

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

6. Taylor expanded in u around -inf 97.6%

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

   1. mul-1-neg 97.6%

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

   2. unsub-neg 97.6%

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

   3. mul-1-neg 97.6%

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

   4. unsub-neg 97.6%

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

8. Simplified 97.6%

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

9. Final simplification 97.6%

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

Alternative 9: 47.0% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. fma-define 97.6%

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

   2. associate-*l* 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in normAngle around 0 97.4%

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

6. Taylor expanded in u around 0 90.1%

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

7. Taylor expanded in u around 0 46.0%

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

8. Final simplification 46.0%

   \[\leadsto n0\_i \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024035 
(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)))