Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.2%

Time: 14.3s

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: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma u (- (/ n1_i (/ (sin normAngle) normAngle)) n0_i) 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), n0_i);
}

Julia

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

Derivation

1. Initial program 96.9%

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

   1. fma-def 96.9%

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

   2. associate-*r/ 97.1%

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

   3. *-rgt-identity 97.1%

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

   4. associate-*r/ 97.7%

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

   5. *-rgt-identity 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in u around 0 90.5%

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

   1. fma-def 90.5%

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

   2. associate-/l* 91.7%

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

   3. associate-/r/ 91.3%

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

   4. associate-/l* 99.2%

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

   5. associate-/r/ 98.8%

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

6. Simplified 98.8%

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

7. Taylor expanded in normAngle around 0 99.0%

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

8. Taylor expanded in n0_i around 0 85.6%

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

   1. +-commutative 85.6%

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

   2. *-commutative 85.6%

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

   3. associate-*l/ 97.4%

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

   4. distribute-rgt-in 97.6%

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

   5. *-commutative 97.6%

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

   6. associate-+r+ 97.6%

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

   7. +-commutative 97.6%

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

   8. *-lft-identity 97.6%

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

   9. distribute-lft-in 97.6%

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

   10. *-lft-identity 97.6%

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

   11. mul-1-neg 97.6%

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

   12. distribute-rgt-neg-in 97.6%

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

   13. mul-1-neg 97.6%

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

   14. associate-+r+ 97.7%

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

   15. +-commutative 97.7%

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

   16. associate-*l/ 85.8%

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

   17. *-commutative 85.8%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 99.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((n1_i / (sinf(normAngle) / 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 / (sin(normangle) / 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(sin(normAngle) / normAngle)) - n0_i)))
end

MATLAB

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

Derivation

1. Initial program 96.9%

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

   1. fma-def 96.9%

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

   2. associate-*r/ 97.1%

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

   3. *-rgt-identity 97.1%

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

   4. associate-*r/ 97.7%

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

   5. *-rgt-identity 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in u around 0 90.5%

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

   1. fma-def 90.5%

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

   2. associate-/l* 91.7%

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

   3. associate-/r/ 91.3%

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

   4. associate-/l* 99.2%

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

   5. associate-/r/ 98.8%

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

6. Simplified 98.8%

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

7. Taylor expanded in normAngle around 0 99.0%

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

8. Taylor expanded in u around 0 91.5%

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

   1. +-commutative 91.5%

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

   2. mul-1-neg 91.5%

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

   3. unsub-neg 91.5%

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

   4. associate-/l* 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

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 96.9%

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

   1. fma-def 96.9%

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

   2. associate-*r/ 97.1%

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

   3. *-rgt-identity 97.1%

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

   4. associate-*r/ 97.7%

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

   5. *-rgt-identity 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in u around 0 90.5%

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

   1. fma-def 90.5%

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

   2. associate-/l* 91.7%

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

   3. associate-/r/ 91.3%

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

   4. associate-/l* 99.2%

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

   5. associate-/r/ 98.8%

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

6. Simplified 98.8%

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

7. Taylor expanded in normAngle around 0 99.0%

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

8. Taylor expanded in n0_i around 0 85.6%

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

   1. +-commutative 85.6%

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

   2. *-commutative 85.6%

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

   3. associate-*l/ 97.4%

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

   4. distribute-rgt-in 97.6%

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

   5. *-commutative 97.6%

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

   6. associate-+r+ 97.6%

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

   7. +-commutative 97.6%

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

   8. *-lft-identity 97.6%

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

   9. distribute-lft-in 97.6%

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

   10. *-lft-identity 97.6%

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

   11. mul-1-neg 97.6%

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

   12. distribute-rgt-neg-in 97.6%

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

   13. mul-1-neg 97.6%

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

   14. associate-+r+ 97.7%

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

   15. +-commutative 97.7%

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

   16. associate-*l/ 85.8%

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

   17. *-commutative 85.8%

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

10. Simplified 99.6%

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

11. Taylor expanded in normAngle around 0 98.1%

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

12. Final simplification 98.1%

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

Alternative 4: 70.5% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999967550318 \cdot 10^{-17} \lor \neg \left(n1_i \leq 3.99999992980668 \cdot 10^{-13}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.999999967550318e-17) (not (<= n1_i 3.99999992980668e-13)))
   (* u n1_i)
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.999999967550318e-17f) || !(n1_i <= 3.99999992980668e-13f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-1.999999967550318e-17)) .or. (.not. (n1_i <= 3.99999992980668e-13))) then
        tmp = u * n1_i
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-1.999999967550318e-17)) || !(n1_i <= Float32(3.99999992980668e-13)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-1.999999967550318e-17)) || ~((n1_i <= single(3.99999992980668e-13))))
		tmp = u * n1_i;
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.99999997e-17 or 3.99999993e-13 < n1_i

   1. Initial program 95.3%

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

      1. fma-def 95.4%

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

      2. associate-*r/ 95.4%

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

      3. *-rgt-identity 95.4%

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

      4. associate-*r/ 96.8%

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

      5. *-rgt-identity 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in normAngle around 0 97.8%

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

   5. Taylor expanded in n0_i around 0 71.8%

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

   if -1.99999997e-17 < n1_i < 3.99999993e-13

   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-def 97.9%

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

      2. associate-*r/ 98.3%

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

      3. *-rgt-identity 98.3%

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

      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. Taylor expanded in u around 0 88.0%

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

      1. fma-def 88.0%

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

      2. associate-/l* 90.0%

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

      3. associate-/r/ 90.0%

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

      4. associate-/l* 99.8%

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

      5. associate-/r/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in normAngle around 0 99.5%

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

   8. Taylor expanded in n0_i around inf 77.2%

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

      1. mul-1-neg 77.2%

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

      2. sub-neg 77.2%

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

   10. Simplified 77.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999967550318 \cdot 10^{-17} \lor \neg \left(n1_i \leq 3.99999992980668 \cdot 10^{-13}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 5: 59.8% accurate, 57.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999967550318 \cdot 10^{-17} \lor \neg \left(n1_i \leq 3.99999992980668 \cdot 10^{-13}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.999999967550318e-17) (not (<= n1_i 3.99999992980668e-13)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.999999967550318e-17f) || !(n1_i <= 3.99999992980668e-13f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-1.999999967550318e-17)) .or. (.not. (n1_i <= 3.99999992980668e-13))) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-1.999999967550318e-17)) || !(n1_i <= Float32(3.99999992980668e-13)))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-1.999999967550318e-17)) || ~((n1_i <= single(3.99999992980668e-13))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.99999997e-17 or 3.99999993e-13 < n1_i

   1. Initial program 95.3%

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

      1. fma-def 95.4%

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

      2. associate-*r/ 95.4%

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

      3. *-rgt-identity 95.4%

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

      4. associate-*r/ 96.8%

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

      5. *-rgt-identity 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in normAngle around 0 97.8%

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

   5. Taylor expanded in n0_i around 0 71.8%

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

   if -1.99999997e-17 < n1_i < 3.99999993e-13

   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-def 97.9%

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

      2. associate-*r/ 98.3%

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

      3. *-rgt-identity 98.3%

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

      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. Taylor expanded in u around 0 62.5%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999967550318 \cdot 10^{-17} \lor \neg \left(n1_i \leq 3.99999992980668 \cdot 10^{-13}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 6: 83.4% accurate, 59.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -5.00000006675716e-11) (* n0_i (- 1.0 u)) (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -5.00000006675716e-11f) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = n0_i + (u * n1_i);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-5.00000006675716e-11)) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = n0_i + (u * n1_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-5.00000006675716e-11))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(n0_i + Float32(u * n1_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-5.00000006675716e-11))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -5.00000007e-11

   1. Initial program 96.2%

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

      1. fma-def 96.2%

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

      2. associate-*r/ 96.3%

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

      3. *-rgt-identity 96.3%

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

      4. associate-*r/ 99.3%

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

      5. *-rgt-identity 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in u around 0 98.7%

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

      1. fma-def 98.7%

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

      2. associate-/l* 99.9%

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

      3. associate-/r/ 96.7%

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

      4. associate-/l* 96.8%

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

      5. associate-/r/ 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in normAngle around 0 100.0%

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

   8. Taylor expanded in n0_i around inf 96.3%

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

      1. mul-1-neg 96.3%

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

      2. sub-neg 96.3%

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

   10. Simplified 96.3%

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

   if -5.00000007e-11 < n0_i

   1. Initial program 97.0%

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

      1. fma-def 97.1%

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

      2. associate-*r/ 97.2%

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

      3. *-rgt-identity 97.2%

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

      4. associate-*r/ 97.5%

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

      5. *-rgt-identity 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in u around 0 89.3%

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

      1. fma-def 89.3%

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

      2. associate-/l* 90.5%

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

      3. associate-/r/ 90.5%

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

      4. associate-/l* 99.6%

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

      5. associate-/r/ 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in n0_i around 0 74.0%

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

   8. Taylor expanded in normAngle around 0 86.5%

      \[\leadsto n0_i + \color{blue}{n1_i \cdot u} \]
   9. Step-by-step derivation

      1. *-commutative 86.5%

         \[\leadsto n0_i + \color{blue}{u \cdot n1_i} \]

   10. Simplified 86.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \]

Alternative 7: 83.4% accurate, 59.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -5.00000006675716e-11) (- n0_i (* u n0_i)) (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -5.00000006675716e-11f) {
		tmp = n0_i - (u * n0_i);
	} else {
		tmp = n0_i + (u * n1_i);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-5.00000006675716e-11)) then
        tmp = n0_i - (u * n0_i)
    else
        tmp = n0_i + (u * n1_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-5.00000006675716e-11))
		tmp = Float32(n0_i - Float32(u * n0_i));
	else
		tmp = Float32(n0_i + Float32(u * n1_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-5.00000006675716e-11))
		tmp = n0_i - (u * n0_i);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -5.00000007e-11

   1. Initial program 96.2%

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

      1. fma-def 96.2%

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

      2. associate-*r/ 96.3%

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

      3. *-rgt-identity 96.3%

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

      4. associate-*r/ 99.3%

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

      5. *-rgt-identity 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in u around 0 98.7%

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

      1. fma-def 98.7%

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

      2. associate-/l* 99.9%

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

      3. associate-/r/ 96.7%

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

      4. associate-/l* 96.8%

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

      5. associate-/r/ 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in normAngle around 0 100.0%

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

   8. Taylor expanded in n0_i around inf 96.8%

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

      1. mul-1-neg 96.8%

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

      2. distribute-lft-neg-out 96.8%

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

      3. *-commutative 96.8%

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

   10. Simplified 96.8%

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

   if -5.00000007e-11 < n0_i

   1. Initial program 97.0%

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

      1. fma-def 97.1%

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

      2. associate-*r/ 97.2%

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

      3. *-rgt-identity 97.2%

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

      4. associate-*r/ 97.5%

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

      5. *-rgt-identity 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in u around 0 89.3%

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

      1. fma-def 89.3%

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

      2. associate-/l* 90.5%

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

      3. associate-/r/ 90.5%

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

      4. associate-/l* 99.6%

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

      5. associate-/r/ 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in n0_i around 0 74.0%

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

   8. Taylor expanded in normAngle around 0 86.5%

      \[\leadsto n0_i + \color{blue}{n1_i \cdot u} \]
   9. Step-by-step derivation

      1. *-commutative 86.5%

         \[\leadsto n0_i + \color{blue}{u \cdot n1_i} \]

   10. Simplified 86.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \]

Alternative 8: 98.0% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

   1. fma-def 96.9%

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

   2. associate-*r/ 97.1%

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

   3. *-rgt-identity 97.1%

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

   4. associate-*r/ 97.7%

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

   5. *-rgt-identity 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in normAngle around 0 97.7%

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

5. Taylor expanded in u around -inf 98.0%

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

   1. mul-1-neg 98.0%

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

   2. unsub-neg 98.0%

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

   3. mul-1-neg 98.0%

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

   4. unsub-neg 98.0%

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

7. Simplified 98.0%

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

8. Final simplification 98.0%

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

Alternative 9: 46.9% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

   1. fma-def 96.9%

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

   2. associate-*r/ 97.1%

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

   3. *-rgt-identity 97.1%

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

   4. associate-*r/ 97.7%

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

   5. *-rgt-identity 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in u around 0 46.3%

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

5. Final simplification 46.3%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023301 
(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)))