Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.4% → 98.1%

Time: 15.1s

Alternatives: 7

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 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 97.3%

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

   1. *-commutative 97.3%

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

   2. associate-*l* 86.9%

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

   3. *-commutative 86.9%

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

   4. associate-*l* 76.7%

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

   5. distribute-lft-out 76.8%

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

   6. associate-*l/ 76.9%

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

   7. *-lft-identity 76.9%

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

   8. fma-def 76.9%

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

   9. *-commutative 76.9%

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

   10. distribute-rgt-out-- 77.0%

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

   11. *-lft-identity 77.0%

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

3. Simplified 77.0%

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

4. Taylor expanded in normAngle around 0 98.2%

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

   1. *-commutative 98.2%

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

   2. fma-def 98.3%

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

6. Simplified 98.3%

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

7. Taylor expanded in u around 0 98.4%

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

   1. *-commutative 98.4%

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

   2. fma-def 98.6%

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

   3. mul-1-neg 98.6%

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

   4. unsub-neg 98.6%

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 2: 86.1% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -9.999999887266023 \cdot 10^{-27} \lor \neg \left(n1_i \leq 4.999999999099794 \cdot 10^{-24}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -9.999999887266023e-27)
         (not (<= n1_i 4.999999999099794e-24)))
   (+ n0_i (* u n1_i))
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -9.999999887266023e-27f) || !(n1_i <= 4.999999999099794e-24f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-9.999999887266023e-27)) .or. (.not. (n1_i <= 4.999999999099794e-24))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-9.999999887266023e-27)) || !(n1_i <= Float32(4.999999999099794e-24)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-9.999999887266023e-27)) || ~((n1_i <= single(4.999999999099794e-24))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -9.99999989e-27 or 5e-24 < n1_i

   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%

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

      2. associate-*l* 82.4%

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

      3. *-commutative 82.4%

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

      4. associate-*l* 78.6%

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

      5. distribute-lft-out 78.7%

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

      6. associate-*l/ 78.8%

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

      7. *-lft-identity 78.8%

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

      8. fma-def 78.8%

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

      9. *-commutative 78.8%

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

      10. distribute-rgt-out-- 78.9%

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

      11. *-lft-identity 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in normAngle around 0 98.3%

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

   5. Taylor expanded in u around 0 87.5%

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

   if -9.99999989e-27 < n1_i < 5e-24

   1. Initial program 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*l* 94.7%

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

      3. *-commutative 94.7%

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

      4. associate-*l* 73.4%

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

      5. distribute-lft-out 73.4%

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

      6. associate-*l/ 73.7%

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

      7. *-lft-identity 73.7%

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

      8. fma-def 73.7%

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

      9. *-commutative 73.7%

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

      10. distribute-rgt-out-- 73.8%

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

      11. *-lft-identity 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in normAngle around 0 98.0%

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

      1. *-commutative 98.0%

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

      2. fma-def 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in n1_i around 0 91.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -9.999999887266023 \cdot 10^{-27} \lor \neg \left(n1_i \leq 4.999999999099794 \cdot 10^{-24}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 3: 86.2% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -9.999999887266023 \cdot 10^{-27} \lor \neg \left(n1_i \leq 4.999999999099794 \cdot 10^{-24}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -9.999999887266023e-27)
         (not (<= n1_i 4.999999999099794e-24)))
   (+ n0_i (* u n1_i))
   (- n0_i (* u n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -9.999999887266023e-27f) || !(n1_i <= 4.999999999099794e-24f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i - (u * n0_i);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-9.999999887266023e-27)) .or. (.not. (n1_i <= 4.999999999099794e-24))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i - (u * n0_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-9.999999887266023e-27)) || !(n1_i <= Float32(4.999999999099794e-24)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i - Float32(u * n0_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-9.999999887266023e-27)) || ~((n1_i <= single(4.999999999099794e-24))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i - (u * n0_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -9.99999989e-27 or 5e-24 < n1_i

   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%

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

      2. associate-*l* 82.4%

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

      3. *-commutative 82.4%

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

      4. associate-*l* 78.6%

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

      5. distribute-lft-out 78.7%

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

      6. associate-*l/ 78.8%

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

      7. *-lft-identity 78.8%

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

      8. fma-def 78.8%

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

      9. *-commutative 78.8%

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

      10. distribute-rgt-out-- 78.9%

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

      11. *-lft-identity 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in normAngle around 0 98.3%

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

   5. Taylor expanded in u around 0 87.5%

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

   if -9.99999989e-27 < n1_i < 5e-24

   1. Initial program 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*l* 94.7%

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

      3. *-commutative 94.7%

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

      4. associate-*l* 73.4%

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

      5. distribute-lft-out 73.4%

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

      6. associate-*l/ 73.7%

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

      7. *-lft-identity 73.7%

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

      8. fma-def 73.7%

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

      9. *-commutative 73.7%

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

      10. distribute-rgt-out-- 73.8%

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

      11. *-lft-identity 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in normAngle around 0 98.0%

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

      1. *-commutative 98.0%

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

      2. fma-def 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in u around 0 98.5%

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

      1. *-commutative 98.5%

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

      2. fma-def 98.7%

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

      3. mul-1-neg 98.7%

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

      4. unsub-neg 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in n1_i around 0 91.9%

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

       1. +-commutative 91.9%

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

       2. mul-1-neg 91.9%

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

       3. unsub-neg 91.9%

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

   12. Simplified 91.9%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -9.999999887266023 \cdot 10^{-27} \lor \neg \left(n1_i \leq 4.999999999099794 \cdot 10^{-24}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \end{array} \]

Alternative 4: 70.1% accurate, 45.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 6.000000233523199 \cdot 10^{-16}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -4.999999858590343e-10)
   (* u n1_i)
   (if (<= n1_i 6.000000233523199e-16) (* n0_i (- 1.0 u)) (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -4.999999858590343e-10f) {
		tmp = u * n1_i;
	} else if (n1_i <= 6.000000233523199e-16f) {
		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 (n1_i <= (-4.999999858590343e-10)) then
        tmp = u * n1_i
    else if (n1_i <= 6.000000233523199e-16) 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 (n1_i <= Float32(-4.999999858590343e-10))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(6.000000233523199e-16))
		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 (n1_i <= single(-4.999999858590343e-10))
		tmp = u * n1_i;
	elseif (n1_i <= single(6.000000233523199e-16))
		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 n1_i < -4.99999986e-10 or 6.0000002e-16 < n1_i

   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%

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

      2. associate-*l* 84.3%

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

      3. *-commutative 84.3%

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

      4. associate-*l* 83.3%

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

      5. distribute-lft-out 83.5%

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

      6. associate-*l/ 83.5%

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

      7. *-lft-identity 83.5%

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

      8. fma-def 83.5%

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

      9. *-commutative 83.5%

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

      10. distribute-rgt-out-- 83.5%

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

      11. *-lft-identity 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in normAngle around 0 98.1%

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

      1. *-commutative 98.1%

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

      2. fma-def 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in n1_i around inf 72.7%

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

      1. *-commutative 72.7%

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

   9. Simplified 72.7%

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

   if -4.99999986e-10 < n1_i < 6.0000002e-16

   1. Initial program 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l* 88.0%

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

      3. *-commutative 88.0%

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

      4. associate-*l* 73.8%

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

      5. distribute-lft-out 73.8%

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

      6. associate-*l/ 74.0%

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

      7. *-lft-identity 74.0%

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

      8. fma-def 74.0%

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

      9. *-commutative 74.0%

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

      10. distribute-rgt-out-- 74.2%

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

      11. *-lft-identity 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in normAngle around 0 98.2%

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

      1. *-commutative 98.2%

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

      2. fma-def 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in n1_i around 0 76.1%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 6.000000233523199 \cdot 10^{-16}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 5: 59.7% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 6.000000233523199 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -4.999999858590343e-10)
   (* u n1_i)
   (if (<= n1_i 6.000000233523199e-16) n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -4.999999858590343e-10f) {
		tmp = u * n1_i;
	} else if (n1_i <= 6.000000233523199e-16f) {
		tmp = 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 (n1_i <= (-4.999999858590343e-10)) then
        tmp = u * n1_i
    else if (n1_i <= 6.000000233523199e-16) then
        tmp = 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 (n1_i <= Float32(-4.999999858590343e-10))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(6.000000233523199e-16))
		tmp = 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 (n1_i <= single(-4.999999858590343e-10))
		tmp = u * n1_i;
	elseif (n1_i <= single(6.000000233523199e-16))
		tmp = n0_i;
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.99999986e-10 or 6.0000002e-16 < n1_i

   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%

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

      2. associate-*l* 84.3%

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

      3. *-commutative 84.3%

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

      4. associate-*l* 83.3%

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

      5. distribute-lft-out 83.5%

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

      6. associate-*l/ 83.5%

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

      7. *-lft-identity 83.5%

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

      8. fma-def 83.5%

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

      9. *-commutative 83.5%

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

      10. distribute-rgt-out-- 83.5%

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

      11. *-lft-identity 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in normAngle around 0 98.1%

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

      1. *-commutative 98.1%

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

      2. fma-def 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in n1_i around inf 72.7%

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

      1. *-commutative 72.7%

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

   9. Simplified 72.7%

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

   if -4.99999986e-10 < n1_i < 6.0000002e-16

   1. Initial program 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l* 88.0%

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

      3. *-commutative 88.0%

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

      4. associate-*l* 73.8%

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

      5. distribute-lft-out 73.8%

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

      6. associate-*l/ 74.0%

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

      7. *-lft-identity 74.0%

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

      8. fma-def 74.0%

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

      9. *-commutative 74.0%

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

      10. distribute-rgt-out-- 74.2%

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

      11. *-lft-identity 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in normAngle around 0 98.2%

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

      1. *-commutative 98.2%

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

      2. fma-def 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in u around 0 98.5%

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

      1. *-commutative 98.5%

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

      2. fma-def 98.7%

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

      3. mul-1-neg 98.7%

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

      4. unsub-neg 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in u around 0 59.4%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 6.000000233523199 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 6: 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 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%

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

   2. associate-*l* 86.9%

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

   3. *-commutative 86.9%

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

   4. associate-*l* 76.7%

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

   5. distribute-lft-out 76.8%

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

   6. associate-*l/ 76.9%

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

   7. *-lft-identity 76.9%

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

   8. fma-def 76.9%

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

   9. *-commutative 76.9%

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

   10. distribute-rgt-out-- 77.0%

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

   11. *-lft-identity 77.0%

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

3. Simplified 77.0%

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

4. Taylor expanded in normAngle around 0 98.2%

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

   1. *-commutative 98.2%

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

   2. fma-def 98.3%

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

6. Simplified 98.3%

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

7. Taylor expanded in u around -inf 98.4%

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

   1. +-commutative 98.4%

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

   2. mul-1-neg 98.4%

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

   3. unsub-neg 98.4%

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

   4. neg-mul-1 98.4%

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

   5. +-commutative 98.4%

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

   6. unsub-neg 98.4%

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

9. Simplified 98.4%

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

10. Final simplification 98.4%

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

Alternative 7: 47.5% 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 97.3%

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

   1. *-commutative 97.3%

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

   2. associate-*l* 86.9%

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

   3. *-commutative 86.9%

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

   4. associate-*l* 76.7%

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

   5. distribute-lft-out 76.8%

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

   6. associate-*l/ 76.9%

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

   7. *-lft-identity 76.9%

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

   8. fma-def 76.9%

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

   9. *-commutative 76.9%

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

   10. distribute-rgt-out-- 77.0%

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

   11. *-lft-identity 77.0%

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

3. Simplified 77.0%

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

4. Taylor expanded in normAngle around 0 98.2%

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

   1. *-commutative 98.2%

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

   2. fma-def 98.3%

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

6. Simplified 98.3%

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

7. Taylor expanded in u around 0 98.4%

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

   1. *-commutative 98.4%

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

   2. fma-def 98.6%

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

   3. mul-1-neg 98.6%

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

   4. unsub-neg 98.6%

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

9. Simplified 98.6%

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

10. Taylor expanded in u around 0 46.8%

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

11. Final simplification 46.8%

    \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023274 
(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)))