Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.4%

Time: 19.1s

Alternatives: 8

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-*l* 85.6%

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

   3. *-commutative 85.6%

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

   4. associate-*l* 72.6%

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

   5. distribute-lft-out 72.7%

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

   6. associate-*l/ 72.8%

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

   7. *-lft-identity 72.8%

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

   8. fma-def 72.8%

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

   9. *-commutative 72.8%

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

   10. distribute-rgt-out-- 72.8%

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

   11. *-lft-identity 72.8%

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

3. Simplified 72.8%

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

4. Taylor expanded in u around 0 87.5%

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

   1. +-commutative 87.5%

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

   2. fma-def 87.6%

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

   3. +-commutative 87.6%

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

   4. mul-1-neg 87.6%

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

   5. unsub-neg 87.6%

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

   6. associate-/l* 95.8%

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

   7. associate-/l* 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

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

2. Taylor expanded in normAngle around 0 97.3%

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

3. Taylor expanded in u around 0 97.7%

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

4. Taylor expanded in u around -inf 89.9%

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

   1. mul-1-neg 89.9%

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

   2. unsub-neg 89.9%

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

   3. mul-1-neg 89.9%

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

   4. unsub-neg 89.9%

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

   5. *-commutative 89.9%

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

   6. associate-/l* 99.2%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

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

Alternative 3: 98.7% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ n0_i + u \cdot \left(\left(n1_i + 0.16666666666666666 \cdot \left(n1_i \cdot \left(normAngle \cdot normAngle\right)\right)\right) - n0_i\right) \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((n1_i + (0.16666666666666666f * (n1_i * (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 + (0.16666666666666666e0 * (n1_i * (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(Float32(0.16666666666666666) * Float32(n1_i * Float32(normAngle * normAngle)))) - n0_i)))
end

MATLAB

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

Derivation

1. Initial program 97.2%

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

2. Taylor expanded in normAngle around 0 97.3%

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

3. Taylor expanded in u around 0 97.7%

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

4. Taylor expanded in u around -inf 89.9%

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

   1. mul-1-neg 89.9%

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

   2. unsub-neg 89.9%

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

   3. mul-1-neg 89.9%

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

   4. unsub-neg 89.9%

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

   5. *-commutative 89.9%

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

   6. associate-/l* 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in normAngle around 0 98.9%

   \[\leadsto n0_i - u \cdot \left(n0_i - \color{blue}{\left(n1_i + 0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)}\right) \]
8. Step-by-step derivation

   1. unpow2 98.9%

      \[\leadsto n0_i - u \cdot \left(n0_i - \left(n1_i + 0.16666666666666666 \cdot \left(n1_i \cdot \color{blue}{\left(normAngle \cdot normAngle\right)}\right)\right)\right) \]

9. Simplified 98.9%

   \[\leadsto n0_i - u \cdot \left(n0_i - \color{blue}{\left(n1_i + 0.16666666666666666 \cdot \left(n1_i \cdot \left(normAngle \cdot normAngle\right)\right)\right)}\right) \]

10. Final simplification 98.9%

    \[\leadsto n0_i + u \cdot \left(\left(n1_i + 0.16666666666666666 \cdot \left(n1_i \cdot \left(normAngle \cdot normAngle\right)\right)\right) - n0_i\right) \]

Alternative 4: 69.0% accurate, 45.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -1.999999936531045e-20)
   (* u n1_i)
   (if (<= n1_i 4.999999980020986e-13) (* 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 <= -1.999999936531045e-20f) {
		tmp = u * n1_i;
	} else if (n1_i <= 4.999999980020986e-13f) {
		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 <= (-1.999999936531045e-20)) then
        tmp = u * n1_i
    else if (n1_i <= 4.999999980020986e-13) 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(-1.999999936531045e-20))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(4.999999980020986e-13))
		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(-1.999999936531045e-20))
		tmp = u * n1_i;
	elseif (n1_i <= single(4.999999980020986e-13))
		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 < -1.99999994e-20 or 4.99999998e-13 < n1_i

   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. *-commutative 96.2%

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

      2. associate-*l* 81.1%

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

      3. *-commutative 81.1%

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

      4. associate-*l* 80.8%

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

      5. distribute-lft-out 80.9%

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

      6. associate-*l/ 81.0%

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

      7. *-lft-identity 81.0%

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

      8. fma-def 81.0%

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

      9. *-commutative 81.0%

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

      10. distribute-rgt-out-- 81.1%

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

      11. *-lft-identity 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in normAngle around 0 97.5%

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

   5. Taylor expanded in n0_i around 0 69.7%

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

      1. *-commutative 69.7%

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

   7. Simplified 69.7%

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

   if -1.99999994e-20 < n1_i < 4.99999998e-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. *-commutative 97.9%

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

      2. associate-*l* 88.6%

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

      3. *-commutative 88.6%

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

      4. associate-*l* 67.0%

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

      5. distribute-lft-out 67.0%

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

      6. associate-*l/ 67.2%

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

      7. *-lft-identity 67.2%

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

      8. fma-def 67.2%

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

      9. *-commutative 67.2%

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

      10. distribute-rgt-out-- 67.1%

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

      11. *-lft-identity 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in normAngle around 0 98.2%

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

   5. Taylor expanded in n0_i around inf 79.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 5: 69.1% accurate, 45.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -1.999999936531045e-20)
   (* u n1_i)
   (if (<= n1_i 4.999999980020986e-13) (- n0_i (* u n0_i)) (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -1.999999936531045e-20f) {
		tmp = u * n1_i;
	} else if (n1_i <= 4.999999980020986e-13f) {
		tmp = n0_i - (u * n0_i);
	} else {
		tmp = u * n1_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n1_i <= (-1.999999936531045e-20)) then
        tmp = u * n1_i
    else if (n1_i <= 4.999999980020986e-13) then
        tmp = n0_i - (u * n0_i)
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-1.999999936531045e-20))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(4.999999980020986e-13))
		tmp = Float32(n0_i - Float32(u * n0_i));
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n1_i <= single(-1.999999936531045e-20))
		tmp = u * n1_i;
	elseif (n1_i <= single(4.999999980020986e-13))
		tmp = n0_i - (u * n0_i);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.99999994e-20 or 4.99999998e-13 < n1_i

   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. *-commutative 96.2%

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

      2. associate-*l* 81.1%

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

      3. *-commutative 81.1%

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

      4. associate-*l* 80.8%

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

      5. distribute-lft-out 80.9%

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

      6. associate-*l/ 81.0%

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

      7. *-lft-identity 81.0%

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

      8. fma-def 81.0%

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

      9. *-commutative 81.0%

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

      10. distribute-rgt-out-- 81.1%

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

      11. *-lft-identity 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in normAngle around 0 97.5%

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

   5. Taylor expanded in n0_i around 0 69.7%

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

      1. *-commutative 69.7%

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

   7. Simplified 69.7%

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

   if -1.99999994e-20 < n1_i < 4.99999998e-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. *-commutative 97.9%

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

      2. associate-*l* 88.6%

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

      3. *-commutative 88.6%

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

      4. associate-*l* 67.0%

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

      5. distribute-lft-out 67.0%

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

      6. associate-*l/ 67.2%

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

      7. *-lft-identity 67.2%

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

      8. fma-def 67.2%

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

      9. *-commutative 67.2%

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

      10. distribute-rgt-out-- 67.1%

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

      11. *-lft-identity 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in u around 0 86.3%

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

      1. +-commutative 86.3%

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

      2. fma-def 86.3%

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

      3. +-commutative 86.3%

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

      4. mul-1-neg 86.3%

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

      5. unsub-neg 86.3%

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

      6. associate-/l* 93.2%

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

      7. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in normAngle around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. fma-def 98.7%

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

   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 80.0%

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

       1. mul-1-neg 80.0%

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

       2. unsub-neg 80.0%

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

   12. Simplified 80.0%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 6: 59.6% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -3.99999987306209 \cdot 10^{-21}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -3.99999987306209e-21)
   (* u n1_i)
   (if (<= n1_i 1.999999936531045e-19) n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -3.99999987306209e-21f) {
		tmp = u * n1_i;
	} else if (n1_i <= 1.999999936531045e-19f) {
		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 <= (-3.99999987306209e-21)) then
        tmp = u * n1_i
    else if (n1_i <= 1.999999936531045e-19) 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(-3.99999987306209e-21))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(1.999999936531045e-19))
		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(-3.99999987306209e-21))
		tmp = u * n1_i;
	elseif (n1_i <= single(1.999999936531045e-19))
		tmp = n0_i;
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -3.9999999e-21 or 1.99999994e-19 < n1_i

   1. Initial program 95.8%

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

      1. *-commutative 95.8%

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

      2. associate-*l* 81.7%

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

      3. *-commutative 81.7%

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

      4. associate-*l* 80.4%

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

      5. distribute-lft-out 80.4%

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

      6. associate-*l/ 80.4%

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

      7. *-lft-identity 80.4%

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

      8. fma-def 80.5%

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

      9. *-commutative 80.5%

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

      10. distribute-rgt-out-- 80.5%

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

      11. *-lft-identity 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in normAngle around 0 97.7%

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

   5. Taylor expanded in n0_i around 0 67.1%

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

      1. *-commutative 67.1%

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

   7. Simplified 67.1%

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

   if -3.9999999e-21 < n1_i < 1.99999994e-19

   1. Initial program 98.4%

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

      1. *-commutative 98.4%

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

      2. associate-*l* 89.0%

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

      3. *-commutative 89.0%

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

      4. associate-*l* 65.9%

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

      5. distribute-lft-out 65.9%

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

      6. associate-*l/ 66.1%

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

      7. *-lft-identity 66.1%

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

      8. fma-def 66.1%

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

      9. *-commutative 66.1%

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

      10. distribute-rgt-out-- 66.1%

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

      11. *-lft-identity 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in u around 0 86.2%

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

      1. +-commutative 86.2%

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

      2. fma-def 86.2%

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

      3. +-commutative 86.2%

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

      4. mul-1-neg 86.2%

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

      5. unsub-neg 86.2%

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

      6. associate-/l* 92.6%

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

      7. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in normAngle around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. fma-def 98.6%

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

   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 65.7%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -3.99999987306209 \cdot 10^{-21}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 7: 98.1% 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 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-*l* 85.6%

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

   3. *-commutative 85.6%

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

   4. associate-*l* 72.6%

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

   5. distribute-lft-out 72.7%

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

   6. associate-*l/ 72.8%

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

   7. *-lft-identity 72.8%

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

   8. fma-def 72.8%

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

   9. *-commutative 72.8%

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

   10. distribute-rgt-out-- 72.8%

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

   11. *-lft-identity 72.8%

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

3. Simplified 72.8%

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

4. Taylor expanded in normAngle around 0 97.9%

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

5. Taylor expanded in u around -inf 98.2%

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

   1. mul-1-neg 98.2%

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

   2. unsub-neg 98.2%

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

   3. mul-1-neg 98.2%

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

   4. unsub-neg 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 8: 47.1% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-*l* 85.6%

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

   3. *-commutative 85.6%

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

   4. associate-*l* 72.6%

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

   5. distribute-lft-out 72.7%

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

   6. associate-*l/ 72.8%

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

   7. *-lft-identity 72.8%

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

   8. fma-def 72.8%

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

   9. *-commutative 72.8%

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

   10. distribute-rgt-out-- 72.8%

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

   11. *-lft-identity 72.8%

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

3. Simplified 72.8%

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

4. Taylor expanded in u around 0 87.5%

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

   1. +-commutative 87.5%

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

   2. fma-def 87.6%

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

   3. +-commutative 87.6%

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

   4. mul-1-neg 87.6%

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

   5. unsub-neg 87.6%

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

   6. associate-/l* 95.8%

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

   7. associate-/l* 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in normAngle around 0 98.2%

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

   1. +-commutative 98.2%

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

   2. fma-def 98.4%

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

9. Simplified 98.4%

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

10. Taylor expanded in u around 0 47.0%

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

11. Final simplification 47.0%

    \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023283 
(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)))