Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 98.4%

Time: 15.8s

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: 97.2% 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: 98.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ {normAngle}^{2} \cdot \left(\left(u \cdot 0.16666666666666666\right) \cdot n1\_i\right) + \left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* (pow normAngle 2.0) (* (* u 0.16666666666666666) n1_i))
  (+ (* n0_i (- 1.0 u)) (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (powf(normAngle, 2.0f) * ((u * 0.16666666666666666f) * n1_i)) + ((n0_i * (1.0f - u)) + (u * n1_i));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = ((normangle ** 2.0e0) * ((u * 0.16666666666666666e0) * n1_i)) + ((n0_i * (1.0e0 - u)) + (u * n1_i))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32((normAngle ^ Float32(2.0)) * Float32(Float32(u * Float32(0.16666666666666666)) * n1_i)) + Float32(Float32(n0_i * Float32(Float32(1.0) - u)) + Float32(u * n1_i)))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = ((normAngle ^ single(2.0)) * ((u * single(0.16666666666666666)) * n1_i)) + ((n0_i * (single(1.0) - u)) + (u * n1_i));
end

Derivation

1. Initial program 97.5%

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

   1. associate-*l* 97.4%

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

   2. cancel-sign-sub 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*r* 82.0%

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

   5. associate-*r/ 82.0%

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

   6. *-rgt-identity 82.0%

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

   7. sin-neg 82.0%

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

   8. distribute-lft-neg-out 82.0%

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

   9. associate-*l* 82.1%

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

   10. *-commutative 82.1%

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

   11. distribute-lft-neg-out 82.1%

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

   12. distribute-rgt-neg-out 82.1%

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

   13. associate-*r/ 82.3%

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

3. Simplified 76.6%

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

5. Taylor expanded in normAngle around 0 74.4%

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

6. Taylor expanded in normAngle around 0 96.6%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({normAngle}^{2} \cdot \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right)\right) + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right)} \]

7. Taylor expanded in n0_i around 0 98.6%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(n1\_i \cdot \left({normAngle}^{2} \cdot u\right)\right)} + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]
8. Step-by-step derivation

   1. *-commutative 98.6%

      \[\leadsto \color{blue}{\left(n1\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) \cdot 0.16666666666666666} + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

   2. *-commutative 98.6%

      \[\leadsto \color{blue}{\left(\left({normAngle}^{2} \cdot u\right) \cdot n1\_i\right)} \cdot 0.16666666666666666 + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

   3. associate-*r* 98.6%

      \[\leadsto \color{blue}{\left({normAngle}^{2} \cdot \left(u \cdot n1\_i\right)\right)} \cdot 0.16666666666666666 + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

   4. *-commutative 98.6%

      \[\leadsto \left({normAngle}^{2} \cdot \color{blue}{\left(n1\_i \cdot u\right)}\right) \cdot 0.16666666666666666 + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

   5. associate-*l* 98.6%

      \[\leadsto \color{blue}{{normAngle}^{2} \cdot \left(\left(n1\_i \cdot u\right) \cdot 0.16666666666666666\right)} + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

   6. metadata-eval 98.6%

      \[\leadsto {normAngle}^{2} \cdot \left(\left(n1\_i \cdot u\right) \cdot \color{blue}{\left(--0.16666666666666666\right)}\right) + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

   7. distribute-rgt-neg-in 98.6%

      \[\leadsto {normAngle}^{2} \cdot \color{blue}{\left(-\left(n1\_i \cdot u\right) \cdot -0.16666666666666666\right)} + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

   8. *-commutative 98.6%

      \[\leadsto {normAngle}^{2} \cdot \left(-\color{blue}{-0.16666666666666666 \cdot \left(n1\_i \cdot u\right)}\right) + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

   9. distribute-lft-neg-in 98.6%

      \[\leadsto {normAngle}^{2} \cdot \color{blue}{\left(\left(--0.16666666666666666\right) \cdot \left(n1\_i \cdot u\right)\right)} + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

   10. metadata-eval 98.6%

       \[\leadsto {normAngle}^{2} \cdot \left(\color{blue}{0.16666666666666666} \cdot \left(n1\_i \cdot u\right)\right) + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

   11. *-commutative 98.6%

       \[\leadsto {normAngle}^{2} \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(u \cdot n1\_i\right)}\right) + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

   12. associate-*r* 98.6%

       \[\leadsto {normAngle}^{2} \cdot \color{blue}{\left(\left(0.16666666666666666 \cdot u\right) \cdot n1\_i\right)} + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

   13. *-commutative 98.6%

       \[\leadsto {normAngle}^{2} \cdot \left(\color{blue}{\left(u \cdot 0.16666666666666666\right)} \cdot n1\_i\right) + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

9. Simplified 98.6%

   \[\leadsto \color{blue}{{normAngle}^{2} \cdot \left(\left(u \cdot 0.16666666666666666\right) \cdot n1\_i\right)} + \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right) \]

10. Final simplification 98.6%

    \[\leadsto {normAngle}^{2} \cdot \left(\left(u \cdot 0.16666666666666666\right) \cdot n1\_i\right) + \left(n0\_i \cdot \left(1 - u\right) + u \cdot n1\_i\right) \]
11. Add Preprocessing

Alternative 2: 98.2% 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 97.5%

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

3. Taylor expanded in u around 0 93.7%

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

   1. mul-1-neg 93.7%

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

   2. unsub-neg 93.7%

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

   3. associate-/l* 97.8%

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

   4. associate-/l* 97.8%

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

5. Simplified 97.8%

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

6. Taylor expanded in normAngle around 0 97.7%

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

   1. sub-neg 97.7%

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

   2. associate-+r+ 97.8%

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

   3. mul-1-neg 97.8%

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

   4. associate-*r* 97.8%

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

   5. distribute-rgt-in 97.7%

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

   6. +-commutative 97.7%

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

   7. fma-define 97.9%

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

   8. mul-1-neg 97.9%

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

   9. unsub-neg 97.9%

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

8. Simplified 97.9%

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

9. Final simplification 97.9%

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

Alternative 3: 70.6% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -2.000000033724767 \cdot 10^{-16} \lor \neg \left(n1\_i \leq 9.9999998245167 \cdot 10^{-15}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -2.000000033724767e-16) (not (<= n1_i 9.9999998245167e-15)))
   (* 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 <= -2.000000033724767e-16f) || !(n1_i <= 9.9999998245167e-15f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-2.000000033724767e-16)) .or. (.not. (n1_i <= 9.9999998245167e-15))) then
        tmp = u * n1_i
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-2.000000033724767e-16)) || !(n1_i <= Float32(9.9999998245167e-15)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-2.000000033724767e-16)) || ~((n1_i <= single(9.9999998245167e-15))))
		tmp = u * n1_i;
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -2.00000003e-16 or 9.99999982e-15 < n1_i

   1. Initial program 95.4%

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

      1. associate-*l* 95.4%

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

      2. cancel-sign-sub 95.4%

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

      3. *-commutative 95.4%

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

      4. associate-*r* 92.4%

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

      5. associate-*r/ 92.5%

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

      6. *-rgt-identity 92.5%

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

      7. sin-neg 92.5%

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

      8. distribute-lft-neg-out 92.5%

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

      9. associate-*l* 92.6%

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

      10. *-commutative 92.6%

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

      11. distribute-lft-neg-out 92.6%

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

      12. distribute-rgt-neg-out 92.6%

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

      13. associate-*r/ 93.0%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in normAngle around 0 84.1%

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

   6. Taylor expanded in n0_i around 0 63.8%

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

      1. *-commutative 63.8%

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

      2. *-commutative 63.8%

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

      3. associate-*r* 63.9%

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

   8. Simplified 63.9%

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

   9. Taylor expanded in normAngle around 0 72.5%

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

       1. *-commutative 72.5%

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

   11. Simplified 72.5%

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

   if -2.00000003e-16 < n1_i < 9.99999982e-15

   1. Initial program 98.8%

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

   3. Taylor expanded in u around 0 92.9%

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

      1. mul-1-neg 92.9%

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

      2. unsub-neg 92.9%

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

      3. associate-/l* 99.3%

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

      4. associate-/l* 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in normAngle around 0 99.4%

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

   7. Taylor expanded in n0_i around inf 78.8%

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -2.000000033724767 \cdot 10^{-16} \lor \neg \left(n1\_i \leq 9.9999998245167 \cdot 10^{-15}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.8% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -2.000000033724767 \cdot 10^{-16} \lor \neg \left(n1\_i \leq 9.9999998245167 \cdot 10^{-15}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -2.000000033724767e-16) (not (<= n1_i 9.9999998245167e-15)))
   (* 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 <= -2.000000033724767e-16f) || !(n1_i <= 9.9999998245167e-15f)) {
		tmp = 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 <= (-2.000000033724767e-16)) .or. (.not. (n1_i <= 9.9999998245167e-15))) then
        tmp = 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(-2.000000033724767e-16)) || !(n1_i <= Float32(9.9999998245167e-15)))
		tmp = 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(-2.000000033724767e-16)) || ~((n1_i <= single(9.9999998245167e-15))))
		tmp = 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 < -2.00000003e-16 or 9.99999982e-15 < n1_i

   1. Initial program 95.4%

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

      1. associate-*l* 95.4%

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

      2. cancel-sign-sub 95.4%

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

      3. *-commutative 95.4%

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

      4. associate-*r* 92.4%

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

      5. associate-*r/ 92.5%

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

      6. *-rgt-identity 92.5%

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

      7. sin-neg 92.5%

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

      8. distribute-lft-neg-out 92.5%

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

      9. associate-*l* 92.6%

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

      10. *-commutative 92.6%

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

      11. distribute-lft-neg-out 92.6%

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

      12. distribute-rgt-neg-out 92.6%

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

      13. associate-*r/ 93.0%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in normAngle around 0 84.1%

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

   6. Taylor expanded in n0_i around 0 63.8%

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

      1. *-commutative 63.8%

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

      2. *-commutative 63.8%

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

      3. associate-*r* 63.9%

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

   8. Simplified 63.9%

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

   9. Taylor expanded in normAngle around 0 72.5%

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

       1. *-commutative 72.5%

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

   11. Simplified 72.5%

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

   if -2.00000003e-16 < n1_i < 9.99999982e-15

   1. Initial program 98.8%

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

   3. Taylor expanded in u around 0 92.9%

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

      1. mul-1-neg 92.9%

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

      2. unsub-neg 92.9%

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

      3. associate-/l* 99.3%

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

      4. associate-/l* 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in normAngle around 0 99.4%

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

   7. Taylor expanded in n1_i around 0 79.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -2.000000033724767 \cdot 10^{-16} \lor \neg \left(n1\_i \leq 9.9999998245167 \cdot 10^{-15}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.3% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -2.000000033724767 \cdot 10^{-16} \lor \neg \left(n1\_i \leq 9.9999998245167 \cdot 10^{-15}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -2.000000033724767e-16) (not (<= n1_i 9.9999998245167e-15)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -2.000000033724767e-16f) || !(n1_i <= 9.9999998245167e-15f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-2.000000033724767e-16)) .or. (.not. (n1_i <= 9.9999998245167e-15))) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-2.000000033724767e-16)) || !(n1_i <= Float32(9.9999998245167e-15)))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-2.000000033724767e-16)) || ~((n1_i <= single(9.9999998245167e-15))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -2.00000003e-16 or 9.99999982e-15 < n1_i

   1. Initial program 95.4%

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

      1. associate-*l* 95.4%

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

      2. cancel-sign-sub 95.4%

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

      3. *-commutative 95.4%

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

      4. associate-*r* 92.4%

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

      5. associate-*r/ 92.5%

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

      6. *-rgt-identity 92.5%

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

      7. sin-neg 92.5%

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

      8. distribute-lft-neg-out 92.5%

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

      9. associate-*l* 92.6%

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

      10. *-commutative 92.6%

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

      11. distribute-lft-neg-out 92.6%

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

      12. distribute-rgt-neg-out 92.6%

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

      13. associate-*r/ 93.0%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in normAngle around 0 84.1%

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

   6. Taylor expanded in n0_i around 0 63.8%

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

      1. *-commutative 63.8%

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

      2. *-commutative 63.8%

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

      3. associate-*r* 63.9%

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

   8. Simplified 63.9%

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

   9. Taylor expanded in normAngle around 0 72.5%

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

       1. *-commutative 72.5%

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

   11. Simplified 72.5%

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

   if -2.00000003e-16 < n1_i < 9.99999982e-15

   1. Initial program 98.8%

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

      1. fma-define 98.8%

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

      2. associate-*r/ 99.1%

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

      3. *-rgt-identity 99.1%

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

      4. associate-*r/ 99.3%

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

      5. *-rgt-identity 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in u around 0 63.9%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -2.000000033724767 \cdot 10^{-16} \lor \neg \left(n1\_i \leq 9.9999998245167 \cdot 10^{-15}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

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 97.5%

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

3. Taylor expanded in u around 0 93.7%

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

   1. mul-1-neg 93.7%

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

   2. unsub-neg 93.7%

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

   3. associate-/l* 97.8%

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

   4. associate-/l* 97.8%

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

5. Simplified 97.8%

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

   1. div-inv 98.1%

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

   2. clear-num 98.0%

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

7. Applied egg-rr 98.0%

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

8. Taylor expanded in normAngle around 0 97.7%

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

   1. cancel-sign-sub-inv 97.7%

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

   2. associate-+l+ 97.8%

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

   3. distribute-rgt-in 97.7%

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

   4. unsub-neg 97.7%

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

10. Simplified 97.7%

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

11. Final simplification 97.7%

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

Alternative 7: 47.0% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. fma-define 97.5%

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

   2. associate-*r/ 97.7%

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

   3. *-rgt-identity 97.7%

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

   4. associate-*r/ 98.0%

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

   5. *-rgt-identity 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in u around 0 47.8%

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

6. Final simplification 47.8%

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

Reproduce

herbie shell --seed 2024046 
(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)))