Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 98.0%

Time: 15.0s

Alternatives: 6

Speedup: 60.1×

Specification

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0\_i \land n0\_i \leq 1\right)\right) \land \left(-1 \leq n1\_i \land n1\_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.6%

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

   1. fma-define 95.6%

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

   2. associate-*r/ 96.0%

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

   3. *-rgt-identity 96.0%

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

   4. associate-*r/ 96.4%

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

   5. *-rgt-identity 96.4%

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

3. Simplified 96.4%

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

5. Taylor expanded in normAngle around 0 99.0%

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

Alternative 2: 70.2% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.99999996490334 \cdot 10^{-13} \lor \neg \left(n1\_i \leq 4.999999980020986 \cdot 10^{-13}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.99999996490334e-13) (not (<= n1_i 4.999999980020986e-13)))
   (* u n1_i)
   (* (- 1.0 u) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.99999996490334e-13f) || !(n1_i <= 4.999999980020986e-13f)) {
		tmp = u * n1_i;
	} else {
		tmp = (1.0f - 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 <= (-1.99999996490334e-13)) .or. (.not. (n1_i <= 4.999999980020986e-13))) then
        tmp = u * n1_i
    else
        tmp = (1.0e0 - 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(-1.99999996490334e-13)) || !(n1_i <= Float32(4.999999980020986e-13)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(Float32(Float32(1.0) - 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(-1.99999996490334e-13)) || ~((n1_i <= single(4.999999980020986e-13))))
		tmp = u * n1_i;
	else
		tmp = (single(1.0) - u) * n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.99999996e-13 or 4.99999998e-13 < n1_i

   1. Initial program 92.9%

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

      1. fma-define 92.8%

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

      2. associate-*r/ 93.1%

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

      3. *-rgt-identity 93.1%

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

      4. associate-*r/ 93.1%

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

      5. *-rgt-identity 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in normAngle around 0 99.5%

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

   6. Taylor expanded in u around inf 67.6%

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

      1. *-commutative 67.6%

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

   8. Simplified 67.6%

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

   if -1.99999996e-13 < n1_i < 4.99999998e-13

   1. Initial program 97.1%

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

      1. fma-define 97.1%

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

      2. associate-*r/ 97.5%

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

      3. *-rgt-identity 97.5%

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

      4. associate-*r/ 98.2%

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

      5. *-rgt-identity 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in normAngle around 0 98.7%

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

   6. Taylor expanded in normAngle around inf 74.7%

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

      1. +-commutative 74.7%

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

      2. *-commutative 74.7%

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

      3. *-commutative 74.7%

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

      4. associate-*l/ 98.7%

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

      5. *-commutative 98.7%

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

      6. associate-*l/ 74.7%

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

      7. associate-*r/ 98.5%

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

      8. fma-define 98.6%

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

      9. *-commutative 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in normAngle around 0 98.1%

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

       1. *-commutative 98.1%

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

       2. sub-neg 98.1%

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

       3. distribute-lft-out 98.3%

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

       4. *-rgt-identity 98.3%

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

       5. associate-+l+ 98.4%

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

       6. *-commutative 98.4%

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

       7. distribute-lft-neg-out 98.4%

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

       8. distribute-rgt-neg-in 98.4%

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

       9. distribute-lft-in 98.4%

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

       10. +-commutative 98.4%

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

       11. unsub-neg 98.4%

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

   11. Simplified 98.4%

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

   12. Taylor expanded in n0_i around inf 76.9%

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

       1. mul-1-neg 76.9%

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

       2. sub-neg 76.9%

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

   14. Simplified 76.9%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.99999996490334 \cdot 10^{-13} \lor \neg \left(n1\_i \leq 4.999999980020986 \cdot 10^{-13}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.5% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.2000000467046398 \cdot 10^{-15} \lor \neg \left(n1\_i \leq 1.000000045813705 \cdot 10^{-18}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.2000000467046398e-15)
         (not (<= n1_i 1.000000045813705e-18)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.2000000467046398e-15f) || !(n1_i <= 1.000000045813705e-18f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-1.2000000467046398e-15)) .or. (.not. (n1_i <= 1.000000045813705e-18))) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-1.2000000467046398e-15)) || !(n1_i <= Float32(1.000000045813705e-18)))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-1.2000000467046398e-15)) || ~((n1_i <= single(1.000000045813705e-18))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.2e-15 or 1.00000005e-18 < n1_i

   1. Initial program 93.7%

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

      1. fma-define 93.7%

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

      2. associate-*r/ 93.9%

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

      3. *-rgt-identity 93.9%

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

      4. associate-*r/ 94.1%

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

      5. *-rgt-identity 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in normAngle around 0 98.7%

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

   6. Taylor expanded in u around inf 62.9%

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

      1. *-commutative 62.9%

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

   8. Simplified 62.9%

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

   if -1.2e-15 < n1_i < 1.00000005e-18

   1. Initial program 97.1%

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

      1. fma-define 97.1%

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

      2. associate-*r/ 97.5%

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

      3. *-rgt-identity 97.5%

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

      4. associate-*r/ 98.2%

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

      5. *-rgt-identity 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1\_i\right)} \]
   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 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.2000000467046398 \cdot 10^{-15} \lor \neg \left(n1\_i \leq 1.000000045813705 \cdot 10^{-18}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 95.6%

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

   1. fma-define 95.6%

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

   2. associate-*r/ 96.0%

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

   3. *-rgt-identity 96.0%

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

   4. associate-*r/ 96.4%

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

   5. *-rgt-identity 96.4%

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

3. Simplified 96.4%

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

5. Taylor expanded in normAngle around 0 99.0%

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

6. Taylor expanded in normAngle around inf 81.3%

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

   1. +-commutative 81.3%

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

   2. *-commutative 81.3%

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

   3. *-commutative 81.3%

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

   4. associate-*l/ 99.0%

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

   5. *-commutative 99.0%

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

   6. associate-*l/ 81.3%

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

   7. associate-*r/ 98.8%

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

   8. fma-define 98.8%

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

   9. *-commutative 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in normAngle around 0 98.6%

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

    1. *-commutative 98.6%

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

    2. sub-neg 98.6%

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

    3. distribute-lft-out 98.8%

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

    4. *-rgt-identity 98.8%

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

    5. associate-+l+ 98.9%

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

    6. *-commutative 98.9%

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

    7. distribute-lft-neg-out 98.9%

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

    8. distribute-rgt-neg-in 98.9%

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

    9. distribute-lft-in 98.9%

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

    10. +-commutative 98.9%

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

    11. unsub-neg 98.9%

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

11. Simplified 98.9%

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

Alternative 5: 81.7% accurate, 84.2× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot n1\_i \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (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 = n0_i + (u * n1_i)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 95.6%

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

   1. fma-define 95.6%

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

   2. associate-*r/ 96.0%

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

   3. *-rgt-identity 96.0%

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

   4. associate-*r/ 96.4%

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

   5. *-rgt-identity 96.4%

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

3. Simplified 96.4%

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

5. Taylor expanded in normAngle around 0 99.0%

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

6. Taylor expanded in normAngle around inf 81.3%

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

   1. +-commutative 81.3%

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

   2. *-commutative 81.3%

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

   3. *-commutative 81.3%

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

   4. associate-*l/ 99.0%

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

   5. *-commutative 99.0%

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

   6. associate-*l/ 81.3%

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

   7. associate-*r/ 98.8%

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

   8. fma-define 98.8%

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

   9. *-commutative 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in normAngle around 0 98.6%

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

    1. *-commutative 98.6%

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

    2. sub-neg 98.6%

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

    3. distribute-lft-out 98.8%

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

    4. *-rgt-identity 98.8%

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

    5. associate-+l+ 98.9%

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

    6. *-commutative 98.9%

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

    7. distribute-lft-neg-out 98.9%

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

    8. distribute-rgt-neg-in 98.9%

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

    9. distribute-lft-in 98.9%

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

    10. +-commutative 98.9%

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

    11. unsub-neg 98.9%

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

11. Simplified 98.9%

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

12. Taylor expanded in n1_i around inf 82.4%

    \[\leadsto n0\_i + \color{blue}{n1\_i \cdot u} \]
13. Step-by-step derivation

    1. *-commutative 82.4%

       \[\leadsto n0\_i + \color{blue}{u \cdot n1\_i} \]

14. Simplified 82.4%

    \[\leadsto n0\_i + \color{blue}{u \cdot n1\_i} \]
15. Add Preprocessing

Alternative 6: 47.3% 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 95.6%

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

   1. fma-define 95.6%

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

   2. associate-*r/ 96.0%

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

   3. *-rgt-identity 96.0%

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

   4. associate-*r/ 96.4%

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

   5. *-rgt-identity 96.4%

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

3. Simplified 96.4%

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

5. Taylor expanded in u around 0 47.3%

   \[\leadsto \color{blue}{n0\_i} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024090 
(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)))