Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.1%

Time: 16.2s

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: 99.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * ((n1_i * (normAngle / sin(normAngle))) - 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. +-commutative 97.5%

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

   2. *-commutative 97.5%

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

   3. cancel-sign-sub 97.5%

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

   4. *-commutative 97.5%

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

   5. associate-*r/ 97.7%

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

   6. *-rgt-identity 97.7%

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

   7. associate-*r/ 88.4%

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

   8. associate-*r* 76.5%

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

   9. associate-*r/ 76.5%

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

   10. *-rgt-identity 76.5%

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

   11. div-sub 76.6%

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

3. Simplified 76.6%

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

5. Taylor expanded in u around 0 89.6%

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

   1. +-commutative 89.6%

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

   2. mul-1-neg 89.6%

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

   3. unsub-neg 89.6%

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

   4. associate-/l* 96.0%

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

   5. associate-/l* 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in normAngle around 0 99.3%

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

9. Final simplification 99.3%

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

Alternative 2: 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. +-commutative 95.4%

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

      2. *-commutative 95.4%

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

      3. cancel-sign-sub 95.4%

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

      4. *-commutative 95.4%

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

      5. associate-*r/ 95.8%

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

      6. *-rgt-identity 95.8%

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

      7. associate-*r/ 87.4%

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

      8. associate-*r* 85.4%

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

      9. associate-*r/ 85.5%

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

      10. *-rgt-identity 85.5%

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

      11. div-sub 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in normAngle around 0 95.8%

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

   6. Taylor expanded in n0_i around 0 72.5%

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

      1. *-commutative 72.5%

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

   8. 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. +-commutative 98.8%

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

      2. *-commutative 98.8%

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

      3. cancel-sign-sub 98.8%

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

      4. *-commutative 98.8%

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

      5. associate-*r/ 98.9%

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

      6. *-rgt-identity 98.9%

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

      7. associate-*r/ 89.1%

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

      8. associate-*r* 70.7%

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

      9. associate-*r/ 70.7%

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

      10. *-rgt-identity 70.7%

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

      11. div-sub 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in u around 0 87.9%

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

      1. +-commutative 87.9%

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

      2. mul-1-neg 87.9%

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

      3. unsub-neg 87.9%

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

      4. associate-/l* 94.1%

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

      5. associate-/l* 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in normAngle around 0 99.3%

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

   9. Taylor expanded in n0_i around inf 78.8%

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

       1. neg-mul-1 78.8%

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

       2. sub-neg 78.8%

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

   11. Simplified 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 3: 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. +-commutative 95.4%

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

      2. *-commutative 95.4%

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

      3. cancel-sign-sub 95.4%

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

      4. *-commutative 95.4%

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

      5. associate-*r/ 95.8%

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

      6. *-rgt-identity 95.8%

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

      7. associate-*r/ 87.4%

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

      8. associate-*r* 85.4%

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

      9. associate-*r/ 85.5%

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

      10. *-rgt-identity 85.5%

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

      11. div-sub 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in normAngle around 0 95.8%

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

   6. Taylor expanded in n0_i around 0 72.5%

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

      1. *-commutative 72.5%

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

   8. 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. +-commutative 98.8%

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

      2. *-commutative 98.8%

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

      3. cancel-sign-sub 98.8%

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

      4. *-commutative 98.8%

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

      5. associate-*r/ 98.9%

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

      6. *-rgt-identity 98.9%

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

      7. associate-*r/ 89.1%

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

      8. associate-*r* 70.7%

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

      9. associate-*r/ 70.7%

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

      10. *-rgt-identity 70.7%

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

      11. div-sub 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in u around 0 87.9%

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

      1. +-commutative 87.9%

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

      2. mul-1-neg 87.9%

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

      3. unsub-neg 87.9%

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

      4. associate-/l* 94.1%

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

      5. associate-/l* 99.3%

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

   7. Simplified 99.3%

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

   8. 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 4: 84.0% accurate, 42.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -9.9999998245167e-14) (- n0_i (* n0_i u)) (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -9.9999998245167e-14f) {
		tmp = n0_i - (n0_i * u);
	} else {
		tmp = n0_i + (u * n1_i);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-9.9999998245167e-14)) then
        tmp = n0_i - (n0_i * u)
    else
        tmp = n0_i + (u * n1_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-9.9999998245167e-14))
		tmp = Float32(n0_i - Float32(n0_i * u));
	else
		tmp = Float32(n0_i + Float32(u * n1_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-9.9999998245167e-14))
		tmp = n0_i - (n0_i * u);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -9.99999982e-14

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. *-commutative 98.7%

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

      3. cancel-sign-sub 98.7%

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

      4. *-commutative 98.7%

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

      5. associate-*r/ 98.7%

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

      6. *-rgt-identity 98.7%

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

      7. associate-*r/ 98.7%

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

      8. associate-*r* 90.5%

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

      9. associate-*r/ 90.4%

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

      10. *-rgt-identity 90.4%

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

      11. div-sub 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in normAngle around 0 97.9%

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

   6. Taylor expanded in u around -inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. unsub-neg 98.3%

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

      3. mul-1-neg 98.3%

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

      4. unsub-neg 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in n0_i around inf 94.6%

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

       1. *-commutative 94.6%

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

   11. Simplified 94.6%

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

   if -9.99999982e-14 < n0_i

   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. fma-define 97.2%

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

      2. associate-*l* 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in u around 0 86.4%

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

   6. Taylor expanded in normAngle around 0 86.3%

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

      1. *-commutative 86.3%

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

   8. Simplified 86.3%

      \[\leadsto \color{blue}{n0\_i + u \cdot n1\_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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. Step-by-step derivation

   1. +-commutative 97.5%

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

   2. *-commutative 97.5%

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

   3. cancel-sign-sub 97.5%

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

   4. *-commutative 97.5%

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

   5. associate-*r/ 97.7%

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

   6. *-rgt-identity 97.7%

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

   7. associate-*r/ 88.4%

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

   8. associate-*r* 76.5%

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

   9. associate-*r/ 76.5%

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

   10. *-rgt-identity 76.5%

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

   11. div-sub 76.6%

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

3. Simplified 76.6%

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

5. Taylor expanded in normAngle around 0 97.5%

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

6. Taylor expanded in u around -inf 97.7%

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

   1. mul-1-neg 97.7%

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

   2. unsub-neg 97.7%

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

   3. mul-1-neg 97.7%

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

   4. unsub-neg 97.7%

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

8. Simplified 97.7%

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

9. Final simplification 97.7%

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

Alternative 6: 81.6% 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 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-*l* 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in u around 0 84.9%

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

6. Taylor expanded in normAngle around 0 84.7%

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

   1. *-commutative 84.7%

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

8. Simplified 84.7%

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

9. Final simplification 84.7%

   \[\leadsto n0\_i + u \cdot n1\_i \]
10. 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. +-commutative 97.5%

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

   2. *-commutative 97.5%

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

   3. cancel-sign-sub 97.5%

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

   4. *-commutative 97.5%

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

   5. associate-*r/ 97.7%

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

   6. *-rgt-identity 97.7%

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

   7. associate-*r/ 88.4%

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

   8. associate-*r* 76.5%

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

   9. associate-*r/ 76.5%

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

   10. *-rgt-identity 76.5%

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

   11. div-sub 76.6%

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

3. Simplified 76.6%

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

5. Taylor expanded in u around 0 89.6%

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

   1. +-commutative 89.6%

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

   2. mul-1-neg 89.6%

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

   3. unsub-neg 89.6%

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

   4. associate-/l* 96.0%

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

   5. associate-/l* 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in u around 0 47.8%

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

9. Final simplification 47.8%

   \[\leadsto n0\_i \]
10. 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)))