Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 98.8%

Time: 17.1s

Alternatives: 8

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.3% 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: 98.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 * ((n1_i * ((normangle / sin(normangle)) * (u / n0_i))) + (sin((normangle * (1.0e0 - u))) / sin(normangle)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.8%

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

   1. *-commutative 96.8%

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

   2. associate-*l* 82.6%

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

   3. *-commutative 82.6%

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

   4. associate-*l* 76.5%

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

   5. distribute-lft-out 76.6%

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

3. Simplified 76.6%

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

5. Taylor expanded in u around 0 76.0%

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

   1. *-commutative 76.0%

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

7. Simplified 76.0%

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

8. Taylor expanded in n0_i around inf 64.9%

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

   1. +-commutative 64.9%

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

   2. associate-/l* 65.8%

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

   3. *-commutative 65.8%

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

   4. times-frac 98.3%

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

10. Simplified 98.3%

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

Alternative 2: 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 96.8%

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

3. Taylor expanded in normAngle around 0 95.8%

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

   1. div-inv 96.4%

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

   2. associate-*l/ 87.1%

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

5. Applied egg-rr 87.1%

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

6. Taylor expanded in u around 0 92.3%

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

   1. neg-mul-1 92.3%

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

   2. +-commutative 92.3%

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

   3. unsub-neg 92.3%

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

   4. associate-/l* 97.9%

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

8. Simplified 97.9%

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

Alternative 3: 86.3% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -4.999999999099794 \cdot 10^{-24} \lor \neg \left(n1\_i \leq 2.0000000390829628 \cdot 10^{-25}\right):\\ \;\;\;\;n0\_i + n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -4.999999999099794e-24)
         (not (<= n1_i 2.0000000390829628e-25)))
   (+ n0_i (* n1_i u))
   (- n0_i (* n0_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -4.999999999099794e-24f) || !(n1_i <= 2.0000000390829628e-25f)) {
		tmp = n0_i + (n1_i * u);
	} else {
		tmp = n0_i - (n0_i * 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 <= (-4.999999999099794e-24)) .or. (.not. (n1_i <= 2.0000000390829628e-25))) then
        tmp = n0_i + (n1_i * u)
    else
        tmp = n0_i - (n0_i * u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-4.999999999099794e-24)) || !(n1_i <= Float32(2.0000000390829628e-25)))
		tmp = Float32(n0_i + Float32(n1_i * u));
	else
		tmp = Float32(n0_i - Float32(n0_i * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-4.999999999099794e-24)) || ~((n1_i <= single(2.0000000390829628e-25))))
		tmp = n0_i + (n1_i * u);
	else
		tmp = n0_i - (n0_i * u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -5e-24 or 2.00000004e-25 < n1_i

   1. Initial program 96.3%

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

   3. Taylor expanded in u around 0 89.8%

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

      1. +-commutative 89.8%

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

      2. mul-1-neg 89.8%

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

      3. unsub-neg 89.8%

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

      4. associate-/l* 97.3%

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

      5. associate-*r* 97.3%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in n1_i around inf 80.2%

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

   7. Taylor expanded in normAngle around 0 85.8%

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

   if -5e-24 < n1_i < 2.00000004e-25

   1. Initial program 97.4%

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

   3. Taylor expanded in n0_i around inf 66.3%

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

      1. associate-/l* 91.7%

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

   5. Simplified 91.7%

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

   6. Taylor expanded in normAngle around 0 90.3%

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

   7. Taylor expanded in u around 0 90.7%

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

      1. associate-*r* 90.7%

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

      2. neg-mul-1 90.7%

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

   9. Simplified 90.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -4.999999999099794 \cdot 10^{-24} \lor \neg \left(n1\_i \leq 2.0000000390829628 \cdot 10^{-25}\right):\\ \;\;\;\;n0\_i + n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -4.999999999099794 \cdot 10^{-24} \lor \neg \left(n1\_i \leq 2.0000000390829628 \cdot 10^{-25}\right):\\ \;\;\;\;n0\_i + n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -4.999999999099794e-24)
         (not (<= n1_i 2.0000000390829628e-25)))
   (+ n0_i (* n1_i u))
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -4.999999999099794e-24f) || !(n1_i <= 2.0000000390829628e-25f)) {
		tmp = n0_i + (n1_i * u);
	} 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 <= (-4.999999999099794e-24)) .or. (.not. (n1_i <= 2.0000000390829628e-25))) then
        tmp = n0_i + (n1_i * u)
    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(-4.999999999099794e-24)) || !(n1_i <= Float32(2.0000000390829628e-25)))
		tmp = Float32(n0_i + Float32(n1_i * u));
	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(-4.999999999099794e-24)) || ~((n1_i <= single(2.0000000390829628e-25))))
		tmp = n0_i + (n1_i * u);
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -5e-24 or 2.00000004e-25 < n1_i

   1. Initial program 96.3%

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

   3. Taylor expanded in u around 0 89.8%

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

      1. +-commutative 89.8%

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

      2. mul-1-neg 89.8%

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

      3. unsub-neg 89.8%

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

      4. associate-/l* 97.3%

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

      5. associate-*r* 97.3%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in n1_i around inf 80.2%

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

   7. Taylor expanded in normAngle around 0 85.8%

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

   if -5e-24 < n1_i < 2.00000004e-25

   1. Initial program 97.4%

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

   3. Taylor expanded in n0_i around inf 66.3%

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

      1. associate-/l* 91.7%

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

   5. Simplified 91.7%

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

   6. Taylor expanded in normAngle around 0 90.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -4.999999999099794 \cdot 10^{-24} \lor \neg \left(n1\_i \leq 2.0000000390829628 \cdot 10^{-25}\right):\\ \;\;\;\;n0\_i + n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.5% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.999999936531045 \cdot 10^{-21} \lor \neg \left(n1\_i \leq 4.00000018325482 \cdot 10^{-18}\right):\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.999999936531045e-21) (not (<= n1_i 4.00000018325482e-18)))
   (* n1_i u)
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.999999936531045e-21f) || !(n1_i <= 4.00000018325482e-18f)) {
		tmp = n1_i * u;
	} 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 <= (-1.999999936531045e-21)) .or. (.not. (n1_i <= 4.00000018325482e-18))) then
        tmp = n1_i * u
    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(-1.999999936531045e-21)) || !(n1_i <= Float32(4.00000018325482e-18)))
		tmp = Float32(n1_i * u);
	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(-1.999999936531045e-21)) || ~((n1_i <= single(4.00000018325482e-18))))
		tmp = n1_i * u;
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.9999999e-21 or 4.00000018e-18 < n1_i

   1. Initial program 95.9%

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

      1. *-commutative 95.9%

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

      2. associate-*l* 92.5%

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

      3. *-commutative 92.5%

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

      4. associate-*l* 81.9%

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

      5. distribute-lft-out 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in u around 0 81.0%

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

      1. *-commutative 81.0%

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

   7. Simplified 81.0%

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

   8. Taylor expanded in n0_i around inf 58.9%

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

      1. +-commutative 58.9%

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

      2. associate-/l* 59.5%

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

      3. *-commutative 59.5%

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

      4. times-frac 97.9%

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

   10. Simplified 97.9%

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

   11. Taylor expanded in normAngle around 0 95.6%

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

   12. Taylor expanded in n0_i around 0 62.3%

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

       1. *-commutative 62.3%

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

   14. Simplified 62.3%

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

   if -1.9999999e-21 < n1_i < 4.00000018e-18

   1. Initial program 97.6%

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

   3. Taylor expanded in n0_i around inf 63.0%

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

      1. associate-/l* 85.2%

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

   5. Simplified 85.2%

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

   6. Taylor expanded in normAngle around 0 84.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.999999936531045 \cdot 10^{-21} \lor \neg \left(n1\_i \leq 4.00000018325482 \cdot 10^{-18}\right):\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.9% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.1999999840067324 \cdot 10^{-23} \lor \neg \left(n1\_i \leq 9.999999682655225 \cdot 10^{-22}\right):\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.1999999840067324e-23)
         (not (<= n1_i 9.999999682655225e-22)))
   (* n1_i u)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.1999999840067324e-23f) || !(n1_i <= 9.999999682655225e-22f)) {
		tmp = n1_i * u;
	} 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 <= (-1.1999999840067324e-23)) .or. (.not. (n1_i <= 9.999999682655225e-22))) then
        tmp = n1_i * u
    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(-1.1999999840067324e-23)) || !(n1_i <= Float32(9.999999682655225e-22)))
		tmp = Float32(n1_i * u);
	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(-1.1999999840067324e-23)) || ~((n1_i <= single(9.999999682655225e-22))))
		tmp = n1_i * u;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.19999998e-23 or 9.9999997e-22 < n1_i

   1. Initial program 96.1%

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

      1. *-commutative 96.1%

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

      2. associate-*l* 92.0%

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

      3. *-commutative 92.0%

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

      4. associate-*l* 81.9%

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

      5. distribute-lft-out 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in u around 0 81.1%

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

      1. *-commutative 81.1%

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

   7. Simplified 81.1%

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

   8. Taylor expanded in n0_i around inf 59.1%

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

      1. +-commutative 59.1%

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

      2. associate-/l* 59.8%

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

      3. *-commutative 59.8%

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

      4. times-frac 97.9%

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

   10. Simplified 97.9%

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

   11. Taylor expanded in normAngle around 0 95.5%

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

   12. Taylor expanded in n0_i around 0 61.1%

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

       1. *-commutative 61.1%

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

   14. Simplified 61.1%

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

   if -1.19999998e-23 < n1_i < 9.9999997e-22

   1. Initial program 97.5%

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

   3. Taylor expanded in u around 0 65.5%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.1999999840067324 \cdot 10^{-23} \lor \neg \left(n1\_i \leq 9.999999682655225 \cdot 10^{-22}\right):\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 96.8%

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

3. Taylor expanded in u around 0 89.3%

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

   1. +-commutative 89.3%

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

   2. mul-1-neg 89.3%

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

   3. unsub-neg 89.3%

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

   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-*r* 94.1%

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

5. Simplified 94.1%

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

6. Taylor expanded in normAngle around 0 96.3%

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

Alternative 8: 47.2% 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 96.8%

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

3. Taylor expanded in u around 0 44.6%

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

Reproduce

herbie shell --seed 2024166 
(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)))