Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 98.2%

Time: 14.0s

Alternatives: 10

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: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(u + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1\_i \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* (* (sin (* (- 1.0 u) normAngle)) (/ 1.0 (sin normAngle))) n0_i)
  (*
   (+ u (* (pow normAngle 2.0) (* -0.16666666666666666 (- (pow u 3.0) u))))
   n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((sinf(((1.0f - u) * normAngle)) * (1.0f / sinf(normAngle))) * n0_i) + ((u + (powf(normAngle, 2.0f) * (-0.16666666666666666f * (powf(u, 3.0f) - 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 = ((sin(((1.0e0 - u) * normangle)) * (1.0e0 / sin(normangle))) * n0_i) + ((u + ((normangle ** 2.0e0) * ((-0.16666666666666666e0) * ((u ** 3.0e0) - u)))) * n1_i)
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * Float32(Float32(1.0) / sin(normAngle))) * n0_i) + Float32(Float32(u + Float32((normAngle ^ Float32(2.0)) * Float32(Float32(-0.16666666666666666) * Float32((u ^ Float32(3.0)) - u)))) * n1_i))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = ((sin(((single(1.0) - u) * normAngle)) * (single(1.0) / sin(normAngle))) * n0_i) + ((u + ((normAngle ^ single(2.0)) * (single(-0.16666666666666666) * ((u ^ single(3.0)) - u)))) * n1_i);
end

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in normAngle around 0 98.9%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\left(u + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {u}^{3} - -0.16666666666666666 \cdot u\right)\right)} \cdot n1\_i \]
4. Step-by-step derivation

   1. distribute-lft-out-- 98.9%

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(u + {normAngle}^{2} \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)}\right) \cdot n1\_i \]

5. Simplified 98.9%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\left(u + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right)} \cdot n1\_i \]
6. Add Preprocessing

Alternative 2: 98.7% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \left(n0\_i \cdot 0.3333333333333333 - -0.16666666666666666 \cdot n1\_i\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (+
   (* u (- n1_i n0_i))
   (*
    (* normAngle normAngle)
    (* u (- (* n0_i 0.3333333333333333) (* -0.16666666666666666 n1_i)))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + ((u * (n1_i - n0_i)) + ((normAngle * normAngle) * (u * ((n0_i * 0.3333333333333333f) - (-0.16666666666666666f * 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 - n0_i)) + ((normangle * normangle) * (u * ((n0_i * 0.3333333333333333e0) - ((-0.16666666666666666e0) * n1_i)))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(Float32(u * Float32(n1_i - n0_i)) + Float32(Float32(normAngle * normAngle) * Float32(u * Float32(Float32(n0_i * Float32(0.3333333333333333)) - Float32(Float32(-0.16666666666666666) * n1_i))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + ((u * (n1_i - n0_i)) + ((normAngle * normAngle) * (u * ((n0_i * single(0.3333333333333333)) - (single(-0.16666666666666666) * n1_i)))));
end

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in u around 0 84.5%

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

   1. +-commutative 84.5%

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

   2. mul-1-neg 84.5%

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

   3. unsub-neg 84.5%

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

   4. associate-/l* 93.6%

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

   5. associate-*r* 93.6%

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

5. Simplified 93.6%

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

6. Taylor expanded in normAngle around 0 98.9%

   \[\leadsto n0\_i + \color{blue}{\left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. unpow2 98.9%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right)\right)\right) \]

8. Applied egg-rr 98.9%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right)\right)\right) \]

9. Taylor expanded in n0_i around 0 98.9%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \color{blue}{\left(0.3333333333333333 \cdot n0\_i - -0.16666666666666666 \cdot n1\_i\right)}\right)\right) \]

10. Final simplification 98.9%

    \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \left(n0\_i \cdot 0.3333333333333333 - -0.16666666666666666 \cdot n1\_i\right)\right)\right) \]
11. Add Preprocessing

Alternative 3: 98.6% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in u around 0 84.5%

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

   1. +-commutative 84.5%

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

   2. mul-1-neg 84.5%

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

   3. unsub-neg 84.5%

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

   4. associate-/l* 93.6%

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

   5. associate-*r* 93.6%

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

5. Simplified 93.6%

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

6. Taylor expanded in normAngle around 0 98.9%

   \[\leadsto n0\_i + \color{blue}{\left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. unpow2 98.9%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right)\right)\right) \]

8. Applied egg-rr 98.9%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right)\right)\right) \]

9. Taylor expanded in n0_i around 0 98.8%

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

    1. *-commutative 98.8%

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

    2. *-commutative 98.8%

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

11. Simplified 98.8%

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

12. Final simplification 98.8%

    \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(0.16666666666666666 \cdot \left(u \cdot n1\_i\right)\right)\right) \]
13. Add Preprocessing

Alternative 4: 98.0% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \left(n0\_i \cdot 0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (+
   (* u (- n1_i n0_i))
   (* (* normAngle normAngle) (* u (* n0_i 0.3333333333333333))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + ((u * (n1_i - n0_i)) + ((normAngle * normAngle) * (u * (n0_i * 0.3333333333333333f))));
}

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)) + ((normangle * normangle) * (u * (n0_i * 0.3333333333333333e0))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(Float32(u * Float32(n1_i - n0_i)) + Float32(Float32(normAngle * normAngle) * Float32(u * Float32(n0_i * Float32(0.3333333333333333))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + ((u * (n1_i - n0_i)) + ((normAngle * normAngle) * (u * (n0_i * single(0.3333333333333333)))));
end

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in u around 0 84.5%

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

   1. +-commutative 84.5%

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

   2. mul-1-neg 84.5%

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

   3. unsub-neg 84.5%

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

   4. associate-/l* 93.6%

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

   5. associate-*r* 93.6%

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

5. Simplified 93.6%

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

6. Taylor expanded in normAngle around 0 98.9%

   \[\leadsto n0\_i + \color{blue}{\left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. unpow2 98.9%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right)\right)\right) \]

8. Applied egg-rr 98.9%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right)\right)\right) \]

9. Taylor expanded in n0_i around inf 98.3%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \color{blue}{\left(0.3333333333333333 \cdot n0\_i\right)}\right)\right) \]
10. Step-by-step derivation

    1. *-commutative 98.3%

       \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \color{blue}{\left(n0\_i \cdot 0.3333333333333333\right)}\right)\right) \]

11. Simplified 98.3%

    \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot \color{blue}{\left(n0\_i \cdot 0.3333333333333333\right)}\right)\right) \]
12. Add Preprocessing

Alternative 5: 71.6% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.0000000390829628 \cdot 10^{-24} \lor \neg \left(n0\_i \leq 1.0000000272452012 \cdot 10^{-27}\right):\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -2.0000000390829628e-24)
         (not (<= n0_i 1.0000000272452012e-27)))
   (* (- 1.0 u) n0_i)
   (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -2.0000000390829628e-24f) || !(n0_i <= 1.0000000272452012e-27f)) {
		tmp = (1.0f - u) * n0_i;
	} else {
		tmp = u * n1_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n0_i <= (-2.0000000390829628e-24)) .or. (.not. (n0_i <= 1.0000000272452012e-27))) then
        tmp = (1.0e0 - u) * n0_i
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-2.0000000390829628e-24)) || !(n0_i <= Float32(1.0000000272452012e-27)))
		tmp = Float32(Float32(Float32(1.0) - u) * n0_i);
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-2.0000000390829628e-24)) || ~((n0_i <= single(1.0000000272452012e-27))))
		tmp = (single(1.0) - u) * n0_i;
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -2.00000004e-24 or 1.00000003e-27 < n0_i

   1. Initial program 97.1%

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

   3. Taylor expanded in u around 0 90.1%

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

      1. +-commutative 90.1%

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

      2. mul-1-neg 90.1%

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

      3. unsub-neg 90.1%

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

      4. associate-/l* 92.3%

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

      5. associate-*r* 92.3%

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

   5. Simplified 92.3%

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

   6. Taylor expanded in normAngle around 0 98.6%

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

   7. Taylor expanded in n0_i around inf 75.5%

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

      1. mul-1-neg 75.5%

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

      2. unsub-neg 75.5%

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

   9. Simplified 75.5%

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

   if -2.00000004e-24 < n0_i < 1.00000003e-27

   1. Initial program 96.6%

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

   3. Taylor expanded in n0_i around 0 53.9%

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

      1. associate-/l* 73.0%

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

      2. *-commutative 73.0%

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

   5. Simplified 73.0%

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

   6. Taylor expanded in normAngle around 0 73.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.0000000390829628 \cdot 10^{-24} \lor \neg \left(n0\_i \leq 1.0000000272452012 \cdot 10^{-27}\right):\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.2% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.4500000211417337 \cdot 10^{-15} \lor \neg \left(n1\_i \leq 5.999999920033662 \cdot 10^{-24}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.4500000211417337e-15)
         (not (<= n1_i 5.999999920033662e-24)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.4500000211417337e-15f) || !(n1_i <= 5.999999920033662e-24f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-1.4500000211417337e-15)) .or. (.not. (n1_i <= 5.999999920033662e-24))) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-1.4500000211417337e-15)) || !(n1_i <= Float32(5.999999920033662e-24)))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-1.4500000211417337e-15)) || ~((n1_i <= single(5.999999920033662e-24))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.45000002e-15 or 5.99999992e-24 < n1_i

   1. Initial program 96.5%

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

   3. Taylor expanded in n0_i around 0 56.7%

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

      1. associate-/l* 67.5%

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

      2. *-commutative 67.5%

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

   5. Simplified 67.5%

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

   6. Taylor expanded in normAngle around 0 67.7%

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

   if -1.45000002e-15 < n1_i < 5.99999992e-24

   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. Add Preprocessing

   3. Taylor expanded in u around 0 63.0%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.4500000211417337 \cdot 10^{-15} \lor \neg \left(n1\_i \leq 5.999999920033662 \cdot 10^{-24}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 46.8× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((1.0f - u) * 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 = ((1.0e0 - u) * n0_i) + (u * n1_i)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in normAngle around 0 98.2%

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

4. Final simplification 98.2%

   \[\leadsto \left(1 - u\right) \cdot n0\_i + u \cdot n1\_i \]
5. Add Preprocessing

Alternative 8: 97.9% 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 96.9%

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

3. Taylor expanded in u around 0 84.5%

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

   1. +-commutative 84.5%

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

   2. mul-1-neg 84.5%

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

   3. unsub-neg 84.5%

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

   4. associate-/l* 93.6%

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

   5. associate-*r* 93.6%

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

5. Simplified 93.6%

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

6. Taylor expanded in normAngle around 0 98.2%

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

Alternative 9: 81.7% accurate, 84.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in u around 0 84.5%

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

   1. +-commutative 84.5%

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

   2. mul-1-neg 84.5%

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

   3. unsub-neg 84.5%

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

   4. associate-/l* 93.6%

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

   5. associate-*r* 93.6%

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

5. Simplified 93.6%

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

6. Taylor expanded in normAngle around 0 98.2%

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

7. Taylor expanded in n1_i around inf 83.9%

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

   1. *-commutative 83.9%

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

9. Simplified 83.9%

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

Alternative 10: 47.7% 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 96.9%

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

3. Taylor expanded in u around 0 45.2%

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

Reproduce

herbie shell --seed 2024149 
(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)))