Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 98.9%

Time: 15.7s

Alternatives: 6

Speedup: 60.1×

Specification

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t_0\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot t_0\right) \cdot n1_i \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t_0\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot t_0\right) \cdot n1_i \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ n0_i - \left(u \cdot \left(n0_i - n1_i\right) - {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (-
  n0_i
  (-
   (* u (- n0_i n1_i))
   (*
    (pow normAngle 2.0)
    (* u (+ (* n0_i 0.3333333333333333) (* n1_i 0.16666666666666666)))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i - ((u * (n0_i - n1_i)) - (powf(normAngle, 2.0f) * (u * ((n0_i * 0.3333333333333333f) + (n1_i * 0.16666666666666666f)))));
}

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 * (n0_i - n1_i)) - ((normangle ** 2.0e0) * (u * ((n0_i * 0.3333333333333333e0) + (n1_i * 0.16666666666666666e0)))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i - Float32(Float32(u * Float32(n0_i - n1_i)) - Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(Float32(n0_i * Float32(0.3333333333333333)) + Float32(n1_i * Float32(0.16666666666666666)))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i - ((u * (n0_i - n1_i)) - ((normAngle ^ single(2.0)) * (u * ((n0_i * single(0.3333333333333333)) + (n1_i * single(0.16666666666666666))))));
end

Derivation

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. Step-by-step derivation

   1. *-commutative 96.6%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. associate-*l* 80.1%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. *-commutative 80.1%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

   4. associate-*l* 73.5%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   5. distribute-lft-out 73.5%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Simplified 73.5%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

4. Taylor expanded in u around 0 85.9%

   \[\leadsto \color{blue}{n0_i + u \cdot \left(-1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}\right)} \]

5. Taylor expanded in normAngle around 0 98.8%

   \[\leadsto n0_i + \color{blue}{\left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-1 \cdot \left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) - -0.16666666666666666 \cdot n1_i\right)\right)\right)} \]

6. Taylor expanded in n0_i around 0 98.8%

   \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(n1_i \cdot u\right) + 0.3333333333333333 \cdot \left(n0_i \cdot u\right)\right)}\right) \]
7. Step-by-step derivation

   1. +-commutative 98.8%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(n0_i \cdot u\right) + 0.16666666666666666 \cdot \left(n1_i \cdot u\right)\right)}\right) \]

   2. associate-*r* 98.8%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(\color{blue}{\left(0.3333333333333333 \cdot n0_i\right) \cdot u} + 0.16666666666666666 \cdot \left(n1_i \cdot u\right)\right)\right) \]

   3. *-commutative 98.8%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(\color{blue}{\left(n0_i \cdot 0.3333333333333333\right)} \cdot u + 0.16666666666666666 \cdot \left(n1_i \cdot u\right)\right)\right) \]

   4. metadata-eval 98.8%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(\left(n0_i \cdot \color{blue}{\left(--0.3333333333333333\right)}\right) \cdot u + 0.16666666666666666 \cdot \left(n1_i \cdot u\right)\right)\right) \]

   5. distribute-rgt-neg-in 98.8%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(\color{blue}{\left(-n0_i \cdot -0.3333333333333333\right)} \cdot u + 0.16666666666666666 \cdot \left(n1_i \cdot u\right)\right)\right) \]

   6. associate-*r* 98.8%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(\left(-n0_i \cdot -0.3333333333333333\right) \cdot u + \color{blue}{\left(0.16666666666666666 \cdot n1_i\right) \cdot u}\right)\right) \]

   7. *-commutative 98.8%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(\left(-n0_i \cdot -0.3333333333333333\right) \cdot u + \color{blue}{\left(n1_i \cdot 0.16666666666666666\right)} \cdot u\right)\right) \]

   8. distribute-rgt-out 98.8%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(\left(-n0_i \cdot -0.3333333333333333\right) + n1_i \cdot 0.16666666666666666\right)\right)}\right) \]

   9. distribute-rgt-neg-in 98.8%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\color{blue}{n0_i \cdot \left(--0.3333333333333333\right)} + n1_i \cdot 0.16666666666666666\right)\right)\right) \]

   10. metadata-eval 98.8%

       \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot \color{blue}{0.3333333333333333} + n1_i \cdot 0.16666666666666666\right)\right)\right) \]

   11. *-commutative 98.8%

       \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + \color{blue}{0.16666666666666666 \cdot n1_i}\right)\right)\right) \]

8. Simplified 98.8%

   \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right)\right)}\right) \]

9. Taylor expanded in n1_i around 0 98.8%

   \[\leadsto n0_i + \left(\color{blue}{\left(-1 \cdot \left(n0_i \cdot u\right) + n1_i \cdot u\right)} + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right)\right)\right) \]
10. Step-by-step derivation

    1. mul-1-neg 98.8%

       \[\leadsto n0_i + \left(\left(\color{blue}{\left(-n0_i \cdot u\right)} + n1_i \cdot u\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right)\right)\right) \]

    2. +-commutative 98.8%

       \[\leadsto n0_i + \left(\color{blue}{\left(n1_i \cdot u + \left(-n0_i \cdot u\right)\right)} + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right)\right)\right) \]

    3. sub-neg 98.8%

       \[\leadsto n0_i + \left(\color{blue}{\left(n1_i \cdot u - n0_i \cdot u\right)} + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right)\right)\right) \]

    4. distribute-rgt-out-- 98.8%

       \[\leadsto n0_i + \left(\color{blue}{u \cdot \left(n1_i - n0_i\right)} + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right)\right)\right) \]

11. Simplified 98.8%

    \[\leadsto n0_i + \left(\color{blue}{u \cdot \left(n1_i - n0_i\right)} + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right)\right)\right) \]

12. Final simplification 98.8%

    \[\leadsto n0_i - \left(u \cdot \left(n0_i - n1_i\right) - {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right)\right)\right) \]

Alternative 2: 98.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ n0_i + \left({normAngle}^{2} \cdot \left(n1_i \cdot \left(u \cdot 0.16666666666666666\right)\right) - u \cdot \left(n0_i - n1_i\right)\right) \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + ((powf(normAngle, 2.0f) * (n1_i * (u * 0.16666666666666666f))) - (u * (n0_i - 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 + (((normangle ** 2.0e0) * (n1_i * (u * 0.16666666666666666e0))) - (u * (n0_i - n1_i)))
end function

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. *-commutative 96.6%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. associate-*l* 80.1%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. *-commutative 80.1%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

   4. associate-*l* 73.5%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   5. distribute-lft-out 73.5%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Simplified 73.5%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

4. Taylor expanded in u around 0 85.9%

   \[\leadsto \color{blue}{n0_i + u \cdot \left(-1 \cdot \frac{n0_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1_i \cdot normAngle}{\sin normAngle}\right)} \]

5. Taylor expanded in normAngle around 0 98.8%

   \[\leadsto n0_i + \color{blue}{\left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-1 \cdot \left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) - -0.16666666666666666 \cdot n1_i\right)\right)\right)} \]

6. Taylor expanded in n0_i around 0 98.7%

   \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(n1_i \cdot u\right)\right)}\right) \]
7. Step-by-step derivation

   1. associate-*r* 98.7%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(\left(0.16666666666666666 \cdot n1_i\right) \cdot u\right)}\right) \]

   2. *-commutative 98.7%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(\color{blue}{\left(n1_i \cdot 0.16666666666666666\right)} \cdot u\right)\right) \]

   3. associate-*l* 98.7%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(n1_i \cdot \left(0.16666666666666666 \cdot u\right)\right)}\right) \]

8. Simplified 98.7%

   \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(n1_i \cdot \left(0.16666666666666666 \cdot u\right)\right)}\right) \]

9. Taylor expanded in n1_i around 0 98.7%

   \[\leadsto n0_i + \left(\color{blue}{\left(-1 \cdot \left(n0_i \cdot u\right) + n1_i \cdot u\right)} + {normAngle}^{2} \cdot \left(n1_i \cdot \left(0.16666666666666666 \cdot u\right)\right)\right) \]
10. Step-by-step derivation

    1. mul-1-neg 98.8%

       \[\leadsto n0_i + \left(\left(\color{blue}{\left(-n0_i \cdot u\right)} + n1_i \cdot u\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right)\right)\right) \]

    2. +-commutative 98.8%

       \[\leadsto n0_i + \left(\color{blue}{\left(n1_i \cdot u + \left(-n0_i \cdot u\right)\right)} + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right)\right)\right) \]

    3. sub-neg 98.8%

       \[\leadsto n0_i + \left(\color{blue}{\left(n1_i \cdot u - n0_i \cdot u\right)} + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right)\right)\right) \]

    4. distribute-rgt-out-- 98.8%

       \[\leadsto n0_i + \left(\color{blue}{u \cdot \left(n1_i - n0_i\right)} + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + 0.16666666666666666 \cdot n1_i\right)\right)\right) \]

11. Simplified 98.7%

    \[\leadsto n0_i + \left(\color{blue}{u \cdot \left(n1_i - n0_i\right)} + {normAngle}^{2} \cdot \left(n1_i \cdot \left(0.16666666666666666 \cdot u\right)\right)\right) \]

12. Final simplification 98.7%

    \[\leadsto n0_i + \left({normAngle}^{2} \cdot \left(n1_i \cdot \left(u \cdot 0.16666666666666666\right)\right) - u \cdot \left(n0_i - n1_i\right)\right) \]

Alternative 3: 60.5% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -1.7999999681214895 \cdot 10^{-22}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.7999999681214895e-22)
   n0_i
   (if (<= n0_i 9.999999998199587e-24) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.7999999681214895e-22f) {
		tmp = n0_i;
	} else if (n0_i <= 9.999999998199587e-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 (n0_i <= (-1.7999999681214895e-22)) then
        tmp = n0_i
    else if (n0_i <= 9.999999998199587e-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 (n0_i <= Float32(-1.7999999681214895e-22))
		tmp = n0_i;
	elseif (n0_i <= Float32(9.999999998199587e-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 (n0_i <= single(-1.7999999681214895e-22))
		tmp = n0_i;
	elseif (n0_i <= single(9.999999998199587e-24))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.79999997e-22 or 1e-23 < n0_i

   1. Initial program 98.0%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. Taylor expanded in u around 0 75.9%

      \[\leadsto \color{blue}{n0_i} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. Taylor expanded in u around 0 74.3%

      \[\leadsto \color{blue}{n0_i + \frac{n1_i \cdot \left(normAngle \cdot u\right)}{\sin normAngle}} \]
   4. Step-by-step derivation

      1. associate-*r* 74.3%

         \[\leadsto n0_i + \frac{\color{blue}{\left(n1_i \cdot normAngle\right) \cdot u}}{\sin normAngle} \]

      2. associate-*l/ 76.6%

         \[\leadsto n0_i + \color{blue}{\frac{n1_i \cdot normAngle}{\sin normAngle} \cdot u} \]

      3. associate-/l* 76.8%

         \[\leadsto n0_i + \color{blue}{\frac{n1_i}{\frac{\sin normAngle}{normAngle}}} \cdot u \]

      4. *-commutative 76.8%

         \[\leadsto n0_i + \color{blue}{u \cdot \frac{n1_i}{\frac{\sin normAngle}{normAngle}}} \]

      5. associate-/l* 76.6%

         \[\leadsto n0_i + u \cdot \color{blue}{\frac{n1_i \cdot normAngle}{\sin normAngle}} \]

      6. associate-*r/ 76.8%

         \[\leadsto n0_i + u \cdot \color{blue}{\left(n1_i \cdot \frac{normAngle}{\sin normAngle}\right)} \]

   5. Simplified 76.8%

      \[\leadsto \color{blue}{n0_i + u \cdot \left(n1_i \cdot \frac{normAngle}{\sin normAngle}\right)} \]

   6. Taylor expanded in n0_i around inf 61.2%

      \[\leadsto \color{blue}{n0_i} \]

   if -1.79999997e-22 < n0_i < 1e-23

   1. Initial program 94.8%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. *-commutative 94.8%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      2. associate-*l* 74.9%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      3. *-commutative 74.9%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

      4. associate-*l* 60.9%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

      5. distribute-lft-out 60.9%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   3. Simplified 60.9%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 96.9%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

   5. Taylor expanded in n0_i around 0 65.4%

      \[\leadsto \color{blue}{n1_i \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 65.4%

         \[\leadsto \color{blue}{u \cdot n1_i} \]

   7. Simplified 65.4%

      \[\leadsto \color{blue}{u \cdot n1_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -1.7999999681214895 \cdot 10^{-22}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 4: 98.1% 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 96.6%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. *-commutative 96.6%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. associate-*l* 80.1%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. *-commutative 80.1%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

   4. associate-*l* 73.5%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   5. distribute-lft-out 73.5%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Simplified 73.5%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

4. Taylor expanded in normAngle around 0 97.6%

   \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

5. Taylor expanded in u around -inf 97.8%

   \[\leadsto \color{blue}{n0_i + -1 \cdot \left(u \cdot \left(n0_i + -1 \cdot n1_i\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 97.8%

      \[\leadsto n0_i + \color{blue}{\left(-u \cdot \left(n0_i + -1 \cdot n1_i\right)\right)} \]

   2. unsub-neg 97.8%

      \[\leadsto \color{blue}{n0_i - u \cdot \left(n0_i + -1 \cdot n1_i\right)} \]

   3. mul-1-neg 97.8%

      \[\leadsto n0_i - u \cdot \left(n0_i + \color{blue}{\left(-n1_i\right)}\right) \]

   4. unsub-neg 97.8%

      \[\leadsto n0_i - u \cdot \color{blue}{\left(n0_i - n1_i\right)} \]

7. Simplified 97.8%

   \[\leadsto \color{blue}{n0_i - u \cdot \left(n0_i - n1_i\right)} \]

8. Final simplification 97.8%

   \[\leadsto n0_i + u \cdot \left(n1_i - n0_i\right) \]

Alternative 5: 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 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. Taylor expanded in u around 0 80.8%

   \[\leadsto \color{blue}{n0_i} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

3. Taylor expanded in normAngle around 0 81.8%

   \[\leadsto \color{blue}{n0_i + n1_i \cdot u} \]
4. Step-by-step derivation

   1. *-commutative 81.8%

      \[\leadsto n0_i + \color{blue}{u \cdot n1_i} \]

5. Simplified 81.8%

   \[\leadsto \color{blue}{n0_i + u \cdot n1_i} \]

6. Final simplification 81.8%

   \[\leadsto n0_i + u \cdot n1_i \]

Alternative 6: 47.3% accurate, 421.0× speedup

Math

\[\begin{array}{l} \\ n0_i \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. Taylor expanded in u around 0 80.8%

   \[\leadsto \color{blue}{n0_i} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

3. Taylor expanded in u around 0 71.8%

   \[\leadsto \color{blue}{n0_i + \frac{n1_i \cdot \left(normAngle \cdot u\right)}{\sin normAngle}} \]
4. Step-by-step derivation

   1. associate-*r* 71.8%

      \[\leadsto n0_i + \frac{\color{blue}{\left(n1_i \cdot normAngle\right) \cdot u}}{\sin normAngle} \]

   2. associate-*l/ 74.6%

      \[\leadsto n0_i + \color{blue}{\frac{n1_i \cdot normAngle}{\sin normAngle} \cdot u} \]

   3. associate-/l* 83.2%

      \[\leadsto n0_i + \color{blue}{\frac{n1_i}{\frac{\sin normAngle}{normAngle}}} \cdot u \]

   4. *-commutative 83.2%

      \[\leadsto n0_i + \color{blue}{u \cdot \frac{n1_i}{\frac{\sin normAngle}{normAngle}}} \]

   5. associate-/l* 74.6%

      \[\leadsto n0_i + u \cdot \color{blue}{\frac{n1_i \cdot normAngle}{\sin normAngle}} \]

   6. associate-*r/ 83.2%

      \[\leadsto n0_i + u \cdot \color{blue}{\left(n1_i \cdot \frac{normAngle}{\sin normAngle}\right)} \]

5. Simplified 83.2%

   \[\leadsto \color{blue}{n0_i + u \cdot \left(n1_i \cdot \frac{normAngle}{\sin normAngle}\right)} \]

6. Taylor expanded in n0_i around inf 46.2%

   \[\leadsto \color{blue}{n0_i} \]

7. Final simplification 46.2%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023320 
(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)))