Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.3%

Time: 13.8s

Alternatives: 9

Speedup: 45.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

Alternative 2: 99.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle, normAngle \cdot 0.3333333333333333, -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   n0_i
   (fma normAngle (* normAngle 0.3333333333333333) -1.0)
   (* normAngle (/ n1_i (sin normAngle))))
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, fmaf(n0_i, fmaf(normAngle, (normAngle * 0.3333333333333333f), -1.0f), (normAngle * (n1_i / sinf(normAngle)))), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, fma(n0_i, fma(normAngle, Float32(normAngle * Float32(0.3333333333333333)), Float32(-1.0)), Float32(normAngle * Float32(n1_i / sin(normAngle)))), n0_i)
end

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in normAngle around 0

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

8. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle, \color{blue}{normAngle \cdot 0.3333333333333333}, -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]
9. Add Preprocessing

Alternative 3: 99.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle, normAngle \cdot 0.3333333333333333, -1\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, n1\_i \cdot \mathsf{fma}\left(0.00205026455026455, normAngle \cdot normAngle, 0.019444444444444445\right), n1\_i \cdot 0.16666666666666666\right), n1\_i\right)\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   n0_i
   (fma normAngle (* normAngle 0.3333333333333333) -1.0)
   (fma
    (* normAngle normAngle)
    (fma
     (* normAngle normAngle)
     (*
      n1_i
      (fma 0.00205026455026455 (* normAngle normAngle) 0.019444444444444445))
     (* n1_i 0.16666666666666666))
    n1_i))
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, fmaf(n0_i, fmaf(normAngle, (normAngle * 0.3333333333333333f), -1.0f), fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), (n1_i * fmaf(0.00205026455026455f, (normAngle * normAngle), 0.019444444444444445f)), (n1_i * 0.16666666666666666f)), n1_i)), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, fma(n0_i, fma(normAngle, Float32(normAngle * Float32(0.3333333333333333)), Float32(-1.0)), fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), Float32(n1_i * fma(Float32(0.00205026455026455), Float32(normAngle * normAngle), Float32(0.019444444444444445))), Float32(n1_i * Float32(0.16666666666666666))), n1_i)), n0_i)
end

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in normAngle around 0

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

8. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle, \color{blue}{normAngle \cdot 0.3333333333333333}, -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

9. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle, normAngle \cdot \frac{1}{3}, \color{blue}{-1}\right), n1\_i + {normAngle}^{2} \cdot \left({normAngle}^{2} \cdot \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{5040} \cdot n1\_i + \left(\frac{1}{720} \cdot n1\_i + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right)\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right), n0\_i\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.6%

    \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle, normAngle \cdot 0.3333333333333333, \color{blue}{-1}\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\mathsf{fma}\left(n1\_i, 0.0011904761904761906, n1\_i \cdot -0.0032407407407407406\right), \left(-normAngle\right) \cdot normAngle, n1\_i \cdot 0.019444444444444445\right), n1\_i \cdot 0.16666666666666666\right), n1\_i\right)\right), n0\_i\right) \]

12. Taylor expanded in n1_i around 0

    \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle, normAngle \cdot \frac{1}{3}, -1\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, n1\_i \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {normAngle}^{2}\right), n1\_i \cdot \frac{1}{6}\right), n1\_i\right)\right), n0\_i\right) \]
13. Step-by-step derivation

14. Applied rewrites 99.6%

    \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle, normAngle \cdot 0.3333333333333333, -1\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, n1\_i \cdot \mathsf{fma}\left(0.00205026455026455, normAngle \cdot normAngle, 0.019444444444444445\right), n1\_i \cdot 0.16666666666666666\right), n1\_i\right)\right), n0\_i\right) \]
15. Add Preprocessing

Alternative 4: 99.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle, normAngle \cdot 0.3333333333333333, -1\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, n1\_i \cdot 0.019444444444444445, n1\_i \cdot 0.16666666666666666\right), n1\_i\right)\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   n0_i
   (fma normAngle (* normAngle 0.3333333333333333) -1.0)
   (fma
    (* normAngle normAngle)
    (fma
     (* normAngle normAngle)
     (* n1_i 0.019444444444444445)
     (* n1_i 0.16666666666666666))
    n1_i))
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, fmaf(n0_i, fmaf(normAngle, (normAngle * 0.3333333333333333f), -1.0f), fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), (n1_i * 0.019444444444444445f), (n1_i * 0.16666666666666666f)), n1_i)), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, fma(n0_i, fma(normAngle, Float32(normAngle * Float32(0.3333333333333333)), Float32(-1.0)), fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), Float32(n1_i * Float32(0.019444444444444445)), Float32(n1_i * Float32(0.16666666666666666))), n1_i)), n0_i)
end

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in normAngle around 0

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

8. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle, \color{blue}{normAngle \cdot 0.3333333333333333}, -1\right), normAngle \cdot \frac{n1\_i}{\sin normAngle}\right), n0\_i\right) \]

9. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle, normAngle \cdot \frac{1}{3}, \color{blue}{-1}\right), n1\_i + {normAngle}^{2} \cdot \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right), n0\_i\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.4%

    \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n0\_i, \mathsf{fma}\left(normAngle, normAngle \cdot 0.3333333333333333, \color{blue}{-1}\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, n1\_i \cdot 0.019444444444444445, n1\_i \cdot 0.16666666666666666\right), n1\_i\right)\right), n0\_i\right) \]
12. Add Preprocessing

Alternative 5: 99.0% accurate, 14.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, n1\_i + \color{blue}{\left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(-1 \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) - \frac{-1}{6} \cdot n1\_i\right)\right)}, n0\_i\right) \]
7. Step-by-step derivation

8. Applied rewrites 99.2%

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle, \color{blue}{normAngle \cdot \mathsf{fma}\left(n1\_i, 0.16666666666666666, n0\_i \cdot 0.3333333333333333\right)}, n1\_i - n0\_i\right), n0\_i\right) \]
9. Add Preprocessing

Alternative 6: 68.8% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{elif}\;n1\_i \leq 2.4999999206638063 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(n0\_i, -u, n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -1.999999936531045e-20)
   (* u n1_i)
   (if (<= n1_i 2.4999999206638063e-20) (fma n0_i (- u) n0_i) (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -1.999999936531045e-20f) {
		tmp = u * n1_i;
	} else if (n1_i <= 2.4999999206638063e-20f) {
		tmp = fmaf(n0_i, -u, n0_i);
	} else {
		tmp = u * n1_i;
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-1.999999936531045e-20))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(2.4999999206638063e-20))
		tmp = fma(n0_i, Float32(-u), n0_i);
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.99999994e-20 or 2.49999992e-20 < n1_i

   1. Initial program 96.4%

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

   3. Taylor expanded in normAngle around 0

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

      1. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in n0_i around 0

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

   8. Applied rewrites 62.1%

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

   if -1.99999994e-20 < n1_i < 2.49999992e-20

   1. Initial program 98.2%

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

   3. Taylor expanded in normAngle around 0

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

      1. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in n0_i around inf

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

   8. Applied rewrites 84.8%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{elif}\;n1\_i \leq 2.4999999206638063 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(n0\_i, -u, n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.8% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{elif}\;n1\_i \leq 2.4999999206638063 \cdot 10^{-20}:\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -1.999999936531045e-20)
   (* u n1_i)
   (if (<= n1_i 2.4999999206638063e-20) (- n0_i (* u n0_i)) (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -1.999999936531045e-20f) {
		tmp = u * n1_i;
	} else if (n1_i <= 2.4999999206638063e-20f) {
		tmp = n0_i - (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 (n1_i <= (-1.999999936531045e-20)) then
        tmp = u * n1_i
    else if (n1_i <= 2.4999999206638063e-20) then
        tmp = n0_i - (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 (n1_i <= Float32(-1.999999936531045e-20))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(2.4999999206638063e-20))
		tmp = Float32(n0_i - Float32(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 (n1_i <= single(-1.999999936531045e-20))
		tmp = u * n1_i;
	elseif (n1_i <= single(2.4999999206638063e-20))
		tmp = n0_i - (u * n0_i);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.99999994e-20 or 2.49999992e-20 < n1_i

   1. Initial program 96.4%

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

   3. Taylor expanded in normAngle around 0

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

      1. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in n0_i around 0

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

   8. Applied rewrites 62.1%

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

   if -1.99999994e-20 < n1_i < 2.49999992e-20

   1. Initial program 98.2%

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

   3. Taylor expanded in normAngle around 0

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

      1. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in n0_i around inf

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

   8. Applied rewrites 84.8%

      \[\leadsto \mathsf{fma}\left(n0\_i, \color{blue}{-u}, n0\_i\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 84.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{elif}\;n1\_i \leq 2.4999999206638063 \cdot 10^{-20}:\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.3% accurate, 45.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(n1_i - n0_i), u, n0_i)
end

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. mul-1-neg N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. *-rgt-identity N/A

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

   9. associate-+l+ N/A

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

   10. distribute-rgt-in N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(n1\_i + \color{blue}{\left(\mathsf{neg}\left(n0\_i\right)\right)}, u, n0\_i\right) \]

   14. unsub-neg N/A

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

   15. lower--.f32 98.5

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

8. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right)} \]
9. Add Preprocessing

Alternative 9: 37.9% accurate, 76.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 98.2

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in n0_i around 0

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

8. Applied rewrites 37.5%

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

9. Final simplification 37.5%

   \[\leadsto u \cdot n1\_i \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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)))