Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 99.4%

Time: 14.1s

Alternatives: 8

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: 97.1% 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: 99.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   n0_i
   (/ (* (cos normAngle) (- 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, ((cosf(normAngle) * -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(cos(normAngle) * Float32(-normAngle)) / sin(normAngle)), Float32(normAngle * Float32(n1_i / sin(normAngle)))), n0_i)
end

Derivation

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 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 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* normAngle normAngle)
   (fma
    (* normAngle normAngle)
    (* n1_i 0.019444444444444445)
    (fma n0_i 0.3333333333333333 (* n1_i 0.16666666666666666)))
   (- 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((normAngle * normAngle), (n1_i * 0.019444444444444445f), fmaf(n0_i, 0.3333333333333333f, (n1_i * 0.16666666666666666f))), (n1_i - n0_i)), n0_i);
}

Julia

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

Derivation

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 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 99.5%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(-1 \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(-1 \cdot \left(\frac{1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \frac{1}{120} \cdot n0\_i\right)\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right)}, n0\_i\right) \]

8. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, -\left(\left(n0\_i \cdot 0.041666666666666664 - \mathsf{fma}\left(n0\_i, 0.008333333333333333, n0\_i \cdot 0.05555555555555555\right)\right) + n1\_i \cdot -0.019444444444444445\right), \mathsf{fma}\left(n0\_i, 0.3333333333333333, n1\_i \cdot 0.16666666666666666\right)\right), n1\_i - n0\_i\right)}, n0\_i\right) \]

9. Taylor expanded in n0_i around 0

   \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{\frac{7}{360} \cdot n1\_i}, \mathsf{fma}\left(n0\_i, \frac{1}{3}, n1\_i \cdot \frac{1}{6}\right)\right), n1\_i - n0\_i\right), n0\_i\right) \]
10. Step-by-step derivation

    1. lower-*.f32 99.5

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

11. Applied rewrites 99.5%

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

12. Final simplification 99.5%

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

Alternative 3: 99.1% accurate, 14.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 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 99.5%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(u, \left(-1 \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) - \frac{-1}{6} \cdot n1\_i\right) \cdot \color{blue}{\left(normAngle \cdot normAngle\right)} + \left(-1 \cdot n0\_i + n1\_i\right), n0\_i\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left(\left(-1 \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) - \frac{-1}{6} \cdot n1\_i\right) \cdot normAngle\right) \cdot normAngle} + \left(-1 \cdot n0\_i + n1\_i\right), n0\_i\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(u, \left(\left(-1 \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) - \frac{-1}{6} \cdot n1\_i\right) \cdot normAngle\right) \cdot normAngle + \color{blue}{\left(n1\_i + -1 \cdot n0\_i\right)}, n0\_i\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(\left(-1 \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) - \frac{-1}{6} \cdot n1\_i\right) \cdot normAngle, normAngle, n1\_i + -1 \cdot n0\_i\right)}, n0\_i\right) \]

8. Applied rewrites 99.4%

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

9. Final simplification 99.4%

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

Alternative 4: 98.9% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\left(u + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot {u}^{3} - \frac{-1}{6} \cdot u\right)\right)} \cdot n1\_i \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\mathsf{fma}\left({normAngle}^{2}, \frac{-1}{6} \cdot {u}^{3} - \frac{-1}{6} \cdot u, u\right)} \cdot n1\_i \]

   3. unpow2 N/A

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, \frac{-1}{6} \cdot {u}^{3} - \frac{-1}{6} \cdot u, u\right) \cdot n1\_i \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, \frac{-1}{6} \cdot {u}^{3} - \frac{-1}{6} \cdot u, u\right) \cdot n1\_i \]

   5. sub-neg N/A

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{\frac{-1}{6} \cdot {u}^{3} + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot u\right)\right)}, u\right) \cdot n1\_i \]

   6. *-commutative N/A

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{{u}^{3} \cdot \frac{-1}{6}} + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot u\right)\right), u\right) \cdot n1\_i \]

   7. cube-mult N/A

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{\left(u \cdot \left(u \cdot u\right)\right)} \cdot \frac{-1}{6} + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot u\right)\right), u\right) \cdot n1\_i \]

   8. unpow2 N/A

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \left(u \cdot \color{blue}{{u}^{2}}\right) \cdot \frac{-1}{6} + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot u\right)\right), u\right) \cdot n1\_i \]

   9. associate-*l* N/A

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \color{blue}{u \cdot \left({u}^{2} \cdot \frac{-1}{6}\right)} + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot u\right)\right), u\right) \cdot n1\_i \]

   10. *-commutative N/A

       \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \left({u}^{2} \cdot \frac{-1}{6}\right) + \left(\mathsf{neg}\left(\color{blue}{u \cdot \frac{-1}{6}}\right)\right), u\right) \cdot n1\_i \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \left({u}^{2} \cdot \frac{-1}{6}\right) + \color{blue}{u \cdot \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right)}, u\right) \cdot n1\_i \]

   12. metadata-eval N/A

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

   13. distribute-lft-out N/A

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

   14. lower-*.f32 N/A

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

   15. lower-fma.f32 N/A

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

   16. unpow2 N/A

       \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(\color{blue}{u \cdot u}, \frac{-1}{6}, \frac{1}{6}\right), u\right) \cdot n1\_i \]

   17. lower-*.f32 98.8

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

      \[\leadsto n0\_i \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(u\right)\right)\right)} + \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, \frac{-1}{6}, \frac{1}{6}\right), u\right) \cdot n1\_i \]

   2. +-commutative N/A

      \[\leadsto n0\_i \cdot \color{blue}{\left(\left(\mathsf{neg}\left(u\right)\right) + 1\right)} + \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, \frac{-1}{6}, \frac{1}{6}\right), u\right) \cdot n1\_i \]

   3. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(n0\_i \cdot \left(\mathsf{neg}\left(u\right)\right) + n0\_i \cdot 1\right)} + \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, \frac{-1}{6}, \frac{1}{6}\right), u\right) \cdot n1\_i \]

   4. *-rgt-identity N/A

      \[\leadsto \left(n0\_i \cdot \left(\mathsf{neg}\left(u\right)\right) + \color{blue}{n0\_i}\right) + \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, \frac{-1}{6}, \frac{1}{6}\right), u\right) \cdot n1\_i \]

   5. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(n0\_i, \mathsf{neg}\left(u\right), n0\_i\right)} + \mathsf{fma}\left(normAngle \cdot normAngle, u \cdot \mathsf{fma}\left(u \cdot u, \frac{-1}{6}, \frac{1}{6}\right), u\right) \cdot n1\_i \]

   6. lower-neg.f32 98.9

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f32 N/A

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

    5. +-commutative N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n1\_i, \color{blue}{\mathsf{fma}\left({normAngle}^{2}, \frac{1}{6}, 1\right)}, -1 \cdot n0\_i\right), n0\_i\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, \frac{1}{6}, 1\right), -1 \cdot n0\_i\right), n0\_i\right) \]

    9. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(\color{blue}{normAngle \cdot normAngle}, \frac{1}{6}, 1\right), -1 \cdot n0\_i\right), n0\_i\right) \]

    10. mul-1-neg N/A

        \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \frac{1}{6}, 1\right), \color{blue}{\mathsf{neg}\left(n0\_i\right)}\right), n0\_i\right) \]

    11. lower-neg.f32 99.2

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

11. Applied rewrites 99.2%

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

Alternative 5: 84.2% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(n1\_i, u, n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n0\_i, -u, n0\_i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i 9.9999998245167e-15) (fma n1_i u n0_i) (fma n0_i (- u) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= 9.9999998245167e-15f) {
		tmp = fmaf(n1_i, u, n0_i);
	} else {
		tmp = fmaf(n0_i, -u, n0_i);
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(9.9999998245167e-15))
		tmp = fma(n1_i, u, n0_i);
	else
		tmp = fma(n0_i, Float32(-u), n0_i);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n0_i < 9.99999982e-15

   1. Initial program 95.8%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in normAngle around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 84.9

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

   8. Applied rewrites 84.9%

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

   if 9.99999982e-15 < n0_i

   1. Initial program 98.8%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   2. 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 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in n0_i around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-neg.f32 88.8

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

   8. Applied rewrites 88.8%

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

Alternative 6: 98.2% 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 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 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 99.5%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 98.8

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

8. Applied rewrites 98.8%

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

Alternative 7: 82.1% accurate, 65.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 u around 0

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

5. Applied rewrites 80.3%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 82.3

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

8. Applied rewrites 82.3%

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

Alternative 8: 38.3% 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 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 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in n0_i around 0

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

   1. lower-*.f32 36.8

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

8. Applied rewrites 36.8%

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

9. Final simplification 36.8%

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

Reproduce

herbie shell --seed 2024214 
(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)))