Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.1%

Time: 14.2s

Alternatives: 10

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: 99.1% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* normAngle normAngle)
   (fma
    (* normAngle normAngle)
    (* 0.019444444444444445 n1_i)
    (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), (0.019444444444444445f * n1_i), 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(Float32(0.019444444444444445) * n1_i), fma(n0_i, Float32(0.3333333333333333), Float32(n1_i * Float32(0.16666666666666666)))), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in 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 99.1%

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

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 99.2%

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

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 99.2

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

11. Applied rewrites 99.2%

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

Alternative 2: 99.0% accurate, 14.3× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   normAngle
   (* normAngle (fma 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(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(normAngle, Float32(normAngle * fma(n0_i, Float32(0.3333333333333333), Float32(n1_i * Float32(0.16666666666666666)))), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in 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 99.1%

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

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. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

8. Applied rewrites 99.1%

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

Alternative 3: 98.8% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in 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 99.1%

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

6. Taylor expanded in normAngle around 0

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

8. Applied rewrites 99.1%

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

9. Taylor expanded in n0_i around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 99.1

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

11. Applied rewrites 99.1%

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

12. Final simplification 99.1%

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

Alternative 4: 98.3% accurate, 17.7× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in 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 99.1%

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

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 99.2%

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

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 99.2

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

11. Applied rewrites 99.2%

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

12. Taylor expanded in n1_i around 0

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

    1. lower-*.f32 98.6

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

14. Applied rewrites 98.6%

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

15. Final simplification 98.6%

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

Alternative 5: 85.5% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(u, n1\_i, n0\_i\right)\\ \mathbf{elif}\;n1\_i \leq 1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(n0\_i, -u, n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, n1\_i, n0\_i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -4.999999841327613e-21)
   (fma u n1_i n0_i)
   (if (<= n1_i 1.0000000272452012e-27)
     (fma n0_i (- u) n0_i)
     (fma u n1_i n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -4.999999841327613e-21f) {
		tmp = fmaf(u, n1_i, n0_i);
	} else if (n1_i <= 1.0000000272452012e-27f) {
		tmp = fmaf(n0_i, -u, n0_i);
	} else {
		tmp = fmaf(u, n1_i, n0_i);
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-4.999999841327613e-21))
		tmp = fma(u, n1_i, n0_i);
	elseif (n1_i <= Float32(1.0000000272452012e-27))
		tmp = fma(n0_i, Float32(-u), n0_i);
	else
		tmp = fma(u, n1_i, n0_i);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.99999984e-21 or 1.00000003e-27 < n1_i

   1. Initial program 96.8%

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

   3. Taylor expanded in normAngle around 0

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 88.8%

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

      1. lift-*.f32 N/A

         \[\leadsto n0\_i \cdot 1 + \color{blue}{u \cdot n1\_i} \]

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. lower-fma.f32 88.8

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

   10. Applied rewrites 88.8%

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

   if -4.99999984e-21 < n1_i < 1.00000003e-27

   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 98.6

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

   5. Applied rewrites 98.6%

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

   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. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f32 90.0

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

   8. Applied rewrites 90.0%

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

Alternative 6: 85.0% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(u, n1\_i, n0\_i\right)\\ \mathbf{elif}\;n1\_i \leq 9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, n1\_i, n0\_i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -4.999999841327613e-21)
   (fma u n1_i n0_i)
   (if (<= n1_i 9.999999796611898e-32) (* n0_i (- 1.0 u)) (fma u n1_i n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -4.999999841327613e-21f) {
		tmp = fmaf(u, n1_i, n0_i);
	} else if (n1_i <= 9.999999796611898e-32f) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = fmaf(u, n1_i, n0_i);
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-4.999999841327613e-21))
		tmp = fma(u, n1_i, n0_i);
	elseif (n1_i <= Float32(9.999999796611898e-32))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = fma(u, n1_i, n0_i);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.99999984e-21 or 9.9999998e-32 < n1_i

   1. Initial program 96.9%

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

   3. Taylor expanded in normAngle around 0

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 88.4%

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

      1. lift-*.f32 N/A

         \[\leadsto n0\_i \cdot 1 + \color{blue}{u \cdot n1\_i} \]

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. lower-fma.f32 88.4

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

   10. Applied rewrites 88.4%

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

   if -4.99999984e-21 < n1_i < 9.9999998e-32

   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 n0_i around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. distribute-lft-neg-in N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-fma.f32 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f32 N/A

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

      15. lower-sin.f32 90.7

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in normAngle around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f32 90.3

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

   8. Applied rewrites 90.3%

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

Alternative 7: 60.6% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.4999999523982838 \cdot 10^{-21}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.4999999523982838e-21)
   n0_i
   (if (<= n0_i 4.999999841327613e-22) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.4999999523982838e-21f) {
		tmp = n0_i;
	} else if (n0_i <= 4.999999841327613e-22f) {
		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 <= (-1.4999999523982838e-21)) then
        tmp = n0_i
    else if (n0_i <= 4.999999841327613e-22) 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(-1.4999999523982838e-21))
		tmp = n0_i;
	elseif (n0_i <= Float32(4.999999841327613e-22))
		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(-1.4999999523982838e-21))
		tmp = n0_i;
	elseif (n0_i <= single(4.999999841327613e-22))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.5e-21 or 4.9999998e-22 < n0_i

   1. Initial program 98.4%

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

   3. Taylor expanded in n0_i around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. distribute-lft-neg-in N/A

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

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

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

      11. mul-1-neg N/A

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

      12. lower-fma.f32 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f32 N/A

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

      15. lower-sin.f32 86.3

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in u around 0

      \[\leadsto n0\_i \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.7%

      \[\leadsto n0\_i \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 67.7

         \[\leadsto \color{blue}{n0\_i} \]

   10. Applied rewrites 67.7%

       \[\leadsto \color{blue}{n0\_i} \]

   if -1.5e-21 < n0_i < 4.9999998e-22

   1. Initial program 96.8%

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

   3. Taylor expanded in normAngle around 0

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in n0_i around 0

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

      1. lower-*.f32 64.0

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

   8. Applied rewrites 64.0%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.4999999523982838 \cdot 10^{-21}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.1% 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 97.6%

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

3. Taylor expanded in 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 99.1%

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

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 98.6

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

8. Applied rewrites 98.6%

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

Alternative 9: 81.5% accurate, 65.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in 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 u around 0

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

8. Applied rewrites 82.4%

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

   1. lift-*.f32 N/A

      \[\leadsto n0\_i \cdot 1 + \color{blue}{u \cdot n1\_i} \]

   2. *-rgt-identity N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lower-fma.f32 82.5

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

10. Applied rewrites 82.5%

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

Alternative 10: 47.5% accurate, 459.0× 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 97.6%

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

3. Taylor expanded in n0_i around inf

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. *-commutative N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. distribute-lft-neg-in N/A

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

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

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

   11. mul-1-neg N/A

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

   12. lower-fma.f32 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f32 N/A

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

   15. lower-sin.f32 60.0

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

5. Applied rewrites 60.0%

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

6. Taylor expanded in u around 0

   \[\leadsto n0\_i \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 48.4%

   \[\leadsto n0\_i \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 48.4

      \[\leadsto \color{blue}{n0\_i} \]

10. Applied rewrites 48.4%

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

Reproduce

herbie shell --seed 2024216 
(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)))