Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.4%

Time: 12.3s

Alternatives: 11

Speedup: 45.9×

Specification

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

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: 99.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\frac{normAngle}{\sin normAngle}, n1\_i, -\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, normAngle \cdot normAngle, -0.3333333333333333\right) \cdot n0\_i, normAngle \cdot normAngle, n0\_i\right)\right), u, n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (fma
   (/ normAngle (sin normAngle))
   n1_i
   (-
    (fma
     (*
      (fma -0.022222222222222223 (* normAngle normAngle) -0.3333333333333333)
      n0_i)
     (* normAngle normAngle)
     n0_i)))
  u
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(fmaf((normAngle / sinf(normAngle)), n1_i, -fmaf((fmaf(-0.022222222222222223f, (normAngle * normAngle), -0.3333333333333333f) * n0_i), (normAngle * normAngle), n0_i)), u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(fma(Float32(normAngle / sin(normAngle)), n1_i, Float32(-fma(Float32(fma(Float32(-0.022222222222222223), Float32(normAngle * normAngle), Float32(-0.3333333333333333)) * n0_i), Float32(normAngle * normAngle), n0_i))), u, n0_i)
end

Derivation

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 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. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(\frac{normAngle}{\sin normAngle} \cdot n1\_i - \left(n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{2} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \frac{1}{120} \cdot n0\_i\right)\right)\right) - \frac{-1}{6} \cdot n0\_i\right)\right), u, n0\_i\right) \]
7. Step-by-step derivation

8. Applied rewrites 99.3%

   \[\leadsto \mathsf{fma}\left(\frac{normAngle}{\sin normAngle} \cdot n1\_i - \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot n0\_i - \mathsf{fma}\left(0.008333333333333333, n0\_i, n0\_i \cdot 0.05555555555555555\right), normAngle \cdot normAngle, n0\_i \cdot -0.3333333333333333\right), normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \]

9. Taylor expanded in n0_i around 0

   \[\leadsto \mathsf{fma}\left(\frac{normAngle}{\sin normAngle} \cdot n1\_i - \mathsf{fma}\left(n0\_i \cdot \left(\frac{-1}{45} \cdot {normAngle}^{2} - \frac{1}{3}\right), normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.3%

    \[\leadsto \mathsf{fma}\left(\frac{normAngle}{\sin normAngle} \cdot n1\_i - \mathsf{fma}\left(\mathsf{fma}\left(normAngle \cdot normAngle, -0.022222222222222223, -0.3333333333333333\right) \cdot n0\_i, normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \]
12. Step-by-step derivation

13. Applied rewrites 99.3%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{normAngle}{\sin normAngle}, n1\_i, -\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, normAngle \cdot normAngle, -0.3333333333333333\right) \cdot n0\_i, normAngle \cdot normAngle, n0\_i\right)\right), u, n0\_i\right) \]
14. Add Preprocessing

Alternative 2: 99.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{normAngle}{\sin normAngle} \cdot n1\_i - \mathsf{fma}\left(\mathsf{fma}\left(normAngle \cdot normAngle, -0.022222222222222223, -0.3333333333333333\right) \cdot n0\_i, normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (-
   (* (/ normAngle (sin normAngle)) n1_i)
   (fma
    (*
     (fma (* normAngle normAngle) -0.022222222222222223 -0.3333333333333333)
     n0_i)
    (* normAngle normAngle)
    n0_i))
  u
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf((((normAngle / sinf(normAngle)) * n1_i) - fmaf((fmaf((normAngle * normAngle), -0.022222222222222223f, -0.3333333333333333f) * n0_i), (normAngle * normAngle), n0_i)), u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(Float32(Float32(normAngle / sin(normAngle)) * n1_i) - fma(Float32(fma(Float32(normAngle * normAngle), Float32(-0.022222222222222223), Float32(-0.3333333333333333)) * n0_i), Float32(normAngle * normAngle), n0_i)), u, n0_i)
end

Derivation

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 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. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(\frac{normAngle}{\sin normAngle} \cdot n1\_i - \left(n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{2} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \frac{1}{120} \cdot n0\_i\right)\right)\right) - \frac{-1}{6} \cdot n0\_i\right)\right), u, n0\_i\right) \]
7. Step-by-step derivation

8. Applied rewrites 99.3%

   \[\leadsto \mathsf{fma}\left(\frac{normAngle}{\sin normAngle} \cdot n1\_i - \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot n0\_i - \mathsf{fma}\left(0.008333333333333333, n0\_i, n0\_i \cdot 0.05555555555555555\right), normAngle \cdot normAngle, n0\_i \cdot -0.3333333333333333\right), normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \]

9. Taylor expanded in n0_i around 0

   \[\leadsto \mathsf{fma}\left(\frac{normAngle}{\sin normAngle} \cdot n1\_i - \mathsf{fma}\left(n0\_i \cdot \left(\frac{-1}{45} \cdot {normAngle}^{2} - \frac{1}{3}\right), normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.3%

    \[\leadsto \mathsf{fma}\left(\frac{normAngle}{\sin normAngle} \cdot n1\_i - \mathsf{fma}\left(\mathsf{fma}\left(normAngle \cdot normAngle, -0.022222222222222223, -0.3333333333333333\right) \cdot n0\_i, normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \]
12. Add Preprocessing

Alternative 3: 99.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 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. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in normAngle around 0

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

8. Applied rewrites 99.3%

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

9. Final simplification 99.3%

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

Alternative 4: 99.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-normAngle\right) \cdot normAngle, \mathsf{fma}\left(n1\_i, 0.0011904761904761906, -0.0032407407407407406 \cdot n1\_i\right), 0.019444444444444445 \cdot n1\_i\right), normAngle \cdot normAngle, 0.16666666666666666 \cdot n1\_i\right), normAngle \cdot normAngle, n1\_i\right) - \mathsf{fma}\left(\mathsf{fma}\left(normAngle \cdot normAngle, -0.022222222222222223, -0.3333333333333333\right) \cdot n0\_i, normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (-
   (fma
    (fma
     (fma
      (* (- normAngle) normAngle)
      (fma n1_i 0.0011904761904761906 (* -0.0032407407407407406 n1_i))
      (* 0.019444444444444445 n1_i))
     (* normAngle normAngle)
     (* 0.16666666666666666 n1_i))
    (* normAngle normAngle)
    n1_i)
   (fma
    (*
     (fma (* normAngle normAngle) -0.022222222222222223 -0.3333333333333333)
     n0_i)
    (* normAngle normAngle)
    n0_i))
  u
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf((fmaf(fmaf(fmaf((-normAngle * normAngle), fmaf(n1_i, 0.0011904761904761906f, (-0.0032407407407407406f * n1_i)), (0.019444444444444445f * n1_i)), (normAngle * normAngle), (0.16666666666666666f * n1_i)), (normAngle * normAngle), n1_i) - fmaf((fmaf((normAngle * normAngle), -0.022222222222222223f, -0.3333333333333333f) * n0_i), (normAngle * normAngle), n0_i)), u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(fma(fma(fma(Float32(Float32(-normAngle) * normAngle), fma(n1_i, Float32(0.0011904761904761906), Float32(Float32(-0.0032407407407407406) * n1_i)), Float32(Float32(0.019444444444444445) * n1_i)), Float32(normAngle * normAngle), Float32(Float32(0.16666666666666666) * n1_i)), Float32(normAngle * normAngle), n1_i) - fma(Float32(fma(Float32(normAngle * normAngle), Float32(-0.022222222222222223), Float32(-0.3333333333333333)) * n0_i), Float32(normAngle * normAngle), n0_i)), u, n0_i)
end

Derivation

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 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. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(\frac{normAngle}{\sin normAngle} \cdot n1\_i - \left(n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{2} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \frac{1}{120} \cdot n0\_i\right)\right)\right) - \frac{-1}{6} \cdot n0\_i\right)\right), u, n0\_i\right) \]
7. Step-by-step derivation

8. Applied rewrites 99.3%

   \[\leadsto \mathsf{fma}\left(\frac{normAngle}{\sin normAngle} \cdot n1\_i - \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot n0\_i - \mathsf{fma}\left(0.008333333333333333, n0\_i, n0\_i \cdot 0.05555555555555555\right), normAngle \cdot normAngle, n0\_i \cdot -0.3333333333333333\right), normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \]

9. Taylor expanded in n0_i around 0

   \[\leadsto \mathsf{fma}\left(\frac{normAngle}{\sin normAngle} \cdot n1\_i - \mathsf{fma}\left(n0\_i \cdot \left(\frac{-1}{45} \cdot {normAngle}^{2} - \frac{1}{3}\right), normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.3%

    \[\leadsto \mathsf{fma}\left(\frac{normAngle}{\sin normAngle} \cdot n1\_i - \mathsf{fma}\left(\mathsf{fma}\left(normAngle \cdot normAngle, -0.022222222222222223, -0.3333333333333333\right) \cdot n0\_i, normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \]

12. Taylor expanded in normAngle around 0

    \[\leadsto \mathsf{fma}\left(\left(n1\_i + {normAngle}^{2} \cdot \left({normAngle}^{2} \cdot \left(-1 \cdot \left({normAngle}^{2} \cdot \left(\frac{-1}{5040} \cdot n1\_i + \left(\frac{1}{720} \cdot n1\_i + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right)\right) - \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right) - \frac{-1}{6} \cdot n1\_i\right)\right) - \mathsf{fma}\left(\mathsf{fma}\left(normAngle \cdot normAngle, \frac{-1}{45}, \frac{-1}{3}\right) \cdot n0\_i, normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \]
13. Step-by-step derivation

14. Applied rewrites 99.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-normAngle\right) \cdot normAngle, \mathsf{fma}\left(n1\_i, 0.0011904761904761906, -0.0032407407407407406 \cdot n1\_i\right), 0.019444444444444445 \cdot n1\_i\right), normAngle \cdot normAngle, 0.16666666666666666 \cdot n1\_i\right), normAngle \cdot normAngle, n1\_i\right) - \mathsf{fma}\left(\mathsf{fma}\left(normAngle \cdot normAngle, -0.022222222222222223, -0.3333333333333333\right) \cdot n0\_i, normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \]
15. Add Preprocessing

Alternative 5: 99.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, n0\_i, \mathsf{fma}\left(0.019444444444444445, n1\_i, 0.06388888888888888 \cdot n0\_i\right)\right) \cdot u, normAngle \cdot normAngle, \mathsf{fma}\left(0.3333333333333333, n0\_i, 0.16666666666666666 \cdot n1\_i\right) \cdot u\right), normAngle \cdot normAngle, \mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (fma
   (*
    (fma
     -0.041666666666666664
     n0_i
     (fma 0.019444444444444445 n1_i (* 0.06388888888888888 n0_i)))
    u)
   (* normAngle normAngle)
   (* (fma 0.3333333333333333 n0_i (* 0.16666666666666666 n1_i)) u))
  (* normAngle normAngle)
  (fma (- n1_i n0_i) u n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(fmaf((fmaf(-0.041666666666666664f, n0_i, fmaf(0.019444444444444445f, n1_i, (0.06388888888888888f * n0_i))) * u), (normAngle * normAngle), (fmaf(0.3333333333333333f, n0_i, (0.16666666666666666f * n1_i)) * u)), (normAngle * normAngle), fmaf((n1_i - n0_i), u, n0_i));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(fma(Float32(fma(Float32(-0.041666666666666664), n0_i, fma(Float32(0.019444444444444445), n1_i, Float32(Float32(0.06388888888888888) * n0_i))) * u), Float32(normAngle * normAngle), Float32(fma(Float32(0.3333333333333333), n0_i, Float32(Float32(0.16666666666666666) * n1_i)) * u)), Float32(normAngle * normAngle), fma(Float32(n1_i - n0_i), u, n0_i))
end

Derivation

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 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. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in normAngle around 0

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in normAngle around 0

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

11. Applied rewrites 99.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, n0\_i, \mathsf{fma}\left(0.019444444444444445, n1\_i, 0.06388888888888888 \cdot n0\_i\right)\right) \cdot u, normAngle \cdot normAngle, \mathsf{fma}\left(0.3333333333333333, n0\_i, 0.16666666666666666 \cdot n1\_i\right) \cdot u\right), \color{blue}{normAngle \cdot normAngle}, \mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right)\right) \]
12. Add Preprocessing

Alternative 6: 99.2% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 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. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in normAngle around 0

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in n0_i around 0

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

11. Applied rewrites 99.2%

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

Alternative 7: 99.1% accurate, 14.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 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. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in normAngle around 0

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

8. Applied rewrites 99.0%

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

Alternative 8: 70.1% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot n0\_i\\ \mathbf{if}\;n0\_i \leq -1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 5.000000015855384 \cdot 10^{-31}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) n0_i)))
   (if (<= n0_i -1.9999999774532045e-26)
     t_0
     (if (<= n0_i 5.000000015855384e-31) (* n1_i u) t_0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = (1.0f - u) * n0_i;
	float tmp;
	if (n0_i <= -1.9999999774532045e-26f) {
		tmp = t_0;
	} else if (n0_i <= 5.000000015855384e-31f) {
		tmp = n1_i * u;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = (1.0e0 - u) * n0_i
    if (n0_i <= (-1.9999999774532045e-26)) then
        tmp = t_0
    else if (n0_i <= 5.000000015855384e-31) then
        tmp = n1_i * u
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(Float32(1.0) - u) * n0_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-1.9999999774532045e-26))
		tmp = t_0;
	elseif (n0_i <= Float32(5.000000015855384e-31))
		tmp = Float32(n1_i * u);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	t_0 = (single(1.0) - u) * n0_i;
	tmp = single(0.0);
	if (n0_i <= single(-1.9999999774532045e-26))
		tmp = t_0;
	elseif (n0_i <= single(5.000000015855384e-31))
		tmp = n1_i * u;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.99999998e-26 or 5e-31 < n0_i

   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. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 98.5

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

   5. Applied rewrites 98.5%

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

   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 77.2%

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

   if -1.99999998e-26 < n0_i < 5e-31

   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. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in n0_i around 0

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

   8. Applied rewrites 70.3%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{elif}\;n0\_i \leq 5.000000015855384 \cdot 10^{-31}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.1% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;1 \cdot n0\_i\\ \mathbf{elif}\;n0\_i \leq 5.000000015855384 \cdot 10^{-31}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;1 \cdot n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.9999999774532045e-26)
   (* 1.0 n0_i)
   (if (<= n0_i 5.000000015855384e-31) (* n1_i u) (* 1.0 n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.9999999774532045e-26f) {
		tmp = 1.0f * n0_i;
	} else if (n0_i <= 5.000000015855384e-31f) {
		tmp = n1_i * u;
	} else {
		tmp = 1.0f * 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 <= (-1.9999999774532045e-26)) then
        tmp = 1.0e0 * n0_i
    else if (n0_i <= 5.000000015855384e-31) then
        tmp = n1_i * u
    else
        tmp = 1.0e0 * 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(-1.9999999774532045e-26))
		tmp = Float32(Float32(1.0) * n0_i);
	elseif (n0_i <= Float32(5.000000015855384e-31))
		tmp = Float32(n1_i * u);
	else
		tmp = Float32(Float32(1.0) * 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(-1.9999999774532045e-26))
		tmp = single(1.0) * n0_i;
	elseif (n0_i <= single(5.000000015855384e-31))
		tmp = n1_i * u;
	else
		tmp = single(1.0) * n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.99999998e-26 or 5e-31 < n0_i

   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. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lower-fma.f32 98.3

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

      5. lift-*.f32 N/A

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

      6. lift-/.f32 N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f32 98.4

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 98.4

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 N/A

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in n0_i around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lower-sin.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-sin.f32 77.3

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

   7. Applied rewrites 77.3%

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

   8. Taylor expanded in u around 0

      \[\leadsto 1 \cdot n0\_i \]
   9. Step-by-step derivation

   10. Applied rewrites 60.4%

       \[\leadsto 1 \cdot n0\_i \]

   if -1.99999998e-26 < n0_i < 5e-31

   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. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in n0_i around 0

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

   8. Applied rewrites 70.3%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;1 \cdot n0\_i\\ \mathbf{elif}\;n0\_i \leq 5.000000015855384 \cdot 10^{-31}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;1 \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 98.2

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f32 98.4

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 98.4

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
6. 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. mul-1-neg N/A

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

   9. *-rgt-identity N/A

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

   10. associate-+l+ N/A

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

   11. distribute-rgt-in N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. lower--.f32 98.2

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

7. Applied rewrites 98.2%

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

Alternative 11: 38.1% accurate, 76.5× speedup

Math

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

FPCore

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

C

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

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 = n1_i * u
end function

Julia

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

MATLAB

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

Derivation

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. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 98.0

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

5. Applied rewrites 98.0%

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

6. Taylor expanded in n0_i around 0

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

8. Applied rewrites 34.1%

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

9. Final simplification 34.1%

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

Reproduce

herbie shell --seed 2024250 
(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)))