Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.0% → 99.2%

Time: 15.0s

Alternatives: 13

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: 97.0% 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: 99.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(u, \left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot 0.019444444444444445 + \left(\mathsf{fma}\left(n0\_i, 0.008333333333333333, n0\_i \cdot 0.05555555555555555\right) - n0\_i \cdot 0.041666666666666664\right)\right), u \cdot \mathsf{fma}\left(n1\_i, 0.16666666666666666, \mathsf{fma}\left(n0\_i, 0.3333333333333333, -0.5 \cdot \left(u \cdot n0\_i\right)\right)\right)\right), \mathsf{fma}\left(u, n1\_i - n0\_i, n0\_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (* normAngle normAngle)
  (fma
   u
   (*
    (* normAngle normAngle)
    (+
     (* n1_i 0.019444444444444445)
     (-
      (fma n0_i 0.008333333333333333 (* n0_i 0.05555555555555555))
      (* n0_i 0.041666666666666664))))
   (*
    u
    (fma
     n1_i
     0.16666666666666666
     (fma n0_i 0.3333333333333333 (* -0.5 (* u n0_i))))))
  (fma u (- n1_i n0_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf((normAngle * normAngle), fmaf(u, ((normAngle * normAngle) * ((n1_i * 0.019444444444444445f) + (fmaf(n0_i, 0.008333333333333333f, (n0_i * 0.05555555555555555f)) - (n0_i * 0.041666666666666664f)))), (u * fmaf(n1_i, 0.16666666666666666f, fmaf(n0_i, 0.3333333333333333f, (-0.5f * (u * n0_i)))))), fmaf(u, (n1_i - n0_i), n0_i));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(normAngle * normAngle), fma(u, Float32(Float32(normAngle * normAngle) * Float32(Float32(n1_i * Float32(0.019444444444444445)) + Float32(fma(n0_i, Float32(0.008333333333333333), Float32(n0_i * Float32(0.05555555555555555))) - Float32(n0_i * Float32(0.041666666666666664))))), Float32(u * fma(n1_i, Float32(0.16666666666666666), fma(n0_i, Float32(0.3333333333333333), Float32(Float32(-0.5) * Float32(u * n0_i)))))), fma(u, Float32(n1_i - n0_i), n0_i))
end

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in normAngle around 0

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\left(u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot {u}^{3} + {normAngle}^{2} \cdot \left(\frac{1}{120} \cdot {u}^{5} - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot {u}^{3} - \frac{-1}{6} \cdot u\right) + \frac{1}{120} \cdot u\right)\right)\right) - \frac{-1}{6} \cdot u\right)\right)} \cdot n1\_i \]

5. Applied rewrites 98.6%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(u, \mathsf{fma}\left(u \cdot u, -0.16666666666666666, 0.16666666666666666\right), \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(0.008333333333333333, {u}^{5} - u, -0.027777777777777776 \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right)\right), u\right)} \cdot n1\_i \]

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + n1\_i \cdot \left(1 + {normAngle}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {normAngle}^{2}\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + n1\_i \cdot \left(1 + {normAngle}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {normAngle}^{2}\right)\right)\right)\right) + n0\_i} \]

   2. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, -1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + n1\_i \cdot \left(1 + {normAngle}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {normAngle}^{2}\right)\right)\right), n0\_i\right)} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, 0.019444444444444445, 0.16666666666666666\right), 1\right), \mathsf{fma}\left(-n0\_i, \frac{normAngle \cdot \cos normAngle}{\sin normAngle}, \left(n0\_i \cdot -0.5\right) \cdot \left(normAngle \cdot \left(normAngle \cdot u\right)\right)\right)\right), n0\_i\right)} \]

9. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-1 \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot u\right) + \frac{1}{6} \cdot n1\_i\right)\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-1 \cdot \left(\frac{1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \frac{1}{120} \cdot n0\_i\right)\right) + \frac{7}{360} \cdot n1\_i\right)\right)\right)\right)} \]

11. Applied rewrites 99.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(u, \left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot 0.019444444444444445 - \left(n0\_i \cdot 0.041666666666666664 - \mathsf{fma}\left(n0\_i, 0.008333333333333333, n0\_i \cdot 0.05555555555555555\right)\right)\right), u \cdot \mathsf{fma}\left(n1\_i, 0.16666666666666666, \mathsf{fma}\left(n0\_i, 0.3333333333333333, -0.5 \cdot \left(n0\_i \cdot u\right)\right)\right)\right), \mathsf{fma}\left(u, n1\_i - n0\_i, n0\_i\right)\right)} \]

12. Final simplification 99.4%

    \[\leadsto \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(u, \left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot 0.019444444444444445 + \left(\mathsf{fma}\left(n0\_i, 0.008333333333333333, n0\_i \cdot 0.05555555555555555\right) - n0\_i \cdot 0.041666666666666664\right)\right), u \cdot \mathsf{fma}\left(n1\_i, 0.16666666666666666, \mathsf{fma}\left(n0\_i, 0.3333333333333333, -0.5 \cdot \left(u \cdot n0\_i\right)\right)\right)\right), \mathsf{fma}\left(u, n1\_i - n0\_i, n0\_i\right)\right) \]
13. Add Preprocessing

Alternative 2: 99.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.3333333333333333, \mathsf{fma}\left(normAngle \cdot normAngle, n1\_i \cdot 0.019444444444444445 + \left(\mathsf{fma}\left(n0\_i, 0.008333333333333333, n0\_i \cdot 0.05555555555555555\right) - n0\_i \cdot 0.041666666666666664\right), \mathsf{fma}\left(n0\_i, u \cdot -0.5, n1\_i \cdot 0.16666666666666666\right)\right)\right), n1\_i - n0\_i\right), n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (fma
   (* normAngle normAngle)
   (fma
    n0_i
    0.3333333333333333
    (fma
     (* normAngle normAngle)
     (+
      (* n1_i 0.019444444444444445)
      (-
       (fma n0_i 0.008333333333333333 (* n0_i 0.05555555555555555))
       (* n0_i 0.041666666666666664)))
     (fma n0_i (* u -0.5) (* 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, fmaf((normAngle * normAngle), ((n1_i * 0.019444444444444445f) + (fmaf(n0_i, 0.008333333333333333f, (n0_i * 0.05555555555555555f)) - (n0_i * 0.041666666666666664f))), fmaf(n0_i, (u * -0.5f), (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(n0_i, Float32(0.3333333333333333), fma(Float32(normAngle * normAngle), Float32(Float32(n1_i * Float32(0.019444444444444445)) + Float32(fma(n0_i, Float32(0.008333333333333333), Float32(n0_i * Float32(0.05555555555555555))) - Float32(n0_i * Float32(0.041666666666666664)))), fma(n0_i, Float32(u * Float32(-0.5)), Float32(n1_i * Float32(0.16666666666666666))))), Float32(n1_i - n0_i)), n0_i)
end

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in normAngle around 0

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\left(u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot {u}^{3} + {normAngle}^{2} \cdot \left(\frac{1}{120} \cdot {u}^{5} - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot {u}^{3} - \frac{-1}{6} \cdot u\right) + \frac{1}{120} \cdot u\right)\right)\right) - \frac{-1}{6} \cdot u\right)\right)} \cdot n1\_i \]

5. Applied rewrites 98.6%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(u, \mathsf{fma}\left(u \cdot u, -0.16666666666666666, 0.16666666666666666\right), \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(0.008333333333333333, {u}^{5} - u, -0.027777777777777776 \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right)\right), u\right)} \cdot n1\_i \]

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + n1\_i \cdot \left(1 + {normAngle}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {normAngle}^{2}\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + n1\_i \cdot \left(1 + {normAngle}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {normAngle}^{2}\right)\right)\right)\right) + n0\_i} \]

   2. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, -1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + n1\_i \cdot \left(1 + {normAngle}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {normAngle}^{2}\right)\right)\right), n0\_i\right)} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(n1\_i, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, 0.019444444444444445, 0.16666666666666666\right), 1\right), \mathsf{fma}\left(-n0\_i, \frac{normAngle \cdot \cos normAngle}{\sin normAngle}, \left(n0\_i \cdot -0.5\right) \cdot \left(normAngle \cdot \left(normAngle \cdot u\right)\right)\right)\right), n0\_i\right)} \]

9. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(-1 \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot u\right) + \left(\frac{1}{6} \cdot n1\_i + {normAngle}^{2} \cdot \left(-1 \cdot \left(\frac{1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot n0\_i - \frac{-1}{6} \cdot n0\_i\right) + \frac{1}{120} \cdot n0\_i\right)\right) + \frac{7}{360} \cdot n1\_i\right)\right)\right)\right)\right)}, n0\_i\right) \]

11. Applied rewrites 99.4%

    \[\leadsto \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.3333333333333333, \mathsf{fma}\left(normAngle \cdot normAngle, n1\_i \cdot 0.019444444444444445 - \left(n0\_i \cdot 0.041666666666666664 - \mathsf{fma}\left(n0\_i, 0.008333333333333333, n0\_i \cdot 0.05555555555555555\right)\right), \mathsf{fma}\left(n0\_i, u \cdot -0.5, n1\_i \cdot 0.16666666666666666\right)\right)\right), n1\_i - n0\_i\right)}, n0\_i\right) \]

12. Final simplification 99.4%

    \[\leadsto \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.3333333333333333, \mathsf{fma}\left(normAngle \cdot normAngle, n1\_i \cdot 0.019444444444444445 + \left(\mathsf{fma}\left(n0\_i, 0.008333333333333333, n0\_i \cdot 0.05555555555555555\right) - n0\_i \cdot 0.041666666666666664\right), \mathsf{fma}\left(n0\_i, u \cdot -0.5, n1\_i \cdot 0.16666666666666666\right)\right)\right), n1\_i - n0\_i\right), n0\_i\right) \]
13. Add Preprocessing

Alternative 3: 99.1% accurate, 8.7× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-*.f32 99.4

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(-1 \cdot \left({normAngle}^{2} \cdot \left(u \cdot \left(\frac{-1}{36} \cdot n1\_i + \frac{1}{120} \cdot n1\_i\right)\right)\right) + u \cdot \left(\frac{1}{3} \cdot n0\_i - \frac{-1}{6} \cdot n1\_i\right)\right)\right)} \]

11. Applied rewrites 99.4%

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

12. Final simplification 99.4%

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

Alternative 4: 99.1% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in normAngle around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-*.f32 99.4

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in normAngle around 0

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

    1. associate-+r+ N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f32 N/A

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

11. Applied rewrites 99.3%

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

Alternative 5: 99.0% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.4%

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

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 99.2%

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

9. Final simplification 99.2%

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

Alternative 6: 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 96.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.4%

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

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 99.2%

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

Alternative 7: 98.8% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.4%

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

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 99.2%

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

9. Taylor expanded in n0_i around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 98.9

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

11. Applied rewrites 98.9%

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

12. Final simplification 98.9%

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

Alternative 8: 98.3% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.4%

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

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 99.2%

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

9. Taylor expanded in n0_i around inf

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

    1. *-commutative N/A

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

    2. lower-*.f32 98.7

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

11. Applied rewrites 98.7%

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

12. Final simplification 98.7%

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

Alternative 9: 69.8% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n0\_i, -u, n0\_i\right)\\ \mathbf{if}\;n0\_i \leq -4.0000000126843074 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (fma n0_i (- u) n0_i)))
   (if (<= n0_i -4.0000000126843074e-28)
     t_0
     (if (<= n0_i 5.000000136226006e-28) (* u n1_i) t_0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = fmaf(n0_i, -u, n0_i);
	float tmp;
	if (n0_i <= -4.0000000126843074e-28f) {
		tmp = t_0;
	} else if (n0_i <= 5.000000136226006e-28f) {
		tmp = u * n1_i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = fma(n0_i, Float32(-u), n0_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-4.0000000126843074e-28))
		tmp = t_0;
	elseif (n0_i <= Float32(5.000000136226006e-28))
		tmp = Float32(u * n1_i);
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.00000001e-28 or 5.00000014e-28 < n0_i

   1. Initial program 97.5%

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

   3. Taylor expanded in normAngle around 0

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

      1. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in n0_i around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-neg.f32 73.4

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

   8. Applied rewrites 73.4%

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

   if -4.00000001e-28 < n0_i < 5.00000014e-28

   1. Initial program 93.6%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in n0_i around 0

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

      1. lower-*.f32 77.4

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

   8. Applied rewrites 77.4%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -4.0000000126843074 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(n0\_i, -u, n0\_i\right)\\ \mathbf{elif}\;n0\_i \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n0\_i, -u, n0\_i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.8% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.000000102130615 \cdot 10^{-19}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{elif}\;n1\_i \leq 2.4499999519773335 \cdot 10^{-17}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -9.000000102130615e-19)
   (* u n1_i)
   (if (<= n1_i 2.4499999519773335e-17) (* n0_i (- 1.0 u)) (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -9.000000102130615e-19f) {
		tmp = u * n1_i;
	} else if (n1_i <= 2.4499999519773335e-17f) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = u * n1_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n1_i <= (-9.000000102130615e-19)) then
        tmp = u * n1_i
    else if (n1_i <= 2.4499999519773335e-17) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-9.000000102130615e-19))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(2.4499999519773335e-17))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n1_i <= single(-9.000000102130615e-19))
		tmp = u * n1_i;
	elseif (n1_i <= single(2.4499999519773335e-17))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -9.0000001e-19 or 2.44999995e-17 < n1_i

   1. Initial program 95.6%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in n0_i around 0

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

      1. lower-*.f32 66.5

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

   8. Applied rewrites 66.5%

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

   if -9.0000001e-19 < n1_i < 2.44999995e-17

   1. Initial program 97.1%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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 82.4

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

   5. Applied rewrites 82.4%

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

   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 81.9

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

   8. Applied rewrites 81.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.000000102130615 \cdot 10^{-19}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{elif}\;n1\_i \leq 2.4499999519773335 \cdot 10^{-17}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.4% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.000000102130615 \cdot 10^{-19}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{elif}\;n1\_i \leq 2.4499999519773335 \cdot 10^{-17}:\\ \;\;\;\;n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -9.000000102130615e-19)
   (* u n1_i)
   (if (<= n1_i 2.4499999519773335e-17) n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -9.000000102130615e-19f) {
		tmp = u * n1_i;
	} else if (n1_i <= 2.4499999519773335e-17f) {
		tmp = n0_i;
	} else {
		tmp = u * n1_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n1_i <= (-9.000000102130615e-19)) then
        tmp = u * n1_i
    else if (n1_i <= 2.4499999519773335e-17) then
        tmp = n0_i
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-9.000000102130615e-19))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(2.4499999519773335e-17))
		tmp = n0_i;
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n1_i <= single(-9.000000102130615e-19))
		tmp = u * n1_i;
	elseif (n1_i <= single(2.4499999519773335e-17))
		tmp = n0_i;
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -9.0000001e-19 or 2.44999995e-17 < n1_i

   1. Initial program 95.6%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in n0_i around 0

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

      1. lower-*.f32 66.5

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

   8. Applied rewrites 66.5%

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

   if -9.0000001e-19 < n1_i < 2.44999995e-17

   1. Initial program 97.1%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 65.4%

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

      1. *-rgt-identity 65.4

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

   10. Applied rewrites 65.4%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.000000102130615 \cdot 10^{-19}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{elif}\;n1\_i \leq 2.4499999519773335 \cdot 10^{-17}:\\ \;\;\;\;n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.2% accurate, 45.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.4%

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

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 98.4

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

8. Applied rewrites 98.4%

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

Alternative 13: 46.7% 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 96.3%

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

3. Taylor expanded in 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 58.2

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

5. Applied rewrites 58.2%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 47.1%

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

   1. *-rgt-identity 47.1

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

10. Applied rewrites 47.1%

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

Reproduce

herbie shell --seed 2024212 
(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)))