Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 99.3%

Time: 14.4s

Alternatives: 9

Speedup: 45.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n0\_i, -2, -n1\_i\right)\\ \mathsf{fma}\left(u, n1\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.08333333333333333, -0.027777777777777776 \cdot \left(n0\_i \cdot 3\right)\right), \left(n0\_i \cdot 3\right) \cdot -0.16666666666666666\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, -0.041666666666666664, -0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right) + -0.027777777777777776 \cdot t\_0, -0.16666666666666666 \cdot t\_0\right)\right), -n0\_i\right), n0\_i\right) \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (fma n0_i -2.0 (- n1_i))))
   (fma
    u
    (+
     n1_i
     (fma
      (* normAngle normAngle)
      (fma
       u
       (fma
        (* normAngle normAngle)
        (fma n0_i 0.08333333333333333 (* -0.027777777777777776 (* n0_i 3.0)))
        (* (* n0_i 3.0) -0.16666666666666666))
       (fma
        (* normAngle normAngle)
        (+
         (fma
          n0_i
          -0.041666666666666664
          (* -0.008333333333333333 (- n1_i n0_i)))
         (* -0.027777777777777776 t_0))
        (* -0.16666666666666666 t_0)))
      (- n0_i)))
    n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = fmaf(n0_i, -2.0f, -n1_i);
	return fmaf(u, (n1_i + fmaf((normAngle * normAngle), fmaf(u, fmaf((normAngle * normAngle), fmaf(n0_i, 0.08333333333333333f, (-0.027777777777777776f * (n0_i * 3.0f))), ((n0_i * 3.0f) * -0.16666666666666666f)), fmaf((normAngle * normAngle), (fmaf(n0_i, -0.041666666666666664f, (-0.008333333333333333f * (n1_i - n0_i))) + (-0.027777777777777776f * t_0)), (-0.16666666666666666f * t_0))), -n0_i)), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = fma(n0_i, Float32(-2.0), Float32(-n1_i))
	return fma(u, Float32(n1_i + fma(Float32(normAngle * normAngle), fma(u, fma(Float32(normAngle * normAngle), fma(n0_i, Float32(0.08333333333333333), Float32(Float32(-0.027777777777777776) * Float32(n0_i * Float32(3.0)))), Float32(Float32(n0_i * Float32(3.0)) * Float32(-0.16666666666666666))), fma(Float32(normAngle * normAngle), Float32(fma(n0_i, Float32(-0.041666666666666664), Float32(Float32(-0.008333333333333333) * Float32(n1_i - n0_i))) + Float32(Float32(-0.027777777777777776) * t_0)), Float32(Float32(-0.16666666666666666) * t_0))), Float32(-n0_i))), n0_i)
end

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in normAngle around 0

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \left({normAngle}^{2} \cdot \left(u \cdot \left(\frac{-1}{6} \cdot \left(n0\_i + 2 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(\frac{1}{12} \cdot n0\_i - \frac{1}{36} \cdot \left(n0\_i + 2 \cdot n0\_i\right)\right)\right)\right) + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot \left(-2 \cdot n0\_i + -1 \cdot n1\_i\right) + {normAngle}^{2} \cdot \left(\frac{-1}{24} \cdot n0\_i - \left(\frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right) + \frac{1}{36} \cdot \left(-2 \cdot n0\_i + -1 \cdot n1\_i\right)\right)\right)\right)\right)\right)\right)} \]

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.08333333333333333, -0.027777777777777776 \cdot \left(3 \cdot n0\_i\right)\right), -0.16666666666666666 \cdot \left(3 \cdot n0\_i\right)\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, -0.041666666666666664, -0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right) + -0.027777777777777776 \cdot \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), -0.16666666666666666 \cdot \mathsf{fma}\left(n0\_i, -2, -n1\_i\right)\right)\right), -n0\_i\right), n0\_i\right)} \]

9. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(u, n1\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(u, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, 0.08333333333333333, -0.027777777777777776 \cdot \left(n0\_i \cdot 3\right)\right), \left(n0\_i \cdot 3\right) \cdot -0.16666666666666666\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, -0.041666666666666664, -0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right) + -0.027777777777777776 \cdot \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), -0.16666666666666666 \cdot \mathsf{fma}\left(n0\_i, -2, -n1\_i\right)\right)\right), -n0\_i\right), n0\_i\right) \]
10. Add Preprocessing

Alternative 2: 99.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n0\_i, -2, -n1\_i\right)\\ \mathsf{fma}\left(u, n1\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, -0.041666666666666664, -0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right) + -0.027777777777777776 \cdot t\_0, -0.16666666666666666 \cdot t\_0\right), -n0\_i\right), n0\_i\right) \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (fma n0_i -2.0 (- n1_i))))
   (fma
    u
    (+
     n1_i
     (fma
      (* normAngle normAngle)
      (fma
       (* normAngle normAngle)
       (+
        (fma
         n0_i
         -0.041666666666666664
         (* -0.008333333333333333 (- n1_i n0_i)))
        (* -0.027777777777777776 t_0))
       (* -0.16666666666666666 t_0))
      (- n0_i)))
    n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = fmaf(n0_i, -2.0f, -n1_i);
	return fmaf(u, (n1_i + fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), (fmaf(n0_i, -0.041666666666666664f, (-0.008333333333333333f * (n1_i - n0_i))) + (-0.027777777777777776f * t_0)), (-0.16666666666666666f * t_0)), -n0_i)), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = fma(n0_i, Float32(-2.0), Float32(-n1_i))
	return fma(u, Float32(n1_i + fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), Float32(fma(n0_i, Float32(-0.041666666666666664), Float32(Float32(-0.008333333333333333) * Float32(n1_i - n0_i))) + Float32(Float32(-0.027777777777777776) * t_0)), Float32(Float32(-0.16666666666666666) * t_0)), Float32(-n0_i))), n0_i)
end

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in normAngle around 0

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

8. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1\_i + \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(n0\_i, -0.041666666666666664, -0.008333333333333333 \cdot \left(n1\_i - n0\_i\right)\right) + -0.027777777777777776 \cdot \mathsf{fma}\left(n0\_i, -2, -n1\_i\right), -0.16666666666666666 \cdot \mathsf{fma}\left(n0\_i, -2, -n1\_i\right)\right), -n0\_i\right), n0\_i\right)} \]
9. Add Preprocessing

Alternative 3: 99.0% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in normAngle around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

8. Applied rewrites 99.3%

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

9. Final simplification 99.3%

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

Alternative 4: 99.0% accurate, 14.3× speedup

Math

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

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, Float32(fma(Float32(normAngle * normAngle), fma(n0_i, Float32(0.3333333333333333), Float32(n1_i * Float32(0.16666666666666666))), n1_i) - n0_i), n0_i)
end

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in normAngle around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in n1_i around 0

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

11. Applied rewrites 99.3%

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

Alternative 5: 60.5% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -5.000000097707407 \cdot 10^{-25}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 7.99999974612418 \cdot 10^{-19}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -5.000000097707407e-25)
   n0_i
   (if (<= n0_i 7.99999974612418e-19) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -5.000000097707407e-25f) {
		tmp = n0_i;
	} else if (n0_i <= 7.99999974612418e-19f) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-5.000000097707407e-25)) then
        tmp = n0_i
    else if (n0_i <= 7.99999974612418e-19) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-5.000000097707407e-25))
		tmp = n0_i;
	elseif (n0_i <= Float32(7.99999974612418e-19))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-5.000000097707407e-25))
		tmp = n0_i;
	elseif (n0_i <= single(7.99999974612418e-19))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -5.0000001e-25 or 7.99999975e-19 < 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 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 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 58.4%

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

      1. *-rgt-identity 58.4

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

   10. Applied rewrites 58.4%

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

   if -5.0000001e-25 < n0_i < 7.99999975e-19

   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 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in n0_i around 0

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

      1. lower-*.f32 68.7

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

   8. Applied rewrites 68.7%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -5.000000097707407 \cdot 10^{-25}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 7.99999974612418 \cdot 10^{-19}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -3.000000026176508 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(n0\_i, -u, n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n1\_i, u, n0\_i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -3.000000026176508e-9) (fma n0_i (- u) n0_i) (fma n1_i u n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -3.000000026176508e-9f) {
		tmp = fmaf(n0_i, -u, n0_i);
	} else {
		tmp = fmaf(n1_i, u, n0_i);
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-3.000000026176508e-9))
		tmp = fma(n0_i, Float32(-u), n0_i);
	else
		tmp = fma(n1_i, u, n0_i);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -3.00000003e-9

   1. Initial program 98.6%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in n0_i around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f32 96.6

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

   8. Applied rewrites 96.6%

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

   if -3.00000003e-9 < n0_i

   1. Initial program 97.8%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)} \]
   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 n0_i around 0

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sin.f32 74.3

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

   8. Applied rewrites 74.3%

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

   9. Taylor expanded in normAngle around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f32 82.3

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

   11. Applied rewrites 82.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(n1\_i, u, n0\_i\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.0% accurate, 45.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in normAngle around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 98.5

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

8. Applied rewrites 98.5%

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

Alternative 8: 81.8% accurate, 65.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in n0_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 73.2

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

8. Applied rewrites 73.2%

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

9. Taylor expanded in normAngle around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f32 80.4

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

11. Applied rewrites 80.4%

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

Alternative 9: 47.2% accurate, 459.0× speedup

Math

\[\begin{array}{l} \\ n0\_i \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in 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 55.3

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

5. Applied rewrites 55.3%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 42.8%

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

   1. *-rgt-identity 42.8

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

10. Applied rewrites 42.8%

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

Reproduce

herbie shell --seed 2024220 
(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)))