Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 99.4%

Time: 22.3s

Alternatives: 14

Speedup: 45.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right)\\ t_1 := n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\\ t_2 := \mathsf{fma}\left(1 - u, t\_0, t\_1\right)\\ t_3 := \mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right)\\ t_4 := {\left(1 - u\right)}^{5}\\ \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, t\_2, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(0.008333333333333333, \mathsf{fma}\left(n0\_i, t\_4, n1\_i \cdot {u}^{5}\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(-0.0001984126984126984, \mathsf{fma}\left(n0\_i, {\left(1 - u\right)}^{7}, n1\_i \cdot {u}^{7}\right), -\mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(n1\_i, 0.008333333333333333 \cdot \left({u}^{5} - u\right), t\_1 \cdot -0.027777777777777776\right) + \mathsf{fma}\left(n0\_i, 0.008333333333333333 \cdot \left(t\_4 + \left(u + -1\right)\right), -0.027777777777777776 \cdot \left(\left(1 - u\right) \cdot t\_0\right)\right), \mathsf{fma}\left(-0.001388888888888889, t\_2, -0.0001984126984126984 \cdot t\_3\right)\right)\right), -\mathsf{fma}\left(0.027777777777777776, t\_2, 0.008333333333333333 \cdot t\_3\right)\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (fma n0_i (* (- 1.0 u) (- 1.0 u)) (- n0_i)))
        (t_1 (* n1_i (* u (fma u u -1.0))))
        (t_2 (fma (- 1.0 u) t_0 t_1))
        (t_3 (fma n0_i (- 1.0 u) (* u n1_i)))
        (t_4 (pow (- 1.0 u) 5.0)))
   (fma
    n0_i
    (- 1.0 u)
    (fma
     u
     n1_i
     (*
      (* normAngle normAngle)
      (fma
       -0.16666666666666666
       t_2
       (*
        (* normAngle normAngle)
        (fma
         0.008333333333333333
         (fma n0_i t_4 (* n1_i (pow u 5.0)))
         (fma
          (* normAngle normAngle)
          (fma
           -0.0001984126984126984
           (fma n0_i (pow (- 1.0 u) 7.0) (* n1_i (pow u 7.0)))
           (-
            (fma
             -0.16666666666666666
             (+
              (fma
               n1_i
               (* 0.008333333333333333 (- (pow u 5.0) u))
               (* t_1 -0.027777777777777776))
              (fma
               n0_i
               (* 0.008333333333333333 (+ t_4 (+ u -1.0)))
               (* -0.027777777777777776 (* (- 1.0 u) t_0))))
             (fma -0.001388888888888889 t_2 (* -0.0001984126984126984 t_3)))))
          (-
           (fma
            0.027777777777777776
            t_2
            (* 0.008333333333333333 t_3))))))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = fmaf(n0_i, ((1.0f - u) * (1.0f - u)), -n0_i);
	float t_1 = n1_i * (u * fmaf(u, u, -1.0f));
	float t_2 = fmaf((1.0f - u), t_0, t_1);
	float t_3 = fmaf(n0_i, (1.0f - u), (u * n1_i));
	float t_4 = powf((1.0f - u), 5.0f);
	return fmaf(n0_i, (1.0f - u), fmaf(u, n1_i, ((normAngle * normAngle) * fmaf(-0.16666666666666666f, t_2, ((normAngle * normAngle) * fmaf(0.008333333333333333f, fmaf(n0_i, t_4, (n1_i * powf(u, 5.0f))), fmaf((normAngle * normAngle), fmaf(-0.0001984126984126984f, fmaf(n0_i, powf((1.0f - u), 7.0f), (n1_i * powf(u, 7.0f))), -fmaf(-0.16666666666666666f, (fmaf(n1_i, (0.008333333333333333f * (powf(u, 5.0f) - u)), (t_1 * -0.027777777777777776f)) + fmaf(n0_i, (0.008333333333333333f * (t_4 + (u + -1.0f))), (-0.027777777777777776f * ((1.0f - u) * t_0)))), fmaf(-0.001388888888888889f, t_2, (-0.0001984126984126984f * t_3)))), -fmaf(0.027777777777777776f, t_2, (0.008333333333333333f * t_3)))))))));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = fma(n0_i, Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)), Float32(-n0_i))
	t_1 = Float32(n1_i * Float32(u * fma(u, u, Float32(-1.0))))
	t_2 = fma(Float32(Float32(1.0) - u), t_0, t_1)
	t_3 = fma(n0_i, Float32(Float32(1.0) - u), Float32(u * n1_i))
	t_4 = Float32(Float32(1.0) - u) ^ Float32(5.0)
	return fma(n0_i, Float32(Float32(1.0) - u), fma(u, n1_i, Float32(Float32(normAngle * normAngle) * fma(Float32(-0.16666666666666666), t_2, Float32(Float32(normAngle * normAngle) * fma(Float32(0.008333333333333333), fma(n0_i, t_4, Float32(n1_i * (u ^ Float32(5.0)))), fma(Float32(normAngle * normAngle), fma(Float32(-0.0001984126984126984), fma(n0_i, (Float32(Float32(1.0) - u) ^ Float32(7.0)), Float32(n1_i * (u ^ Float32(7.0)))), Float32(-fma(Float32(-0.16666666666666666), Float32(fma(n1_i, Float32(Float32(0.008333333333333333) * Float32((u ^ Float32(5.0)) - u)), Float32(t_1 * Float32(-0.027777777777777776))) + fma(n0_i, Float32(Float32(0.008333333333333333) * Float32(t_4 + Float32(u + Float32(-1.0)))), Float32(Float32(-0.027777777777777776) * Float32(Float32(Float32(1.0) - u) * t_0)))), fma(Float32(-0.001388888888888889), t_2, Float32(Float32(-0.0001984126984126984) * t_3))))), Float32(-fma(Float32(0.027777777777777776), t_2, Float32(Float32(0.008333333333333333) * t_3))))))))))
end

Derivation

1. Initial program 98.0%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
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) + \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) + {normAngle}^{2} \cdot \left(\left(\frac{-1}{5040} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{7}\right) + \frac{-1}{5040} \cdot \left(n1\_i \cdot {u}^{7}\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\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) + \frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\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) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right) + \left(\frac{-1}{5040} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \left(\frac{-1}{5040} \cdot \left(n1\_i \cdot u\right) + \left(\frac{1}{120} \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) + \frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\right)\right)\right)\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 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(0.008333333333333333, \mathsf{fma}\left(n0\_i, {\left(1 - u\right)}^{5}, n1\_i \cdot {u}^{5}\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(-0.0001984126984126984, \mathsf{fma}\left(n0\_i, {\left(1 - u\right)}^{7}, n1\_i \cdot {u}^{7}\right), -\mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(n1\_i, 0.008333333333333333 \cdot \left({u}^{5} - u\right), -0.027777777777777776 \cdot \left(n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right)\right) + \mathsf{fma}\left(n0\_i, 0.008333333333333333 \cdot \left({\left(1 - u\right)}^{5} - \left(1 - u\right)\right), -0.027777777777777776 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right)\right)\right), \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), -0.0001984126984126984 \cdot \mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right)\right)\right)\right), -\mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right) \cdot 0.008333333333333333\right)\right)\right)\right)\right)\right)} \]

6. Final simplification 99.1%

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(u, n1\_i, \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), \left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(0.008333333333333333, \mathsf{fma}\left(n0\_i, {\left(1 - u\right)}^{5}, n1\_i \cdot {u}^{5}\right), \mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(-0.0001984126984126984, \mathsf{fma}\left(n0\_i, {\left(1 - u\right)}^{7}, n1\_i \cdot {u}^{7}\right), -\mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(n1\_i, 0.008333333333333333 \cdot \left({u}^{5} - u\right), \left(n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right) \cdot -0.027777777777777776\right) + \mathsf{fma}\left(n0\_i, 0.008333333333333333 \cdot \left({\left(1 - u\right)}^{5} + \left(u + -1\right)\right), -0.027777777777777776 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right)\right)\right), \mathsf{fma}\left(-0.001388888888888889, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), -0.0001984126984126984 \cdot \mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right)\right)\right)\right), -\mathsf{fma}\left(0.027777777777777776, \mathsf{fma}\left(1 - u, \mathsf{fma}\left(n0\_i, \left(1 - u\right) \cdot \left(1 - u\right), -n0\_i\right), n1\_i \cdot \left(u \cdot \mathsf{fma}\left(u, u, -1\right)\right)\right), 0.008333333333333333 \cdot \mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right)\right)\right)\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 2: 99.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 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 99.3%

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

Reproduce

herbie shell --seed 2024219 
(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)))