Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.1%

Time: 6.6s

Alternatives: 9

Speedup: 19.5×

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

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (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)
use fmin_fmax_functions
    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: 97.3% accurate, 1.0× speedup

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

Math

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

FPCore

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

Julia

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

MATLAB

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

Alternative 1: 99.1% accurate, 4.7× speedup

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

Math

\[n0\_i + u \cdot \mathsf{fma}\left(n1\_i, \frac{normAngle}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot normAngle}, -n0\_i\right) \]

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (+
 n0_i
 (*
  u
  (fma
   n1_i
   (/
    normAngle
    (*
     (fma
      (* normAngle normAngle)
      (fma
       (* normAngle normAngle)
       0.008333333333333333
       -0.16666666666666666)
      1.0)
     normAngle))
   (- n0_i)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * fmaf(n1_i, (normAngle / (fmaf((normAngle * normAngle), fmaf((normAngle * normAngle), 0.008333333333333333f, -0.16666666666666666f), 1.0f) * normAngle)), -n0_i));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * fma(n1_i, Float32(normAngle / Float32(fma(Float32(normAngle * normAngle), fma(Float32(normAngle * normAngle), Float32(0.008333333333333333), Float32(-0.16666666666666666)), Float32(1.0)) * normAngle)), Float32(-n0_i))))
end

Derivation

1. Initial program 97.3%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 88.0%

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

5. Taylor expanded in normAngle around 0

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

7. Applied rewrites 90.1%

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

8. Taylor expanded in normAngle around 0

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

10. Applied rewrites 90.0%

    \[\leadsto n0\_i + u \cdot \mathsf{fma}\left(-1, n0\_i, \frac{n1\_i \cdot normAngle}{normAngle \cdot \left(1 + {normAngle}^{2} \cdot \left(0.008333333333333333 \cdot {normAngle}^{2} - 0.16666666666666666\right)\right)}\right) \]
11. Step-by-step derivation

12. Applied rewrites 99.0%

    \[\leadsto n0\_i + u \cdot \mathsf{fma}\left(n1\_i, \frac{normAngle}{\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(normAngle \cdot normAngle, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot normAngle}, -n0\_i\right) \]
13. Add Preprocessing

Alternative 2: 99.0% accurate, 5.4× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.3%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 88.0%

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

5. Taylor expanded in normAngle around 0

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

7. Applied rewrites 90.1%

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

8. Taylor expanded in normAngle around 0

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

10. Applied rewrites 99.0%

    \[\leadsto n0\_i + u \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i + {normAngle}^{2} \cdot \left(-1 \cdot \left({normAngle}^{2} \cdot \mathsf{fma}\left(-0.027777777777777776, n1\_i, 0.008333333333333333 \cdot n1\_i\right)\right) - -0.16666666666666666 \cdot n1\_i\right)\right) \]

12. Applied rewrites 99.1%

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

Alternative 3: 98.9% accurate, 6.1× speedup

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

Math

\[n0\_i + u \cdot \mathsf{fma}\left(n1\_i, \frac{normAngle}{\mathsf{fma}\left(normAngle \cdot normAngle, -0.16666666666666666, 1\right) \cdot normAngle}, -n0\_i\right) \]

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (+
 n0_i
 (*
  u
  (fma
   n1_i
   (/
    normAngle
    (*
     (fma (* normAngle normAngle) -0.16666666666666666 1.0)
     normAngle))
   (- n0_i)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * fmaf(n1_i, (normAngle / (fmaf((normAngle * normAngle), -0.16666666666666666f, 1.0f) * normAngle)), -n0_i));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * fma(n1_i, Float32(normAngle / Float32(fma(Float32(normAngle * normAngle), Float32(-0.16666666666666666), Float32(1.0)) * normAngle)), Float32(-n0_i))))
end

Derivation

1. Initial program 97.3%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 88.0%

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

5. Taylor expanded in normAngle around 0

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

7. Applied rewrites 90.1%

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

8. Taylor expanded in normAngle around 0

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

10. Applied rewrites 89.8%

    \[\leadsto n0\_i + u \cdot \mathsf{fma}\left(-1, n0\_i, \frac{n1\_i \cdot normAngle}{normAngle \cdot \left(1 + -0.16666666666666666 \cdot {normAngle}^{2}\right)}\right) \]
11. Step-by-step derivation

12. Applied rewrites 98.9%

    \[\leadsto n0\_i + u \cdot \mathsf{fma}\left(n1\_i, \frac{normAngle}{\mathsf{fma}\left(normAngle \cdot normAngle, -0.16666666666666666, 1\right) \cdot normAngle}, -n0\_i\right) \]
13. Add Preprocessing

Alternative 4: 98.7% accurate, 7.2× speedup

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

Math

\[\mathsf{fma}\left(n0\_i, 1 - u, \left(n1\_i - \left(-0.16666666666666666 \cdot n1\_i\right) \cdot \left(normAngle \cdot normAngle\right)\right) \cdot u\right) \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(n0_i, (1.0f - u), ((n1_i - ((-0.16666666666666666f * n1_i) * (normAngle * normAngle))) * u));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(n0_i, Float32(Float32(1.0) - u), Float32(Float32(n1_i - Float32(Float32(Float32(-0.16666666666666666) * n1_i) * Float32(normAngle * normAngle))) * u))
end

Derivation

1. Initial program 97.3%

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

2. Taylor expanded in normAngle around 0

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - \left(\frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right) \]
3. Step-by-step derivation

4. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(n1\_i, u, {normAngle}^{2} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, n0\_i \cdot {\left(1 - u\right)}^{3}, -0.16666666666666666 \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - \mathsf{fma}\left(-0.16666666666666666, n0\_i \cdot \left(1 - u\right), -0.16666666666666666 \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right) \]

5. Taylor expanded in u around 0

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

7. Applied rewrites 98.8%

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, u \cdot \left(n1\_i + {normAngle}^{2} \cdot \left(0.5 \cdot n0\_i - \mathsf{fma}\left(-0.16666666666666666, n1\_i, 0.16666666666666666 \cdot n0\_i\right)\right)\right)\right) \]

9. Applied rewrites 98.8%

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \left(n1\_i - \left(-0.16666666666666666 \cdot n1\_i - n0\_i \cdot 0.3333333333333333\right) \cdot \left(normAngle \cdot normAngle\right)\right) \cdot u\right) \]

10. Taylor expanded in n0_i around 0

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

12. Applied rewrites 98.7%

    \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \left(n1\_i - \left(-0.16666666666666666 \cdot n1\_i\right) \cdot \left(normAngle \cdot normAngle\right)\right) \cdot u\right) \]
13. Add Preprocessing

Alternative 5: 98.4% accurate, 8.6× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.3%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 88.0%

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

5. Taylor expanded in normAngle around 0

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

7. Applied rewrites 90.3%

   \[\leadsto n0\_i + u \cdot \mathsf{fma}\left(-1, n0\_i + {normAngle}^{2} \cdot \left(-0.5 \cdot n0\_i - -0.16666666666666666 \cdot n0\_i\right), \frac{n1\_i \cdot normAngle}{\sin normAngle}\right) \]

9. Applied rewrites 90.4%

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

10. Taylor expanded in normAngle around 0

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

12. Applied rewrites 98.4%

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

Alternative 6: 98.3% accurate, 19.5× speedup

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

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (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 97.3%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 88.0%

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

5. Taylor expanded in normAngle around 0

   \[\leadsto n0\_i + u \cdot \left(n1\_i + -1 \cdot n0\_i\right) \]
6. Step-by-step derivation

7. Applied rewrites 98.2%

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

9. Applied rewrites 98.3%

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

Alternative 7: 85.9% accurate, 11.8× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;n1\_i \leq -3.018727824514825 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(n1\_i, u, n0\_i\right)\\ \mathbf{elif}\;n1\_i \leq 2.483693850208526 \cdot 10^{-22}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n1\_i, u, n0\_i\right)\\ \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (if (<= n1_i -3.018727824514825e-27)
  (fma n1_i u n0_i)
  (if (<= n1_i 2.483693850208526e-22)
    (* n0_i (- 1.0 u))
    (fma n1_i u n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -3.018727824514825e-27f) {
		tmp = fmaf(n1_i, u, n0_i);
	} else if (n1_i <= 2.483693850208526e-22f) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = fmaf(n1_i, u, n0_i);
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-3.018727824514825e-27))
		tmp = fma(n1_i, u, n0_i);
	elseif (n1_i <= Float32(2.483693850208526e-22))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = fma(n1_i, u, n0_i);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -3.01872782e-27 or 2.48369385e-22 < n1_i

   1. Initial program 97.3%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 81.0%

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

   5. Taylor expanded in normAngle around 0

      \[\leadsto n0\_i + u \cdot n1\_i \]
   6. Step-by-step derivation

   7. Applied rewrites 81.6%

      \[\leadsto n0\_i + u \cdot n1\_i \]
   8. Step-by-step derivation

   9. Applied rewrites 81.7%

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

   if -3.01872782e-27 < n1_i < 2.48369385e-22

   1. Initial program 97.3%

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

   2. Taylor expanded in normAngle around 0

      \[\leadsto n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u \]
   3. Step-by-step derivation

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in n0_i around -inf

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

   7. Applied rewrites 97.8%

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

   9. Applied rewrites 97.7%

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

   10. Taylor expanded in n0_i around inf

       \[\leadsto n0\_i \cdot \left(1 - u\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 58.4%

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

Alternative 8: 81.7% accurate, 28.0× speedup

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

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (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 97.3%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 81.0%

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

5. Taylor expanded in normAngle around 0

   \[\leadsto n0\_i + u \cdot n1\_i \]
6. Step-by-step derivation

7. Applied rewrites 81.6%

   \[\leadsto n0\_i + u \cdot n1\_i \]
8. Step-by-step derivation

9. Applied rewrites 81.7%

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

Alternative 9: 46.8% accurate, 33.6× speedup

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

Math

\[--1 \cdot n0\_i \]

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (- (* -1.0 n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return -(-1.0f * n0_i);
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
use fmin_fmax_functions
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = -((-1.0e0) * n0_i)
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(-Float32(Float32(-1.0) * n0_i))
end

MATLAB

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

Derivation

1. Initial program 97.3%

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

2. Taylor expanded in normAngle around 0

   \[\leadsto n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u \]
3. Step-by-step derivation

4. Applied rewrites 98.0%

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

5. Taylor expanded in n0_i around -inf

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in u around 0

   \[\leadsto -1 \cdot \left(n0\_i \cdot -1\right) \]
9. Step-by-step derivation

10. Applied rewrites 46.8%

    \[\leadsto -1 \cdot \left(n0\_i \cdot -1\right) \]

12. Applied rewrites 46.8%

    \[\leadsto --1 \cdot n0\_i \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2026084 
(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)))