Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 99.1%

Time: 4.7s

Alternatives: 6

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: 97.1% 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, 6.2× 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

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

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
   (fma -0.16666666666666666 (- n1_i n0_i) (* -0.5 n0_i))
   (* 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(fmaf(-0.16666666666666666f, (n1_i - n0_i), (-0.5f * n0_i)), (normAngle * normAngle), n0_i)), u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(n1_i - fma(fma(Float32(-0.16666666666666666), Float32(n1_i - n0_i), Float32(Float32(-0.5) * n0_i)), Float32(normAngle * normAngle), n0_i)), u, 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. 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

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

7. Applied rewrites 98.9%

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

9. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(n1\_i - \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, n1\_i - n0\_i, -0.5 \cdot n0\_i\right), normAngle \cdot normAngle, n0\_i\right), u, n0\_i\right) \]
10. Add Preprocessing

Alternative 2: 98.9% 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

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

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 (* -0.16666666666666666 n1_i) (* 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), n0_i)), u, n0_i);
}

Julia

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

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

7. Applied rewrites 98.9%

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

9. Applied rewrites 99.1%

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

10. Taylor expanded in n0_i around 0

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

12. Applied rewrites 98.9%

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

Alternative 3: 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

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

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 (* -0.3333333333333333 n0_i) (* normAngle normAngle) n0_i))
 u
 n0_i))

C

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

Julia

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

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

7. Applied rewrites 98.9%

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

9. Applied rewrites 99.1%

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

10. Taylor expanded in n0_i around inf

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

12. Applied rewrites 98.4%

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

Alternative 4: 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 97.1%

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

2. 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 88.1%

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

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 98.1%

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

9. Applied rewrites 98.3%

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

Alternative 5: 84.3% accurate, 16.1× 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

\[\begin{array}{l} \mathbf{if}\;n0\_i \leq 1.5203668808942347 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(n1\_i, u, n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]

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 (<= n0_i 1.5203668808942347e-14)
  (fma n1_i u n0_i)
  (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= 1.5203668808942347e-14f) {
		tmp = fmaf(n1_i, u, n0_i);
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(1.5203668808942347e-14))
		tmp = fma(n1_i, u, n0_i);
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n0_i < 1.52036688e-14

   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. 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 80.8%

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

   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 1.52036688e-14 < n0_i

   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. 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

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in u around inf

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

   10. Applied rewrites 36.6%

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

   11. Taylor expanded in n0_i around inf

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

   13. Applied rewrites 58.7%

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

Alternative 6: 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 97.1%

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

2. 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 80.8%

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

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

Reproduce

herbie shell --seed 2026086 
(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)))