Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 98.1%

Time: 1.1min

Alternatives: 3

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
  (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

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
  (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: 98.1% accurate, 19.5× speedup

Math

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

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 97.3%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 97.7%

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

4. Applied rewrites 97.7%

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

   1. lift-fma.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

      \[\leadsto \left(n0\_i + \left(\mathsf{neg}\left(u\right)\right) \cdot n0\_i\right) + n1\_i \cdot u \]

   6. associate-+l+ N/A

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

   7. add-flip-rev N/A

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

   8. lower-+.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. distribute-rgt-out-- N/A

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

   13. lower-*.f32 N/A

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

   14. lower-neg.f32 N/A

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

   15. lower--.f32 97.9%

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

6. Applied rewrites 97.9%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. sub-flip N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. distribute-lft-neg-in N/A

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

   10. lift--.f32 N/A

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

   11. sub-negate-rev N/A

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

   12. remove-double-neg N/A

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

   13. lower-fma.f32 N/A

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

   14. lower--.f32 98.1%

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

8. Applied rewrites 98.1%

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

Alternative 2: 82.0% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

4. Applied rewrites 97.6%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 81.9%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 82.0%

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

9. Applied rewrites 82.0%

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

Alternative 3: 46.6% accurate, 167.8× speedup

Math

\[n0\_i \]

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)
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 = 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 97.3%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 97.7%

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

4. Applied rewrites 97.7%

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

5. Taylor expanded in u around 0

   \[\leadsto n0\_i \]
6. Step-by-step derivation

7. Applied rewrites 46.6%

   \[\leadsto n0\_i \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025322 
(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)))