Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 97.6%

Time: 16.2s

Alternatives: 4

Speedup: 1.0×

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: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (/ (sin (* (- 1.0 u) normAngle)) (sin normAngle))
  n0_i
  (* (/ (sin (* u normAngle)) (sin normAngle)) n1_i)))

C

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

Julia

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

Derivation

1. Initial program 96.5%

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

   1. fma-define 96.6%

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

   2. associate-*r/ 96.9%

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

   3. *-rgt-identity 96.9%

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

   4. associate-*r/ 97.1%

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

   5. *-rgt-identity 97.1%

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

3. Simplified 97.1%

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

5. Final simplification 97.1%

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

Alternative 2: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = (n0_i * (sin(((1.0e0 - u) * normangle)) * t_0)) + (n1_i * (sin((u * normangle)) * t_0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.5%

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

3. Final simplification 96.5%

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

Alternative 3: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right), n0\_i, \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)}{\sin normAngle} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (/
  (fma (sin (* (- 1.0 u) normAngle)) n0_i (* (sin (* u normAngle)) n1_i))
  (sin normAngle)))

C

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

Julia

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

Derivation

1. Initial program 96.5%

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

   1. associate-*l* 96.3%

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

   2. cancel-sign-sub 96.3%

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

   3. *-commutative 96.3%

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

   4. associate-*r* 82.2%

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

   5. associate-*r/ 82.3%

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

   6. *-rgt-identity 82.3%

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

   7. sin-neg 82.3%

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

   8. distribute-lft-neg-out 82.3%

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

   9. associate-*l* 82.4%

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

   10. *-commutative 82.4%

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

   11. distribute-lft-neg-out 82.4%

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

   12. distribute-rgt-neg-out 82.4%

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

   13. associate-*r/ 82.6%

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

3. Simplified 77.3%

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

5. Final simplification 77.3%

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

Alternative 4: 73.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (1.0f / sinf(normAngle)) * ((sinf((u * normAngle)) * n1_i) + (sinf(((1.0f - u) * normAngle)) * 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 = (1.0e0 / sin(normangle)) * ((sin((u * normangle)) * n1_i) + (sin(((1.0e0 - u) * normangle)) * n0_i))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.5%

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

   1. *-commutative 96.5%

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

   2. associate-*l* 82.4%

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

   3. *-commutative 82.4%

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

   4. associate-*l* 77.2%

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

   5. distribute-lft-out 77.2%

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

3. Simplified 77.2%

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

5. Final simplification 77.2%

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

Reproduce

herbie shell --seed 2024043 
(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)))