Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 98.8%

Time: 14.4s

Alternatives: 9

Speedup: 60.1×

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: 97.3% 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: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(1 - u\right) \cdot n0_i + n1_i \cdot u\right) + \left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) + -0.16666666666666666 \cdot \left(n0_i \cdot \left(u + -1\right) - n1_i \cdot u\right)\right) \cdot {normAngle}^{2} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (+ (* (- 1.0 u) n0_i) (* n1_i u))
  (*
   (+
    (+
     (* -0.16666666666666666 (* n1_i (pow u 3.0)))
     (* -0.16666666666666666 (* (pow (- 1.0 u) 3.0) n0_i)))
    (* -0.16666666666666666 (- (* n0_i (+ u -1.0)) (* n1_i u))))
   (pow normAngle 2.0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (((1.0f - u) * n0_i) + (n1_i * u)) + ((((-0.16666666666666666f * (n1_i * powf(u, 3.0f))) + (-0.16666666666666666f * (powf((1.0f - u), 3.0f) * n0_i))) + (-0.16666666666666666f * ((n0_i * (u + -1.0f)) - (n1_i * u)))) * powf(normAngle, 2.0f));
}

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 - u) * n0_i) + (n1_i * u)) + (((((-0.16666666666666666e0) * (n1_i * (u ** 3.0e0))) + ((-0.16666666666666666e0) * (((1.0e0 - u) ** 3.0e0) * n0_i))) + ((-0.16666666666666666e0) * ((n0_i * (u + (-1.0e0))) - (n1_i * u)))) * (normangle ** 2.0e0))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(Float32(Float32(Float32(1.0) - u) * n0_i) + Float32(n1_i * u)) + Float32(Float32(Float32(Float32(Float32(-0.16666666666666666) * Float32(n1_i * (u ^ Float32(3.0)))) + Float32(Float32(-0.16666666666666666) * Float32((Float32(Float32(1.0) - u) ^ Float32(3.0)) * n0_i))) + Float32(Float32(-0.16666666666666666) * Float32(Float32(n0_i * Float32(u + Float32(-1.0))) - Float32(n1_i * u)))) * (normAngle ^ Float32(2.0))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (((single(1.0) - u) * n0_i) + (n1_i * u)) + ((((single(-0.16666666666666666) * (n1_i * (u ^ single(3.0)))) + (single(-0.16666666666666666) * (((single(1.0) - u) ^ single(3.0)) * n0_i))) + (single(-0.16666666666666666) * ((n0_i * (u + single(-1.0))) - (n1_i * u)))) * (normAngle ^ single(2.0)));
end

Derivation

1. Initial program 97.6%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. +-commutative 97.6%

      \[\leadsto \color{blue}{\left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i} \]

   2. *-commutative 97.6%

      \[\leadsto \color{blue}{n1_i \cdot \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i \]

   3. associate-*r* 85.7%

      \[\leadsto \color{blue}{\left(n1_i \cdot \sin \left(u \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i \]

   4. *-commutative 85.7%

      \[\leadsto \left(n1_i \cdot \sin \left(u \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle} + \color{blue}{n0_i \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} \]

   5. associate-*r* 73.3%

      \[\leadsto \left(n1_i \cdot \sin \left(u \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle} + \color{blue}{\left(n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} \]

   6. distribute-rgt-out 73.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \]

   7. *-commutative 73.3%

      \[\leadsto \color{blue}{\left(n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} \]

   8. associate-*r/ 73.4%

      \[\leadsto \color{blue}{\frac{\left(n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot 1}{\sin normAngle}} \]

   9. associate-/l* 73.4%

      \[\leadsto \color{blue}{\frac{n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)}{\frac{\sin normAngle}{1}}} \]

   10. *-commutative 73.4%

       \[\leadsto \frac{\color{blue}{\sin \left(u \cdot normAngle\right) \cdot n1_i} + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)}{\frac{\sin normAngle}{1}} \]

   11. fma-def 73.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)}}{\frac{\sin normAngle}{1}} \]

   12. *-commutative 73.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, \color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}\right)}{\frac{\sin normAngle}{1}} \]

   13. /-rgt-identity 73.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)}{\color{blue}{\sin normAngle}} \]

3. Simplified 73.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)}{\sin normAngle}} \]

4. Taylor expanded in normAngle around 0 98.4%

   \[\leadsto \color{blue}{\left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)} \]

5. Final simplification 98.4%

   \[\leadsto \left(\left(1 - u\right) \cdot n0_i + n1_i \cdot u\right) + \left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) + -0.16666666666666666 \cdot \left(n0_i \cdot \left(u + -1\right) - n1_i \cdot u\right)\right) \cdot {normAngle}^{2} \]

Alternative 2: 98.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(1 - u\right) \cdot n0_i + n1_i \cdot u\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (+ (* (- 1.0 u) n0_i) (* n1_i u))
  (*
   (pow normAngle 2.0)
   (* u (+ (* n0_i 0.3333333333333333) (* n1_i 0.16666666666666666))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (((1.0f - u) * n0_i) + (n1_i * u)) + (powf(normAngle, 2.0f) * (u * ((n0_i * 0.3333333333333333f) + (n1_i * 0.16666666666666666f))));
}

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 - u) * n0_i) + (n1_i * u)) + ((normangle ** 2.0e0) * (u * ((n0_i * 0.3333333333333333e0) + (n1_i * 0.16666666666666666e0))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(Float32(Float32(Float32(1.0) - u) * n0_i) + Float32(n1_i * u)) + Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(Float32(n0_i * Float32(0.3333333333333333)) + Float32(n1_i * Float32(0.16666666666666666))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (((single(1.0) - u) * n0_i) + (n1_i * u)) + ((normAngle ^ single(2.0)) * (u * ((n0_i * single(0.3333333333333333)) + (n1_i * single(0.16666666666666666)))));
end

Derivation

1. Initial program 97.6%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. +-commutative 97.6%

      \[\leadsto \color{blue}{\left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i} \]

   2. *-commutative 97.6%

      \[\leadsto \color{blue}{n1_i \cdot \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i \]

   3. associate-*r* 85.7%

      \[\leadsto \color{blue}{\left(n1_i \cdot \sin \left(u \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i \]

   4. *-commutative 85.7%

      \[\leadsto \left(n1_i \cdot \sin \left(u \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle} + \color{blue}{n0_i \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} \]

   5. associate-*r* 73.3%

      \[\leadsto \left(n1_i \cdot \sin \left(u \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle} + \color{blue}{\left(n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} \]

   6. distribute-rgt-out 73.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \]

   7. *-commutative 73.3%

      \[\leadsto \color{blue}{\left(n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} \]

   8. associate-*r/ 73.4%

      \[\leadsto \color{blue}{\frac{\left(n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot 1}{\sin normAngle}} \]

   9. associate-/l* 73.4%

      \[\leadsto \color{blue}{\frac{n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)}{\frac{\sin normAngle}{1}}} \]

   10. *-commutative 73.4%

       \[\leadsto \frac{\color{blue}{\sin \left(u \cdot normAngle\right) \cdot n1_i} + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)}{\frac{\sin normAngle}{1}} \]

   11. fma-def 73.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)}}{\frac{\sin normAngle}{1}} \]

   12. *-commutative 73.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, \color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}\right)}{\frac{\sin normAngle}{1}} \]

   13. /-rgt-identity 73.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)}{\color{blue}{\sin normAngle}} \]

3. Simplified 73.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)}{\sin normAngle}} \]

4. Taylor expanded in normAngle around 0 98.4%

   \[\leadsto \color{blue}{\left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)} \]

5. Taylor expanded in u around 0 98.0%

   \[\leadsto \color{blue}{\left(\left(0.5 \cdot n0_i - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right) \cdot u\right)} \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]
6. Step-by-step derivation

   1. *-commutative 98.0%

      \[\leadsto \color{blue}{\left(u \cdot \left(0.5 \cdot n0_i - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right)\right)} \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

   2. cancel-sign-sub-inv 98.0%

      \[\leadsto \left(u \cdot \color{blue}{\left(0.5 \cdot n0_i + \left(--0.16666666666666666\right) \cdot \left(n1_i + -1 \cdot n0_i\right)\right)}\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

   3. *-commutative 98.0%

      \[\leadsto \left(u \cdot \left(\color{blue}{n0_i \cdot 0.5} + \left(--0.16666666666666666\right) \cdot \left(n1_i + -1 \cdot n0_i\right)\right)\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

   4. metadata-eval 98.0%

      \[\leadsto \left(u \cdot \left(n0_i \cdot 0.5 + \color{blue}{0.16666666666666666} \cdot \left(n1_i + -1 \cdot n0_i\right)\right)\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

   5. mul-1-neg 98.0%

      \[\leadsto \left(u \cdot \left(n0_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1_i + \color{blue}{\left(-n0_i\right)}\right)\right)\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

7. Simplified 98.0%

   \[\leadsto \color{blue}{\left(u \cdot \left(n0_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1_i + \left(-n0_i\right)\right)\right)\right)} \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

8. Taylor expanded in n0_i around 0 98.0%

   \[\leadsto \left(u \cdot \color{blue}{\left(0.3333333333333333 \cdot n0_i + 0.16666666666666666 \cdot n1_i\right)}\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

9. Final simplification 98.0%

   \[\leadsto \left(\left(1 - u\right) \cdot n0_i + n1_i \cdot u\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right)\right) \]

Alternative 3: 98.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(1 - u\right) \cdot n0_i + n1_i \cdot u\right) + {normAngle}^{2} \cdot \left(n1_i \cdot \left(u \cdot 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (+ (* (- 1.0 u) n0_i) (* n1_i u))
  (* (pow normAngle 2.0) (* n1_i (* u 0.16666666666666666)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (((1.0f - u) * n0_i) + (n1_i * u)) + (powf(normAngle, 2.0f) * (n1_i * (u * 0.16666666666666666f)));
}

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 - u) * n0_i) + (n1_i * u)) + ((normangle ** 2.0e0) * (n1_i * (u * 0.16666666666666666e0)))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(Float32(Float32(Float32(1.0) - u) * n0_i) + Float32(n1_i * u)) + Float32((normAngle ^ Float32(2.0)) * Float32(n1_i * Float32(u * Float32(0.16666666666666666)))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (((single(1.0) - u) * n0_i) + (n1_i * u)) + ((normAngle ^ single(2.0)) * (n1_i * (u * single(0.16666666666666666))));
end

Derivation

1. Initial program 97.6%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. +-commutative 97.6%

      \[\leadsto \color{blue}{\left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i} \]

   2. *-commutative 97.6%

      \[\leadsto \color{blue}{n1_i \cdot \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i \]

   3. associate-*r* 85.7%

      \[\leadsto \color{blue}{\left(n1_i \cdot \sin \left(u \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i \]

   4. *-commutative 85.7%

      \[\leadsto \left(n1_i \cdot \sin \left(u \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle} + \color{blue}{n0_i \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} \]

   5. associate-*r* 73.3%

      \[\leadsto \left(n1_i \cdot \sin \left(u \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle} + \color{blue}{\left(n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} \]

   6. distribute-rgt-out 73.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \]

   7. *-commutative 73.3%

      \[\leadsto \color{blue}{\left(n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} \]

   8. associate-*r/ 73.4%

      \[\leadsto \color{blue}{\frac{\left(n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot 1}{\sin normAngle}} \]

   9. associate-/l* 73.4%

      \[\leadsto \color{blue}{\frac{n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)}{\frac{\sin normAngle}{1}}} \]

   10. *-commutative 73.4%

       \[\leadsto \frac{\color{blue}{\sin \left(u \cdot normAngle\right) \cdot n1_i} + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)}{\frac{\sin normAngle}{1}} \]

   11. fma-def 73.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)}}{\frac{\sin normAngle}{1}} \]

   12. *-commutative 73.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, \color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i}\right)}{\frac{\sin normAngle}{1}} \]

   13. /-rgt-identity 73.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)}{\color{blue}{\sin normAngle}} \]

3. Simplified 73.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(u \cdot normAngle\right), n1_i, \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)}{\sin normAngle}} \]

4. Taylor expanded in normAngle around 0 98.4%

   \[\leadsto \color{blue}{\left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)} \]

5. Taylor expanded in u around 0 98.0%

   \[\leadsto \color{blue}{\left(\left(0.5 \cdot n0_i - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right) \cdot u\right)} \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]
6. Step-by-step derivation

   1. *-commutative 98.0%

      \[\leadsto \color{blue}{\left(u \cdot \left(0.5 \cdot n0_i - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right)\right)} \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

   2. cancel-sign-sub-inv 98.0%

      \[\leadsto \left(u \cdot \color{blue}{\left(0.5 \cdot n0_i + \left(--0.16666666666666666\right) \cdot \left(n1_i + -1 \cdot n0_i\right)\right)}\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

   3. *-commutative 98.0%

      \[\leadsto \left(u \cdot \left(\color{blue}{n0_i \cdot 0.5} + \left(--0.16666666666666666\right) \cdot \left(n1_i + -1 \cdot n0_i\right)\right)\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

   4. metadata-eval 98.0%

      \[\leadsto \left(u \cdot \left(n0_i \cdot 0.5 + \color{blue}{0.16666666666666666} \cdot \left(n1_i + -1 \cdot n0_i\right)\right)\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

   5. mul-1-neg 98.0%

      \[\leadsto \left(u \cdot \left(n0_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1_i + \color{blue}{\left(-n0_i\right)}\right)\right)\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

7. Simplified 98.0%

   \[\leadsto \color{blue}{\left(u \cdot \left(n0_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1_i + \left(-n0_i\right)\right)\right)\right)} \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

8. Taylor expanded in n0_i around 0 97.9%

   \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(n1_i \cdot u\right)\right)} \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]
9. Step-by-step derivation

   1. associate-*r* 97.9%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot n1_i\right) \cdot u\right)} \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

   2. *-commutative 97.9%

      \[\leadsto \left(\color{blue}{\left(n1_i \cdot 0.16666666666666666\right)} \cdot u\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

   3. associate-*l* 97.9%

      \[\leadsto \color{blue}{\left(n1_i \cdot \left(0.16666666666666666 \cdot u\right)\right)} \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

10. Simplified 97.9%

    \[\leadsto \color{blue}{\left(n1_i \cdot \left(0.16666666666666666 \cdot u\right)\right)} \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) \]

11. Final simplification 97.9%

    \[\leadsto \left(\left(1 - u\right) \cdot n0_i + n1_i \cdot u\right) + {normAngle}^{2} \cdot \left(n1_i \cdot \left(u \cdot 0.16666666666666666\right)\right) \]

Alternative 4: 98.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. fma-def 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

   2. associate-*r/ 97.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   3. *-rgt-identity 97.9%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   4. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

   5. *-rgt-identity 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

4. Taylor expanded in normAngle around 0 97.3%

   \[\leadsto \color{blue}{n1_i \cdot u + \left(1 - u\right) \cdot n0_i} \]

5. Taylor expanded in u around 0 97.6%

   \[\leadsto \color{blue}{\left(n1_i + -1 \cdot n0_i\right) \cdot u + n0_i} \]
6. Step-by-step derivation

   1. *-commutative 97.6%

      \[\leadsto \color{blue}{u \cdot \left(n1_i + -1 \cdot n0_i\right)} + n0_i \]

   2. fma-def 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1_i + -1 \cdot n0_i, n0_i\right)} \]

   3. mul-1-neg 97.7%

      \[\leadsto \mathsf{fma}\left(u, n1_i + \color{blue}{\left(-n0_i\right)}, n0_i\right) \]

   4. unsub-neg 97.7%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1_i - n0_i}, n0_i\right) \]

7. Simplified 97.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right)} \]

8. Final simplification 97.7%

   \[\leadsto \mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \]

Alternative 5: 70.5% accurate, 45.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -3.5000000656357872 \cdot 10^{-15}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -3.5000000656357872e-15)
   (* n1_i u)
   (if (<= n1_i 1.9999999920083944e-12) (* (- 1.0 u) n0_i) (* n1_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -3.5000000656357872e-15f) {
		tmp = n1_i * u;
	} else if (n1_i <= 1.9999999920083944e-12f) {
		tmp = (1.0f - u) * n0_i;
	} else {
		tmp = n1_i * u;
	}
	return tmp;
}

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) :: tmp
    if (n1_i <= (-3.5000000656357872e-15)) then
        tmp = n1_i * u
    else if (n1_i <= 1.9999999920083944e-12) then
        tmp = (1.0e0 - u) * n0_i
    else
        tmp = n1_i * u
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-3.5000000656357872e-15))
		tmp = Float32(n1_i * u);
	elseif (n1_i <= Float32(1.9999999920083944e-12))
		tmp = Float32(Float32(Float32(1.0) - u) * n0_i);
	else
		tmp = Float32(n1_i * u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n1_i <= single(-3.5000000656357872e-15))
		tmp = n1_i * u;
	elseif (n1_i <= single(1.9999999920083944e-12))
		tmp = (single(1.0) - u) * n0_i;
	else
		tmp = n1_i * u;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -3.50000007e-15 or 1.99999999e-12 < 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. Step-by-step derivation

      1. fma-def 97.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

      2. associate-*r/ 97.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      3. *-rgt-identity 97.5%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      4. associate-*r/ 98.0%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

      5. *-rgt-identity 98.0%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 97.1%

      \[\leadsto \color{blue}{n1_i \cdot u + \left(1 - u\right) \cdot n0_i} \]

   5. Taylor expanded in n1_i around inf 68.1%

      \[\leadsto \color{blue}{n1_i \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 68.1%

         \[\leadsto \color{blue}{u \cdot n1_i} \]

   7. Simplified 68.1%

      \[\leadsto \color{blue}{u \cdot n1_i} \]

   if -3.50000007e-15 < n1_i < 1.99999999e-12

   1. Initial program 97.8%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. fma-def 97.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

      2. associate-*r/ 98.2%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      3. *-rgt-identity 98.2%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      4. associate-*r/ 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

      5. *-rgt-identity 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 97.4%

      \[\leadsto \color{blue}{n1_i \cdot u + \left(1 - u\right) \cdot n0_i} \]

   5. Taylor expanded in n1_i around 0 75.5%

      \[\leadsto \color{blue}{\left(1 - u\right) \cdot n0_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -3.5000000656357872 \cdot 10^{-15}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 6: 60.1% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -3.5000000656357872 \cdot 10^{-15}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -3.5000000656357872e-15)
   (* n1_i u)
   (if (<= n1_i 1.9999999920083944e-12) n0_i (* n1_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -3.5000000656357872e-15f) {
		tmp = n1_i * u;
	} else if (n1_i <= 1.9999999920083944e-12f) {
		tmp = n0_i;
	} else {
		tmp = n1_i * u;
	}
	return tmp;
}

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) :: tmp
    if (n1_i <= (-3.5000000656357872e-15)) then
        tmp = n1_i * u
    else if (n1_i <= 1.9999999920083944e-12) then
        tmp = n0_i
    else
        tmp = n1_i * u
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-3.5000000656357872e-15))
		tmp = Float32(n1_i * u);
	elseif (n1_i <= Float32(1.9999999920083944e-12))
		tmp = n0_i;
	else
		tmp = Float32(n1_i * u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n1_i <= single(-3.5000000656357872e-15))
		tmp = n1_i * u;
	elseif (n1_i <= single(1.9999999920083944e-12))
		tmp = n0_i;
	else
		tmp = n1_i * u;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -3.50000007e-15 or 1.99999999e-12 < 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. Step-by-step derivation

      1. fma-def 97.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

      2. associate-*r/ 97.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      3. *-rgt-identity 97.5%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      4. associate-*r/ 98.0%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

      5. *-rgt-identity 98.0%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 97.1%

      \[\leadsto \color{blue}{n1_i \cdot u + \left(1 - u\right) \cdot n0_i} \]

   5. Taylor expanded in n1_i around inf 68.1%

      \[\leadsto \color{blue}{n1_i \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 68.1%

         \[\leadsto \color{blue}{u \cdot n1_i} \]

   7. Simplified 68.1%

      \[\leadsto \color{blue}{u \cdot n1_i} \]

   if -3.50000007e-15 < n1_i < 1.99999999e-12

   1. Initial program 97.8%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. fma-def 97.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

      2. associate-*r/ 98.2%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      3. *-rgt-identity 98.2%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

      4. associate-*r/ 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

      5. *-rgt-identity 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

   4. Taylor expanded in u around 0 58.6%

      \[\leadsto \color{blue}{n0_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -3.5000000656357872 \cdot 10^{-15}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 7: 98.1% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ n0_i + u \cdot \left(n1_i - n0_i\right) \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * (n1_i - 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 = n0_i + (u * (n1_i - n0_i))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. fma-def 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

   2. associate-*r/ 97.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   3. *-rgt-identity 97.9%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   4. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

   5. *-rgt-identity 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

4. Taylor expanded in normAngle around 0 97.3%

   \[\leadsto \color{blue}{n1_i \cdot u + \left(1 - u\right) \cdot n0_i} \]

5. Taylor expanded in u around 0 97.6%

   \[\leadsto \color{blue}{\left(n1_i + -1 \cdot n0_i\right) \cdot u + n0_i} \]
6. Step-by-step derivation

   1. *-commutative 97.6%

      \[\leadsto \color{blue}{u \cdot \left(n1_i + -1 \cdot n0_i\right)} + n0_i \]

   2. fma-def 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1_i + -1 \cdot n0_i, n0_i\right)} \]

   3. mul-1-neg 97.7%

      \[\leadsto \mathsf{fma}\left(u, n1_i + \color{blue}{\left(-n0_i\right)}, n0_i\right) \]

   4. unsub-neg 97.7%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1_i - n0_i}, n0_i\right) \]

7. Simplified 97.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right)} \]

8. Taylor expanded in u around 0 97.6%

   \[\leadsto \color{blue}{\left(n1_i - n0_i\right) \cdot u + n0_i} \]

9. Final simplification 97.6%

   \[\leadsto n0_i + u \cdot \left(n1_i - n0_i\right) \]

Alternative 8: 81.7% accurate, 84.2× speedup

Math

\[\begin{array}{l} \\ n0_i + n1_i \cdot u \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i) :precision binary32 (+ n0_i (* n1_i u)))

C

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

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 = n0_i + (n1_i * u)
end function

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (n1_i * u);
end

Derivation

1. Initial program 97.6%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. +-commutative 97.6%

      \[\leadsto \color{blue}{\left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i} \]

   2. *-commutative 97.6%

      \[\leadsto \color{blue}{n1_i \cdot \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i \]

   3. associate-*r* 85.7%

      \[\leadsto \color{blue}{\left(n1_i \cdot \sin \left(u \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} + \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i \]

   4. *-commutative 85.7%

      \[\leadsto \left(n1_i \cdot \sin \left(u \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle} + \color{blue}{n0_i \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} \]

   5. associate-*r* 73.3%

      \[\leadsto \left(n1_i \cdot \sin \left(u \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle} + \color{blue}{\left(n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} \]

   6. distribute-rgt-out 73.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \]

   7. *-commutative 73.3%

      \[\leadsto \color{blue}{\left(n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} \]

   8. associate-*r/ 73.4%

      \[\leadsto \color{blue}{\frac{\left(n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot 1}{\sin normAngle}} \]

   9. associate-/l* 73.4%

      \[\leadsto \color{blue}{\frac{n1_i \cdot \sin \left(u \cdot normAngle\right) + n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)}{\frac{\sin normAngle}{1}}} \]

3. Simplified 73.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(normAngle - u \cdot normAngle\right), n0_i, \sin \left(u \cdot normAngle\right) \cdot n1_i\right)}{\sin normAngle}} \]
4. Step-by-step derivation

   1. fma-udef 73.5%

      \[\leadsto \frac{\color{blue}{\sin \left(normAngle - u \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i}}{\sin normAngle} \]

5. Applied egg-rr 73.5%

   \[\leadsto \frac{\color{blue}{\sin \left(normAngle - u \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i}}{\sin normAngle} \]

6. Taylor expanded in u around 0 63.0%

   \[\leadsto \frac{\color{blue}{\sin normAngle \cdot n0_i} + \sin \left(u \cdot normAngle\right) \cdot n1_i}{\sin normAngle} \]

7. Taylor expanded in normAngle around 0 81.0%

   \[\leadsto \color{blue}{n1_i \cdot u + n0_i} \]

8. Final simplification 81.0%

   \[\leadsto n0_i + n1_i \cdot u \]

Alternative 9: 47.1% accurate, 421.0× speedup

Math

\[\begin{array}{l} \\ n0_i \end{array} \]

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)
    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 97.6%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. fma-def 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

   2. associate-*r/ 97.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   3. *-rgt-identity 97.9%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right) \]

   4. associate-*r/ 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1_i\right) \]

   5. *-rgt-identity 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1_i\right) \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\right)} \]

4. Taylor expanded in u around 0 44.6%

   \[\leadsto \color{blue}{n0_i} \]

5. Final simplification 44.6%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023258 
(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)))