Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 98.8%

Time: 13.1s

Alternatives: 10

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: 97.2% 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, 3.4× speedup

Math

\[\begin{array}{l} \\ n1_i \cdot u - \left(\left(u \cdot n0_i - n0_i\right) - \left(u \cdot \left(0.5 \cdot n0_i - 0.16666666666666666 \cdot \left(n0_i - n1_i\right)\right)\right) \cdot {normAngle}^{2}\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (-
  (* n1_i u)
  (-
   (- (* u n0_i) n0_i)
   (*
    (* u (- (* 0.5 n0_i) (* 0.16666666666666666 (- n0_i n1_i))))
    (pow normAngle 2.0)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n1_i * u) - (((u * n0_i) - n0_i) - ((u * ((0.5f * n0_i) - (0.16666666666666666f * (n0_i - n1_i)))) * 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 = (n1_i * u) - (((u * n0_i) - n0_i) - ((u * ((0.5e0 * n0_i) - (0.16666666666666666e0 * (n0_i - n1_i)))) * (normangle ** 2.0e0)))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(n1_i * u) - Float32(Float32(Float32(u * n0_i) - n0_i) - Float32(Float32(u * Float32(Float32(Float32(0.5) * n0_i) - Float32(Float32(0.16666666666666666) * Float32(n0_i - n1_i)))) * (normAngle ^ Float32(2.0)))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n1_i * u) - (((u * n0_i) - n0_i) - ((u * ((single(0.5) * n0_i) - (single(0.16666666666666666) * (n0_i - n1_i)))) * (normAngle ^ single(2.0))));
end

Derivation

1. Initial program 97.9%

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

   1. +-commutative 97.9%

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

   2. *-commutative 97.9%

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

   3. associate-*r* 86.2%

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

   4. *-commutative 86.2%

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

   5. associate-*r* 70.4%

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

   6. distribute-rgt-out 70.5%

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

   7. *-commutative 70.5%

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

   8. associate-*r/ 70.7%

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

   9. associate-/l* 70.7%

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

3. Simplified 70.7%

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

4. Taylor expanded in u around 0 70.6%

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

   1. associate-*r* 70.5%

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

   2. *-commutative 70.5%

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

6. Simplified 70.5%

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

7. Taylor expanded in normAngle around 0 98.7%

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

8. Taylor expanded in u around 0 98.6%

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

   1. cancel-sign-sub-inv 98.6%

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

   2. metadata-eval 98.6%

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

   3. neg-mul-1 98.6%

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

   4. sub-neg 98.6%

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

10. Simplified 98.6%

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

11. Taylor expanded in u around 0 98.8%

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

    1. +-commutative 98.8%

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

    2. mul-1-neg 98.8%

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

    3. unsub-neg 98.8%

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

    4. *-commutative 98.8%

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

13. Simplified 98.8%

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

14. Final simplification 98.8%

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

Alternative 2: 98.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n1_i * u) + ((powf(normAngle, 2.0f) * (u * ((n0_i * 0.3333333333333333f) + (n1_i * 0.16666666666666666f)))) + (n0_i * (1.0f - 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 = (n1_i * u) + (((normangle ** 2.0e0) * (u * ((n0_i * 0.3333333333333333e0) + (n1_i * 0.16666666666666666e0)))) + (n0_i * (1.0e0 - u)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. +-commutative 97.9%

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

   2. *-commutative 97.9%

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

   3. associate-*r* 86.2%

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

   4. *-commutative 86.2%

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

   5. associate-*r* 70.4%

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

   6. distribute-rgt-out 70.5%

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

   7. *-commutative 70.5%

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

   8. associate-*r/ 70.7%

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

   9. associate-/l* 70.7%

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

3. Simplified 70.7%

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

4. Taylor expanded in u around 0 70.6%

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

   1. associate-*r* 70.5%

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

   2. *-commutative 70.5%

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

6. Simplified 70.5%

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

7. Taylor expanded in normAngle around 0 98.7%

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

8. Taylor expanded in u around 0 98.6%

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

   1. cancel-sign-sub-inv 98.6%

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

   2. metadata-eval 98.6%

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

   3. neg-mul-1 98.6%

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

   4. sub-neg 98.6%

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

10. Simplified 98.6%

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

11. Taylor expanded in n0_i around 0 98.6%

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

12. Final simplification 98.6%

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

Alternative 3: 98.5% accurate, 3.6× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n1_i * u) + ((n0_i * (1.0f - 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 = (n1_i * u) + ((n0_i * (1.0e0 - u)) + ((normangle ** 2.0e0) * (n1_i * (u * 0.16666666666666666e0))))
end function

Julia

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

Derivation

1. Initial program 97.9%

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

   1. +-commutative 97.9%

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

   2. *-commutative 97.9%

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

   3. associate-*r* 86.2%

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

   4. *-commutative 86.2%

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

   5. associate-*r* 70.4%

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

   6. distribute-rgt-out 70.5%

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

   7. *-commutative 70.5%

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

   8. associate-*r/ 70.7%

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

   9. associate-/l* 70.7%

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

3. Simplified 70.7%

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

4. Taylor expanded in u around 0 70.6%

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

   1. associate-*r* 70.5%

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

   2. *-commutative 70.5%

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

6. Simplified 70.5%

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

7. Taylor expanded in normAngle around 0 98.7%

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

8. Taylor expanded in n0_i around 0 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-*l* 98.4%

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

10. Simplified 98.4%

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

11. Final simplification 98.4%

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

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 97.9%

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

   1. fma-def 98.0%

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

   2. associate-*r/ 98.3%

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

   3. *-rgt-identity 98.3%

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

   4. associate-*r/ 98.4%

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

   5. *-rgt-identity 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in normAngle around 0 98.0%

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

5. Taylor expanded in u around 0 98.2%

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

   1. *-commutative 98.2%

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

   2. fma-def 98.4%

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

   3. mul-1-neg 98.4%

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

   4. unsub-neg 98.4%

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

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Alternative 5: 70.4% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -4.0000000126843074 \cdot 10^{-30} \lor \neg \left(n0_i \leq 4.999999943633011 \cdot 10^{-27}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -4.0000000126843074e-30)
         (not (<= n0_i 4.999999943633011e-27)))
   (* n0_i (- 1.0 u))
   (* n1_i u)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -4.0000000126843074e-30f) || !(n0_i <= 4.999999943633011e-27f)) {
		tmp = n0_i * (1.0f - u);
	} 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 ((n0_i <= (-4.0000000126843074e-30)) .or. (.not. (n0_i <= 4.999999943633011e-27))) then
        tmp = n0_i * (1.0e0 - u)
    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 ((n0_i <= Float32(-4.0000000126843074e-30)) || !(n0_i <= Float32(4.999999943633011e-27)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	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 ((n0_i <= single(-4.0000000126843074e-30)) || ~((n0_i <= single(4.999999943633011e-27))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = n1_i * u;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4e-30 or 4.99999994e-27 < n0_i

   1. Initial program 98.4%

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

      1. fma-def 98.5%

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

      2. associate-*r/ 98.9%

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

      3. *-rgt-identity 98.9%

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

      4. associate-*r/ 98.9%

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

      5. *-rgt-identity 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in normAngle around 0 98.0%

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

   5. Taylor expanded in n1_i around 0 74.0%

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

   if -4e-30 < n0_i < 4.99999994e-27

   1. Initial program 96.7%

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

      1. fma-def 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/ 96.7%

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

      3. *-rgt-identity 96.7%

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

      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. Taylor expanded in normAngle around 0 97.8%

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

   5. Taylor expanded in u around inf 70.8%

      \[\leadsto \color{blue}{n1_i \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -4.0000000126843074 \cdot 10^{-30} \lor \neg \left(n0_i \leq 4.999999943633011 \cdot 10^{-27}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 6: 85.8% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -9.999999974752427 \cdot 10^{-7} \lor \neg \left(n0_i \leq 7.99999985961336 \cdot 10^{-13}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u + n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -9.999999974752427e-7) (not (<= n0_i 7.99999985961336e-13)))
   (* n0_i (- 1.0 u))
   (+ (* n1_i u) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -9.999999974752427e-7f) || !(n0_i <= 7.99999985961336e-13f)) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = (n1_i * u) + n0_i;
	}
	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 ((n0_i <= (-9.999999974752427e-7)) .or. (.not. (n0_i <= 7.99999985961336e-13))) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = (n1_i * u) + n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-9.999999974752427e-7)) || !(n0_i <= Float32(7.99999985961336e-13)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(Float32(n1_i * u) + n0_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-9.999999974752427e-7)) || ~((n0_i <= single(7.99999985961336e-13))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = (n1_i * u) + n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -9.99999997e-7 or 7.99999986e-13 < n0_i

   1. Initial program 98.9%

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

      1. fma-def 99.0%

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

      2. associate-*r/ 99.3%

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

      3. *-rgt-identity 99.3%

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

      4. associate-*r/ 99.4%

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

      5. *-rgt-identity 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in normAngle around 0 99.0%

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

   5. Taylor expanded in n1_i around 0 90.0%

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

   if -9.99999997e-7 < n0_i < 7.99999986e-13

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*r* 81.4%

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

      4. *-commutative 81.4%

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

      5. associate-*r* 61.1%

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

      6. distribute-rgt-out 61.2%

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

      7. *-commutative 61.2%

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

      8. associate-*r/ 61.3%

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

      9. associate-/l* 61.3%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in u around 0 61.2%

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

      1. associate-*r* 61.2%

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

      2. *-commutative 61.2%

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

   6. Simplified 61.2%

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

   7. Taylor expanded in normAngle around inf 61.2%

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

   8. Taylor expanded in u around 0 56.0%

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

   9. Taylor expanded in normAngle around 0 84.2%

      \[\leadsto \color{blue}{n1_i \cdot u + n0_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -9.999999974752427 \cdot 10^{-7} \lor \neg \left(n0_i \leq 7.99999985961336 \cdot 10^{-13}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u + n0_i\\ \end{array} \]

Alternative 7: 85.8% accurate, 45.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{elif}\;n0_i \leq 7.99999985961336 \cdot 10^{-13}:\\ \;\;\;\;n1_i \cdot u + n0_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -9.999999974752427e-7)
   (- n0_i (* u n0_i))
   (if (<= n0_i 7.99999985961336e-13) (+ (* 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 <= -9.999999974752427e-7f) {
		tmp = n0_i - (u * n0_i);
	} else if (n0_i <= 7.99999985961336e-13f) {
		tmp = (n1_i * u) + n0_i;
	} else {
		tmp = n0_i * (1.0f - 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 (n0_i <= (-9.999999974752427e-7)) then
        tmp = n0_i - (u * n0_i)
    else if (n0_i <= 7.99999985961336e-13) then
        tmp = (n1_i * u) + n0_i
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-9.999999974752427e-7))
		tmp = Float32(n0_i - Float32(u * n0_i));
	elseif (n0_i <= Float32(7.99999985961336e-13))
		tmp = Float32(Float32(n1_i * u) + n0_i);
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-9.999999974752427e-7))
		tmp = n0_i - (u * n0_i);
	elseif (n0_i <= single(7.99999985961336e-13))
		tmp = (n1_i * u) + n0_i;
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if n0_i < -9.99999997e-7

   1. Initial program 98.8%

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

      1. fma-def 98.7%

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

      2. associate-*r/ 99.1%

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

      3. *-rgt-identity 99.1%

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

      4. associate-*r/ 99.1%

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

      5. *-rgt-identity 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in normAngle around 0 98.3%

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

   5. Taylor expanded in u around 0 98.5%

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

      1. +-commutative 98.9%

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

      2. mul-1-neg 98.9%

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

      3. unsub-neg 98.9%

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

      4. *-commutative 98.9%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in n1_i around 0 93.1%

      \[\leadsto \color{blue}{n0_i - u \cdot n0_i} \]

   if -9.99999997e-7 < n0_i < 7.99999986e-13

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*r* 81.4%

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

      4. *-commutative 81.4%

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

      5. associate-*r* 61.1%

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

      6. distribute-rgt-out 61.2%

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

      7. *-commutative 61.2%

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

      8. associate-*r/ 61.3%

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

      9. associate-/l* 61.3%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in u around 0 61.2%

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

      1. associate-*r* 61.2%

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

      2. *-commutative 61.2%

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

   6. Simplified 61.2%

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

   7. Taylor expanded in normAngle around inf 61.2%

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

   8. Taylor expanded in u around 0 56.0%

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

   9. Taylor expanded in normAngle around 0 84.2%

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

   if 7.99999986e-13 < n0_i

   1. Initial program 99.0%

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

      1. fma-def 99.2%

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

      2. associate-*r/ 99.5%

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

      3. *-rgt-identity 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-rgt-identity 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in normAngle around 0 99.4%

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

   5. Taylor expanded in n1_i around 0 88.5%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{elif}\;n0_i \leq 7.99999985961336 \cdot 10^{-13}:\\ \;\;\;\;n1_i \cdot u + n0_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 8: 60.5% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -1.4999999523982838 \cdot 10^{-21}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.4999999523982838e-21)
   n0_i
   (if (<= n0_i 4.999999943633011e-27) (* n1_i u) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.4999999523982838e-21f) {
		tmp = n0_i;
	} else if (n0_i <= 4.999999943633011e-27f) {
		tmp = n1_i * u;
	} else {
		tmp = n0_i;
	}
	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 (n0_i <= (-1.4999999523982838e-21)) then
        tmp = n0_i
    else if (n0_i <= 4.999999943633011e-27) then
        tmp = n1_i * u
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-1.4999999523982838e-21))
		tmp = n0_i;
	elseif (n0_i <= Float32(4.999999943633011e-27))
		tmp = Float32(n1_i * u);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-1.4999999523982838e-21))
		tmp = n0_i;
	elseif (n0_i <= single(4.999999943633011e-27))
		tmp = n1_i * u;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.5e-21 or 4.99999994e-27 < n0_i

   1. Initial program 98.4%

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

      1. fma-def 98.5%

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

      2. associate-*r/ 98.9%

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

      3. *-rgt-identity 98.9%

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

      4. associate-*r/ 98.9%

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

      5. *-rgt-identity 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in u around 0 57.7%

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

   if -1.5e-21 < n0_i < 4.99999994e-27

   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. Step-by-step derivation

      1. fma-def 97.1%

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

      2. associate-*r/ 97.2%

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

      3. *-rgt-identity 97.2%

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

      4. associate-*r/ 97.6%

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

      5. *-rgt-identity 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in normAngle around 0 97.0%

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

   5. Taylor expanded in u around inf 64.5%

      \[\leadsto \color{blue}{n1_i \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -1.4999999523982838 \cdot 10^{-21}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 9: 98.1% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. fma-def 98.0%

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

   2. associate-*r/ 98.3%

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

   3. *-rgt-identity 98.3%

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

   4. associate-*r/ 98.4%

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

   5. *-rgt-identity 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in normAngle around 0 98.0%

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

5. Taylor expanded in u around -inf 98.2%

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

   1. +-commutative 98.2%

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

   2. mul-1-neg 98.2%

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

   3. unsub-neg 98.2%

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

   4. +-commutative 98.2%

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

   5. mul-1-neg 98.2%

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

   6. unsub-neg 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 10: 47.6% 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 97.9%

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

   1. fma-def 98.0%

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

   2. associate-*r/ 98.3%

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

   3. *-rgt-identity 98.3%

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

   4. associate-*r/ 98.4%

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

   5. *-rgt-identity 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in u around 0 46.8%

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

5. Final simplification 46.8%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023213 
(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)))