Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.0%

Time: 15.0s

Alternatives: 8

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: 99.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, \left(n1\_i + \left(normAngle \cdot normAngle\right) \cdot \left(0.3333333333333333 \cdot n0\_i - n1\_i \cdot -0.16666666666666666\right)\right) - n0\_i, n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (-
   (+
    n1_i
    (*
     (* normAngle normAngle)
     (- (* 0.3333333333333333 n0_i) (* n1_i -0.16666666666666666))))
   n0_i)
  n0_i))

C

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

Julia

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 81.4%

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

   3. *-commutative 81.4%

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

   4. associate-*l* 75.1%

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

   5. distribute-lft-out 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in u around 0 89.7%

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

   1. +-commutative 89.7%

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

   2. fma-define 89.7%

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

   3. +-commutative 89.7%

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

   4. mul-1-neg 89.7%

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

   5. unsub-neg 89.7%

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

   6. associate-/l* 95.1%

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

   7. *-commutative 95.1%

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

   8. associate-/l* 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in normAngle around 0 99.7%

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left(n1\_i + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right)\right) - n0\_i}, n0\_i\right) \]
9. Step-by-step derivation

   1. unpow2 99.7%

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

10. Applied egg-rr 99.7%

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

11. Taylor expanded in n0_i around 0 99.7%

    \[\leadsto \mathsf{fma}\left(u, \left(n1\_i + \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot n0\_i - -0.16666666666666666 \cdot n1\_i\right)}\right) - n0\_i, n0\_i\right) \]

12. Final simplification 99.7%

    \[\leadsto \mathsf{fma}\left(u, \left(n1\_i + \left(normAngle \cdot normAngle\right) \cdot \left(0.3333333333333333 \cdot n0\_i - n1\_i \cdot -0.16666666666666666\right)\right) - n0\_i, n0\_i\right) \]
13. Add Preprocessing

Alternative 2: 98.8% accurate, 3.7× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  u
  (- (+ n1_i (* (* normAngle normAngle) (* n1_i 0.16666666666666666))) n0_i)
  n0_i))

C

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

Julia

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 81.4%

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

   3. *-commutative 81.4%

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

   4. associate-*l* 75.1%

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

   5. distribute-lft-out 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in u around 0 89.7%

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

   1. +-commutative 89.7%

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

   2. fma-define 89.7%

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

   3. +-commutative 89.7%

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

   4. mul-1-neg 89.7%

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

   5. unsub-neg 89.7%

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

   6. associate-/l* 95.1%

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

   7. *-commutative 95.1%

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

   8. associate-/l* 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in normAngle around 0 99.7%

   \[\leadsto \mathsf{fma}\left(u, \color{blue}{\left(n1\_i + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot n0\_i - \left(-0.5 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right)\right) - n0\_i}, n0\_i\right) \]
9. Step-by-step derivation

   1. unpow2 99.7%

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

10. Applied egg-rr 99.7%

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

11. Taylor expanded in n0_i around 0 99.5%

    \[\leadsto \mathsf{fma}\left(u, \left(n1\_i + \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(0.16666666666666666 \cdot n1\_i\right)}\right) - n0\_i, n0\_i\right) \]
12. Step-by-step derivation

    1. *-commutative 99.5%

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

13. Simplified 99.5%

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

Alternative 3: 99.1% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 81.4%

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

   3. *-commutative 81.4%

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

   4. associate-*l* 75.1%

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

   5. distribute-lft-out 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in u around 0 89.7%

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

6. Taylor expanded in normAngle around 0 92.8%

   \[\leadsto n0\_i + u \cdot \left(-1 \cdot \color{blue}{\left(n0\_i + {normAngle}^{2} \cdot \left(-0.5 \cdot n0\_i - -0.16666666666666666 \cdot n0\_i\right)\right)} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right) \]
7. Step-by-step derivation

   1. distribute-rgt-out-- 92.8%

      \[\leadsto n0\_i + u \cdot \left(-1 \cdot \left(n0\_i + {normAngle}^{2} \cdot \color{blue}{\left(n0\_i \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right) \]

   2. metadata-eval 92.8%

      \[\leadsto n0\_i + u \cdot \left(-1 \cdot \left(n0\_i + {normAngle}^{2} \cdot \left(n0\_i \cdot \color{blue}{-0.3333333333333333}\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right) \]

8. Simplified 92.8%

   \[\leadsto n0\_i + u \cdot \left(-1 \cdot \color{blue}{\left(n0\_i + {normAngle}^{2} \cdot \left(n0\_i \cdot -0.3333333333333333\right)\right)} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right) \]

9. Taylor expanded in normAngle around 0 92.6%

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

    1. neg-mul-1 92.6%

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

11. Simplified 92.6%

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

12. Taylor expanded in n1_i around inf 99.3%

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

    1. +-commutative 99.3%

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

    2. mul-1-neg 99.3%

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

    3. unsub-neg 99.3%

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

14. Simplified 99.3%

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

Alternative 4: 84.7% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -4.99999991225835 \cdot 10^{-15} \lor \neg \left(n0\_i \leq 4.999999841327613 \cdot 10^{-21}\right):\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -4.99999991225835e-15) (not (<= n0_i 4.999999841327613e-21)))
   (- n0_i (* u n0_i))
   (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -4.99999991225835e-15f) || !(n0_i <= 4.999999841327613e-21f)) {
		tmp = n0_i - (u * n0_i);
	} else {
		tmp = n0_i + (u * n1_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 <= (-4.99999991225835e-15)) .or. (.not. (n0_i <= 4.999999841327613e-21))) then
        tmp = n0_i - (u * n0_i)
    else
        tmp = n0_i + (u * n1_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-4.99999991225835e-15)) || !(n0_i <= Float32(4.999999841327613e-21)))
		tmp = Float32(n0_i - Float32(u * n0_i));
	else
		tmp = Float32(n0_i + Float32(u * n1_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-4.99999991225835e-15)) || ~((n0_i <= single(4.999999841327613e-21))))
		tmp = n0_i - (u * n0_i);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.99999991e-15 or 4.99999984e-21 < n0_i

   1. Initial program 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*l* 90.1%

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

      3. *-commutative 90.1%

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

      4. associate-*l* 89.9%

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

      5. distribute-lft-out 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in u around 0 96.9%

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

   6. Taylor expanded in normAngle around 0 99.6%

      \[\leadsto n0\_i + u \cdot \left(-1 \cdot \color{blue}{\left(n0\_i + {normAngle}^{2} \cdot \left(-0.5 \cdot n0\_i - -0.16666666666666666 \cdot n0\_i\right)\right)} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right) \]
   7. Step-by-step derivation

      1. distribute-rgt-out-- 99.6%

         \[\leadsto n0\_i + u \cdot \left(-1 \cdot \left(n0\_i + {normAngle}^{2} \cdot \color{blue}{\left(n0\_i \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right) \]

      2. metadata-eval 99.6%

         \[\leadsto n0\_i + u \cdot \left(-1 \cdot \left(n0\_i + {normAngle}^{2} \cdot \left(n0\_i \cdot \color{blue}{-0.3333333333333333}\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right) \]

   8. Simplified 99.6%

      \[\leadsto n0\_i + u \cdot \left(-1 \cdot \color{blue}{\left(n0\_i + {normAngle}^{2} \cdot \left(n0\_i \cdot -0.3333333333333333\right)\right)} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right) \]

   9. Taylor expanded in normAngle around 0 99.2%

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

       1. neg-mul-1 99.2%

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

   11. Simplified 99.2%

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

   12. Taylor expanded in n0_i around inf 88.6%

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

       1. associate-*r* 88.6%

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

       2. mul-1-neg 88.6%

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

   14. Simplified 88.6%

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

   if -4.99999991e-15 < n0_i < 4.99999984e-21

   1. Initial program 96.5%

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

      1. *-commutative 96.5%

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

      2. associate-*l* 73.2%

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

      3. *-commutative 73.2%

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

      4. associate-*l* 61.5%

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

      5. distribute-lft-out 61.5%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in u around 0 83.0%

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

   6. Taylor expanded in n0_i around 0 70.5%

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

   7. Taylor expanded in normAngle around 0 88.2%

      \[\leadsto n0\_i + \color{blue}{n1\_i \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -4.99999991225835 \cdot 10^{-15} \lor \neg \left(n0\_i \leq 4.999999841327613 \cdot 10^{-21}\right):\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.1% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 7.000000383309403 \cdot 10^{-27}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -5.0000000843119176e-17)
   n0_i
   (if (<= n0_i 7.000000383309403e-27) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -5.0000000843119176e-17f) {
		tmp = n0_i;
	} else if (n0_i <= 7.000000383309403e-27f) {
		tmp = u * n1_i;
	} 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 <= (-5.0000000843119176e-17)) then
        tmp = n0_i
    else if (n0_i <= 7.000000383309403e-27) then
        tmp = u * n1_i
    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(-5.0000000843119176e-17))
		tmp = n0_i;
	elseif (n0_i <= Float32(7.000000383309403e-27))
		tmp = Float32(u * n1_i);
	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(-5.0000000843119176e-17))
		tmp = n0_i;
	elseif (n0_i <= single(7.000000383309403e-27))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -5.00000008e-17 or 7.00000038e-27 < n0_i

   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%

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

      2. associate-*l* 83.6%

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

      3. *-commutative 83.6%

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

      4. associate-*l* 82.0%

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

      5. distribute-lft-out 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in u around 0 64.8%

      \[\leadsto \color{blue}{n0\_i} \]

   if -5.00000008e-17 < n0_i < 7.00000038e-27

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-*l* 78.1%

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

      3. *-commutative 78.1%

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

      4. associate-*l* 65.0%

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

      5. distribute-lft-out 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in n0_i around 0 47.3%

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

      1. associate-/l* 60.6%

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

      2. *-commutative 60.6%

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

   7. Simplified 60.6%

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

      1. clear-num 60.6%

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

      2. associate-/r/ 60.4%

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

      3. *-commutative 60.4%

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

   9. Applied egg-rr 60.4%

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

   10. Taylor expanded in normAngle around 0 61.5%

       \[\leadsto n1\_i \cdot \color{blue}{u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 7.000000383309403 \cdot 10^{-27}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% 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 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 81.4%

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

   3. *-commutative 81.4%

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

   4. associate-*l* 75.1%

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

   5. distribute-lft-out 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in u around 0 89.7%

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

6. Taylor expanded in normAngle around 0 98.8%

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

   1. neg-mul-1 98.8%

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

   2. sub-neg 98.8%

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

8. Simplified 98.8%

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

Alternative 7: 81.9% accurate, 84.2× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot n1\_i \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 81.4%

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

   3. *-commutative 81.4%

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

   4. associate-*l* 75.1%

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

   5. distribute-lft-out 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in u around 0 89.7%

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

6. Taylor expanded in n0_i around 0 72.2%

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

7. Taylor expanded in normAngle around 0 81.7%

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

8. Final simplification 81.7%

   \[\leadsto n0\_i + u \cdot n1\_i \]
9. Add Preprocessing

Alternative 8: 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 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 81.4%

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

   3. *-commutative 81.4%

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

   4. associate-*l* 75.1%

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

   5. distribute-lft-out 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in u around 0 50.5%

   \[\leadsto \color{blue}{n0\_i} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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)))