Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 98.8%

Time: 14.5s

Alternatives: 9

Speedup: 60.1×

Specification

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0\_i \land n0\_i \leq 1\right)\right) \land \left(-1 \leq n1\_i \land n1\_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

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

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: t_0
    t_0 = 1.0e0 / sin(normangle)
    code = ((sin(((1.0e0 - u) * normangle)) * t_0) * n0_i) + ((sin((u * normangle)) * t_0) * n1_i)
end function

Julia

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

MATLAB

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

Initial Program: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: t_0
    t_0 = 1.0e0 / sin(normangle)
    code = ((sin(((1.0e0 - u) * normangle)) * t_0) * n0_i) + ((sin((u * normangle)) * t_0) * n1_i)
end function

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (n0_i * ((u * ((n1_i * (normAngle / n0_i)) - (normAngle * cosf(normAngle)))) / sinf(normAngle)));
}

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 + (n0_i * ((u * ((n1_i * (normangle / n0_i)) - (normangle * cos(normangle)))) / sin(normangle)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l* 97.6%

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

   2. cancel-sign-sub 97.6%

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

   3. *-commutative 97.6%

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

   4. associate-*r* 81.6%

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

   5. associate-*r/ 81.8%

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

   6. *-rgt-identity 81.8%

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

   7. sin-neg 81.8%

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

   8. distribute-lft-neg-out 81.8%

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

   9. associate-*l* 81.9%

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

   10. *-commutative 81.9%

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

   11. distribute-lft-neg-out 81.9%

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

   12. distribute-rgt-neg-out 81.9%

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

   13. associate-*r/ 82.1%

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

3. Simplified 74.8%

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

5. Taylor expanded in n0_i around inf 74.8%

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

6. Taylor expanded in u around 0 83.3%

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

   1. associate-/l* 90.3%

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

   2. +-commutative 90.3%

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

   3. mul-1-neg 90.3%

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

   4. unsub-neg 90.3%

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

   5. associate-/l* 99.1%

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

8. Simplified 99.1%

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

Alternative 2: 98.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l* 97.6%

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

   2. cancel-sign-sub 97.6%

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

   3. *-commutative 97.6%

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

   4. associate-*r* 81.6%

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

   5. associate-*r/ 81.8%

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

   6. *-rgt-identity 81.8%

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

   7. sin-neg 81.8%

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

   8. distribute-lft-neg-out 81.8%

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

   9. associate-*l* 81.9%

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

   10. *-commutative 81.9%

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

   11. distribute-lft-neg-out 81.9%

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

   12. distribute-rgt-neg-out 81.9%

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

   13. associate-*r/ 82.1%

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

3. Simplified 74.8%

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

5. Taylor expanded in n0_i around inf 74.8%

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

6. Taylor expanded in u around 0 83.3%

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

   1. associate-/l* 90.3%

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

   2. +-commutative 90.3%

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

   3. mul-1-neg 90.3%

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

   4. unsub-neg 90.3%

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

   5. associate-/l* 99.1%

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

8. Simplified 99.1%

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

9. Taylor expanded in normAngle around 0 98.8%

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

10. Final simplification 98.8%

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

Alternative 3: 98.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(n1\_i - \left(n0\_i + {normAngle}^{2} \cdot \left(n0\_i \cdot -0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

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

C

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. fma-define 97.8%

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

   2. associate-*r/ 97.9%

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

   3. *-rgt-identity 97.9%

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

   4. associate-*r/ 98.1%

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

   5. *-rgt-identity 98.1%

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

3. Simplified 98.1%

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

5. Taylor expanded in normAngle around 0 98.7%

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

6. Taylor expanded in u around 0 94.6%

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

   1. mul-1-neg 94.6%

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

   2. unsub-neg 94.6%

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

   3. associate-*r* 94.6%

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

8. Simplified 94.6%

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

9. Taylor expanded in normAngle around 0 98.8%

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

    1. distribute-rgt-out-- 98.8%

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

    2. metadata-eval 98.8%

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

11. Simplified 98.8%

    \[\leadsto n0\_i + u \cdot \left(n1\_i - \color{blue}{\left(n0\_i + {normAngle}^{2} \cdot \left(n0\_i \cdot -0.3333333333333333\right)\right)}\right) \]
12. Add Preprocessing

Alternative 4: 85.3% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -7.99999974612418 \cdot 10^{-20} \lor \neg \left(n0\_i \leq 9.9999998245167 \cdot 10^{-14}\right):\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -7.99999974612418e-20) (not (<= n0_i 9.9999998245167e-14)))
   (- n0_i (* n0_i u))
   (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -7.99999974612418e-20f) || !(n0_i <= 9.9999998245167e-14f)) {
		tmp = n0_i - (n0_i * u);
	} 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 <= (-7.99999974612418e-20)) .or. (.not. (n0_i <= 9.9999998245167e-14))) then
        tmp = n0_i - (n0_i * u)
    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(-7.99999974612418e-20)) || !(n0_i <= Float32(9.9999998245167e-14)))
		tmp = Float32(n0_i - Float32(n0_i * u));
	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(-7.99999974612418e-20)) || ~((n0_i <= single(9.9999998245167e-14))))
		tmp = n0_i - (n0_i * u);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -7.99999975e-20 or 9.99999982e-14 < n0_i

   1. Initial program 98.3%

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

      1. fma-define 98.4%

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

      2. associate-*r/ 98.6%

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

      3. *-rgt-identity 98.6%

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

      4. associate-*r/ 98.6%

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

      5. *-rgt-identity 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in normAngle around 0 99.2%

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

   6. Taylor expanded in u around 0 97.2%

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

      1. mul-1-neg 97.2%

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

      2. unsub-neg 97.2%

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

      3. associate-*r* 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in normAngle around 0 99.2%

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

   10. Taylor expanded in n1_i around 0 89.1%

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

       1. mul-1-neg 89.1%

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

       2. *-commutative 89.1%

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

       3. distribute-lft-neg-in 89.1%

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

   12. Simplified 89.1%

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

   if -7.99999975e-20 < n0_i < 9.99999982e-14

   1. Initial program 97.3%

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

      1. fma-define 97.3%

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

      2. associate-*r/ 97.4%

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

      3. *-rgt-identity 97.4%

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

      4. associate-*r/ 97.7%

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

      5. *-rgt-identity 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in normAngle around 0 98.3%

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

   6. Taylor expanded in u around 0 92.4%

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

      1. mul-1-neg 92.4%

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

      2. unsub-neg 92.4%

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

      3. associate-*r* 92.4%

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

   8. Simplified 92.4%

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

   9. Taylor expanded in normAngle around 0 98.1%

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

   10. Taylor expanded in n1_i around inf 86.8%

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

       1. *-commutative 86.8%

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

   12. Simplified 86.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -7.99999974612418 \cdot 10^{-20} \lor \neg \left(n0\_i \leq 9.9999998245167 \cdot 10^{-14}\right):\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -7.99999974612418 \cdot 10^{-20} \lor \neg \left(n0\_i \leq 9.9999998245167 \cdot 10^{-14}\right):\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -7.99999974612418e-20) (not (<= n0_i 9.9999998245167e-14)))
   (* n0_i (- 1.0 u))
   (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -7.99999974612418e-20f) || !(n0_i <= 9.9999998245167e-14f)) {
		tmp = n0_i * (1.0f - u);
	} 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 <= (-7.99999974612418e-20)) .or. (.not. (n0_i <= 9.9999998245167e-14))) then
        tmp = n0_i * (1.0e0 - u)
    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(-7.99999974612418e-20)) || !(n0_i <= Float32(9.9999998245167e-14)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	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(-7.99999974612418e-20)) || ~((n0_i <= single(9.9999998245167e-14))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -7.99999975e-20 or 9.99999982e-14 < n0_i

   1. Initial program 98.3%

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

      1. fma-define 98.4%

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

      2. associate-*r/ 98.6%

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

      3. *-rgt-identity 98.6%

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

      4. associate-*r/ 98.6%

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

      5. *-rgt-identity 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in normAngle around 0 99.2%

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

   6. Taylor expanded in u around 0 97.2%

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

      1. mul-1-neg 97.2%

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

      2. unsub-neg 97.2%

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

      3. associate-*r* 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in normAngle around 0 99.2%

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

   10. Taylor expanded in n0_i around inf 88.6%

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

       1. mul-1-neg 88.6%

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

       2. unsub-neg 88.6%

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

   12. Simplified 88.6%

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

   if -7.99999975e-20 < n0_i < 9.99999982e-14

   1. Initial program 97.3%

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

      1. fma-define 97.3%

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

      2. associate-*r/ 97.4%

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

      3. *-rgt-identity 97.4%

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

      4. associate-*r/ 97.7%

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

      5. *-rgt-identity 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in normAngle around 0 98.3%

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

   6. Taylor expanded in u around 0 92.4%

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

      1. mul-1-neg 92.4%

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

      2. unsub-neg 92.4%

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

      3. associate-*r* 92.4%

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

   8. Simplified 92.4%

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

   9. Taylor expanded in normAngle around 0 98.1%

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

   10. Taylor expanded in n1_i around inf 86.8%

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

       1. *-commutative 86.8%

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

   12. Simplified 86.8%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -7.99999974612418 \cdot 10^{-20} \lor \neg \left(n0\_i \leq 9.9999998245167 \cdot 10^{-14}\right):\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.1% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.9999999774532045 \cdot 10^{-26} \lor \neg \left(n0\_i \leq 1.9999999774532045 \cdot 10^{-26}\right):\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -1.9999999774532045e-26)
         (not (<= n0_i 1.9999999774532045e-26)))
   (* n0_i (- 1.0 u))
   (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -1.9999999774532045e-26f) || !(n0_i <= 1.9999999774532045e-26f)) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = 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 <= (-1.9999999774532045e-26)) .or. (.not. (n0_i <= 1.9999999774532045e-26))) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = 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(-1.9999999774532045e-26)) || !(n0_i <= Float32(1.9999999774532045e-26)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = 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(-1.9999999774532045e-26)) || ~((n0_i <= single(1.9999999774532045e-26))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.99999998e-26 or 1.99999998e-26 < n0_i

   1. Initial program 98.0%

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

      1. fma-define 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/ 98.2%

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

      3. *-rgt-identity 98.2%

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

      4. associate-*r/ 98.3%

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

      5. *-rgt-identity 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in normAngle around 0 98.6%

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

   6. Taylor expanded in u around 0 95.3%

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

      1. mul-1-neg 95.3%

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

      2. unsub-neg 95.3%

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

      3. associate-*r* 95.3%

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

   8. Simplified 95.3%

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

   9. Taylor expanded in normAngle around 0 98.6%

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

   10. Taylor expanded in n0_i around inf 79.0%

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

       1. mul-1-neg 79.0%

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

       2. unsub-neg 79.0%

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

   12. Simplified 79.0%

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

   if -1.99999998e-26 < n0_i < 1.99999998e-26

   1. Initial program 97.3%

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

      1. fma-define 97.3%

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

      2. associate-*r/ 97.4%

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

      3. *-rgt-identity 97.4%

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

      4. associate-*r/ 97.8%

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

      5. *-rgt-identity 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in normAngle around 0 98.9%

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

   6. Taylor expanded in u around inf 71.5%

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

      1. *-commutative 71.5%

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

   8. Simplified 71.5%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.9999999774532045 \cdot 10^{-26} \lor \neg \left(n0\_i \leq 1.9999999774532045 \cdot 10^{-26}\right):\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.5% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 1.0000000195414814 \cdot 10^{-24}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.9999999774532045e-26)
   n0_i
   (if (<= n0_i 1.0000000195414814e-24) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.9999999774532045e-26f) {
		tmp = n0_i;
	} else if (n0_i <= 1.0000000195414814e-24f) {
		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 <= (-1.9999999774532045e-26)) then
        tmp = n0_i
    else if (n0_i <= 1.0000000195414814e-24) 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(-1.9999999774532045e-26))
		tmp = n0_i;
	elseif (n0_i <= Float32(1.0000000195414814e-24))
		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(-1.9999999774532045e-26))
		tmp = n0_i;
	elseif (n0_i <= single(1.0000000195414814e-24))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.99999998e-26 or 1.00000002e-24 < 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. fma-define 97.9%

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

      2. associate-*r/ 98.1%

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

      3. *-rgt-identity 98.1%

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

      4. associate-*r/ 98.2%

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

      5. *-rgt-identity 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in u around 0 62.0%

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

   if -1.99999998e-26 < n0_i < 1.00000002e-24

   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. fma-define 97.5%

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

      2. associate-*r/ 97.6%

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

      3. *-rgt-identity 97.6%

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

      4. associate-*r/ 97.9%

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

      5. *-rgt-identity 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in normAngle around 0 98.5%

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

   6. Taylor expanded in u around inf 69.9%

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

      1. *-commutative 69.9%

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

   8. Simplified 69.9%

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

Alternative 8: 97.9% 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 97.8%

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

   1. fma-define 97.8%

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

   2. associate-*r/ 97.9%

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

   3. *-rgt-identity 97.9%

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

   4. associate-*r/ 98.1%

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

   5. *-rgt-identity 98.1%

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

3. Simplified 98.1%

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

5. Taylor expanded in normAngle around 0 98.7%

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

6. Taylor expanded in u around 0 94.6%

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

   1. mul-1-neg 94.6%

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

   2. unsub-neg 94.6%

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

   3. associate-*r* 94.6%

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

8. Simplified 94.6%

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

9. Taylor expanded in normAngle around 0 98.6%

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

Alternative 9: 46.8% 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 97.8%

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

   1. fma-define 97.8%

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

   2. associate-*r/ 97.9%

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

   3. *-rgt-identity 97.9%

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

   4. associate-*r/ 98.1%

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

   5. *-rgt-identity 98.1%

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

3. Simplified 98.1%

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

5. Taylor expanded in u around 0 48.6%

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

Reproduce

herbie shell --seed 2024113 
(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)))