Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.2%

Time: 18.5s

Alternatives: 10

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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/ 97.7%

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

   3. *-rgt-identity 97.7%

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

   4. associate-*r/ 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 89.8%

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

   1. +-commutative 89.8%

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

   2. mul-1-neg 89.8%

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

   3. unsub-neg 89.8%

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

   4. associate-/l* 94.9%

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

   5. associate-/l* 99.3%

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

   6. associate-/l* 98.9%

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

7. Simplified 98.9%

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

   1. pow1 98.9%

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

   2. clear-num 98.9%

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

   3. un-div-inv 98.9%

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

   4. associate-*r* 94.6%

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

   5. clear-num 94.6%

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

   6. un-div-inv 94.9%

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

   7. quot-tan 94.9%

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

9. Applied egg-rr 94.9%

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

    1. unpow1 94.9%

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

    2. associate-/r/ 94.9%

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

    3. associate-/l* 99.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

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

Alternative 2: 98.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot -0.16666666666666666 - \left(n0\_i \cdot -0.5 + n1\_i \cdot -0.16666666666666666\right)\right)\right)\right) \end{array} \]

FPCore

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

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) * (u * ((n0_i * -0.16666666666666666f) - ((n0_i * -0.5f) + (n1_i * -0.16666666666666666f))))));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i + ((u * (n1_i - n0_i)) + ((normangle ** 2.0e0) * (u * ((n0_i * (-0.16666666666666666e0)) - ((n0_i * (-0.5e0)) + (n1_i * (-0.16666666666666666e0)))))))
end function

Julia

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

MATLAB

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

Derivation

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/ 97.7%

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

   3. *-rgt-identity 97.7%

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

   4. associate-*r/ 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 89.8%

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

   1. +-commutative 89.8%

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

   2. mul-1-neg 89.8%

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

   3. unsub-neg 89.8%

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

   4. associate-/l* 94.9%

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

   5. associate-/l* 99.3%

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

   6. associate-/l* 98.9%

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

7. Simplified 98.9%

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

8. Taylor expanded in normAngle around 0 98.9%

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

9. Final simplification 98.9%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot -0.16666666666666666 - \left(n0\_i \cdot -0.5 + n1\_i \cdot -0.16666666666666666\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 3: 98.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * (n1_i - (n0_i * (normAngle / tanf(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 + (u * (n1_i - (n0_i * (normangle / tan(normangle)))))
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 / tan(normAngle))))))
end

MATLAB

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

Derivation

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/ 97.7%

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

   3. *-rgt-identity 97.7%

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

   4. associate-*r/ 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 89.8%

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

   1. +-commutative 89.8%

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

   2. mul-1-neg 89.8%

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

   3. unsub-neg 89.8%

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

   4. associate-/l* 94.9%

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

   5. associate-/l* 99.3%

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

   6. associate-/l* 98.9%

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

7. Simplified 98.9%

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

   1. pow1 98.9%

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

   2. clear-num 98.9%

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

   3. un-div-inv 98.9%

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

   4. associate-*r* 94.6%

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

   5. clear-num 94.6%

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

   6. un-div-inv 94.9%

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

   7. quot-tan 94.9%

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

9. Applied egg-rr 94.9%

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

    1. unpow1 94.9%

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

    2. associate-/r/ 94.9%

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

    3. associate-/l* 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in normAngle around 0 98.3%

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

13. Final simplification 98.3%

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

Alternative 4: 98.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
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/ 97.7%

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

   3. *-rgt-identity 97.7%

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

   4. associate-*r/ 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 89.8%

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

   1. +-commutative 89.8%

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

   2. mul-1-neg 89.8%

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

   3. unsub-neg 89.8%

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

   4. associate-/l* 94.9%

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

   5. associate-/l* 99.3%

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

   6. associate-/l* 98.9%

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

7. Simplified 98.9%

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

8. Taylor expanded in normAngle around 0 98.1%

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

   1. +-commutative 98.1%

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

   2. fma-define 98.1%

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

10. Simplified 98.1%

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

11. Final simplification 98.1%

    \[\leadsto \mathsf{fma}\left(u, n1\_i - n0\_i, n0\_i\right) \]
12. Add Preprocessing

Alternative 5: 70.7% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -4.99999991225835 \cdot 10^{-15} \lor \neg \left(n1\_i \leq 7.99999985961336 \cdot 10^{-13}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -4.99999991225835e-15) (not (<= n1_i 7.99999985961336e-13)))
   (* u n1_i)
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -4.99999991225835e-15f) || !(n1_i <= 7.99999985961336e-13f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-4.99999991225835e-15)) .or. (.not. (n1_i <= 7.99999985961336e-13))) then
        tmp = u * n1_i
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-4.99999991225835e-15)) || !(n1_i <= Float32(7.99999985961336e-13)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-4.99999991225835e-15)) || ~((n1_i <= single(7.99999985961336e-13))))
		tmp = u * n1_i;
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.99999991e-15 or 7.99999986e-13 < n1_i

   1. Initial program 96.0%

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

      1. fma-define 96.0%

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

      2. associate-*r/ 96.1%

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

      3. *-rgt-identity 96.1%

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

      4. associate-*r/ 96.3%

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

      5. *-rgt-identity 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in n0_i around 0 57.8%

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

   6. Taylor expanded in normAngle around 0 64.4%

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

      1. *-commutative 64.4%

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

   8. Simplified 64.4%

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

   if -4.99999991e-15 < n1_i < 7.99999986e-13

   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 98.3%

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

      2. associate-*r/ 98.5%

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

      3. *-rgt-identity 98.5%

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

      4. associate-*r/ 99.2%

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

      5. *-rgt-identity 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in u around 0 87.8%

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

      1. +-commutative 87.8%

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

      2. mul-1-neg 87.8%

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

      3. unsub-neg 87.8%

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

      4. associate-/l* 93.3%

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

      5. associate-/l* 99.5%

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

      6. associate-/l* 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in normAngle around 0 99.0%

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

   9. Taylor expanded in n0_i around inf 79.2%

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

       1. mul-1-neg 79.2%

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

       2. unsub-neg 79.2%

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

   11. Simplified 79.2%

       \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -4.99999991225835 \cdot 10^{-15} \lor \neg \left(n1\_i \leq 7.99999985961336 \cdot 10^{-13}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.7% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.000000031374395 \cdot 10^{-22} \lor \neg \left(n1\_i \leq 1.0000000195414814 \cdot 10^{-24}\right):\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.000000031374395e-22)
         (not (<= n1_i 1.0000000195414814e-24)))
   (+ n0_i (* u n1_i))
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.000000031374395e-22f) || !(n1_i <= 1.0000000195414814e-24f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-1.000000031374395e-22)) .or. (.not. (n1_i <= 1.0000000195414814e-24))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-1.000000031374395e-22)) || !(n1_i <= Float32(1.0000000195414814e-24)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-1.000000031374395e-22)) || ~((n1_i <= single(1.0000000195414814e-24))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.00000003e-22 or 1.00000002e-24 < n1_i

   1. Initial program 96.5%

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

      1. fma-define 96.5%

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

      2. associate-*r/ 96.5%

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

      3. *-rgt-identity 96.5%

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

      4. associate-*r/ 97.4%

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

      5. *-rgt-identity 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in u around 0 90.1%

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

      1. +-commutative 90.1%

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

      2. mul-1-neg 90.1%

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

      3. unsub-neg 90.1%

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

      4. associate-/l* 97.6%

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

      5. associate-/l* 99.3%

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

      6. associate-/l* 98.6%

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

   7. Simplified 98.6%

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

      1. pow1 98.6%

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

      2. clear-num 98.6%

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

      3. un-div-inv 98.7%

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

      4. associate-*r* 97.0%

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

      5. clear-num 97.0%

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

      6. un-div-inv 97.6%

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

      7. quot-tan 97.6%

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

   9. Applied egg-rr 97.6%

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

       1. unpow1 97.6%

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

       2. associate-/r/ 97.5%

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

       3. associate-/l* 99.2%

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

   11. Simplified 99.2%

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

   12. Taylor expanded in normAngle around 0 97.7%

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

   13. Taylor expanded in n1_i around inf 86.4%

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

   if -1.00000003e-22 < n1_i < 1.00000002e-24

   1. Initial program 98.9%

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

      1. fma-define 99.0%

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

      2. associate-*r/ 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. associate-*r/ 99.2%

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

      5. *-rgt-identity 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in u around 0 89.4%

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

      1. +-commutative 89.4%

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

      2. mul-1-neg 89.4%

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

      3. unsub-neg 89.4%

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

      4. associate-/l* 91.3%

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

      5. associate-/l* 99.4%

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

      6. associate-/l* 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in normAngle around 0 98.6%

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

   9. Taylor expanded in n0_i around inf 90.0%

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

       1. mul-1-neg 90.0%

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

       2. unsub-neg 90.0%

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

   11. Simplified 90.0%

       \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.000000031374395 \cdot 10^{-22} \lor \neg \left(n1\_i \leq 1.0000000195414814 \cdot 10^{-24}\right):\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.8% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.000000031374395 \cdot 10^{-22} \lor \neg \left(n1\_i \leq 1.0000000195414814 \cdot 10^{-24}\right):\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.000000031374395e-22)
         (not (<= n1_i 1.0000000195414814e-24)))
   (+ n0_i (* u n1_i))
   (- n0_i (* n0_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.000000031374395e-22f) || !(n1_i <= 1.0000000195414814e-24f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i - (n0_i * u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-1.000000031374395e-22)) .or. (.not. (n1_i <= 1.0000000195414814e-24))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i - (n0_i * u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-1.000000031374395e-22)) || !(n1_i <= Float32(1.0000000195414814e-24)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i - Float32(n0_i * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-1.000000031374395e-22)) || ~((n1_i <= single(1.0000000195414814e-24))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i - (n0_i * u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.00000003e-22 or 1.00000002e-24 < n1_i

   1. Initial program 96.5%

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

      1. fma-define 96.5%

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

      2. associate-*r/ 96.5%

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

      3. *-rgt-identity 96.5%

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

      4. associate-*r/ 97.4%

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

      5. *-rgt-identity 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in u around 0 90.1%

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

      1. +-commutative 90.1%

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

      2. mul-1-neg 90.1%

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

      3. unsub-neg 90.1%

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

      4. associate-/l* 97.6%

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

      5. associate-/l* 99.3%

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

      6. associate-/l* 98.6%

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

   7. Simplified 98.6%

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

      1. pow1 98.6%

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

      2. clear-num 98.6%

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

      3. un-div-inv 98.7%

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

      4. associate-*r* 97.0%

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

      5. clear-num 97.0%

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

      6. un-div-inv 97.6%

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

      7. quot-tan 97.6%

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

   9. Applied egg-rr 97.6%

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

       1. unpow1 97.6%

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

       2. associate-/r/ 97.5%

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

       3. associate-/l* 99.2%

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

   11. Simplified 99.2%

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

   12. Taylor expanded in normAngle around 0 97.7%

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

   13. Taylor expanded in n1_i around inf 86.4%

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

   if -1.00000003e-22 < n1_i < 1.00000002e-24

   1. Initial program 98.9%

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

      1. fma-define 99.0%

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

      2. associate-*r/ 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. associate-*r/ 99.2%

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

      5. *-rgt-identity 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in u around 0 89.4%

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

      1. +-commutative 89.4%

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

      2. mul-1-neg 89.4%

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

      3. unsub-neg 89.4%

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

      4. associate-/l* 91.3%

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

      5. associate-/l* 99.4%

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

      6. associate-/l* 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in normAngle around 0 98.6%

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

   9. Taylor expanded in n1_i around 0 90.2%

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

       1. associate-*r* 90.2%

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

       2. neg-mul-1 90.2%

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

   11. Simplified 90.2%

       \[\leadsto n0\_i + \color{blue}{\left(-n0\_i\right) \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.000000031374395 \cdot 10^{-22} \lor \neg \left(n1\_i \leq 1.0000000195414814 \cdot 10^{-24}\right):\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.6% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.000000031374395 \cdot 10^{-22} \lor \neg \left(n1\_i \leq 2.99999994735501 \cdot 10^{-14}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.000000031374395e-22) (not (<= n1_i 2.99999994735501e-14)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.000000031374395e-22f) || !(n1_i <= 2.99999994735501e-14f)) {
		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 ((n1_i <= (-1.000000031374395e-22)) .or. (.not. (n1_i <= 2.99999994735501e-14))) 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 ((n1_i <= Float32(-1.000000031374395e-22)) || !(n1_i <= Float32(2.99999994735501e-14)))
		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 ((n1_i <= single(-1.000000031374395e-22)) || ~((n1_i <= single(2.99999994735501e-14))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.00000003e-22 or 2.99999995e-14 < n1_i

   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. fma-define 96.8%

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

      2. associate-*r/ 96.8%

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

      3. *-rgt-identity 96.8%

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

      4. associate-*r/ 97.0%

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

      5. *-rgt-identity 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in n0_i around 0 50.2%

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

   6. Taylor expanded in normAngle around 0 58.8%

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

      1. *-commutative 58.8%

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

   8. Simplified 58.8%

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

   if -1.00000003e-22 < n1_i < 2.99999995e-14

   1. Initial program 98.1%

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

      1. fma-define 98.2%

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

      2. associate-*r/ 98.4%

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

      3. *-rgt-identity 98.4%

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

      4. associate-*r/ 99.2%

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

      5. *-rgt-identity 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in u around 0 68.1%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.000000031374395 \cdot 10^{-22} \lor \neg \left(n1\_i \leq 2.99999994735501 \cdot 10^{-14}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
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/ 97.7%

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

   3. *-rgt-identity 97.7%

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

   4. associate-*r/ 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 89.8%

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

   1. +-commutative 89.8%

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

   2. mul-1-neg 89.8%

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

   3. unsub-neg 89.8%

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

   4. associate-/l* 94.9%

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

   5. associate-/l* 99.3%

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

   6. associate-/l* 98.9%

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

7. Simplified 98.9%

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

8. Taylor expanded in normAngle around 0 98.1%

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

9. Final simplification 98.1%

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

Alternative 10: 47.4% 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 97.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
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/ 97.7%

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

   3. *-rgt-identity 97.7%

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

   4. associate-*r/ 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 49.5%

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

6. Final simplification 49.5%

   \[\leadsto n0\_i \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024078 
(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)))