Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 98.6%

Time: 16.0s

Alternatives: 11

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = u * ((n1_i * (normAngle / sin(normAngle))) + ((n0_i / sin(normAngle)) * (sin((normAngle * (single(1.0) - u))) / u)));
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. *-commutative 97.5%

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

   2. associate-*l* 78.6%

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

   3. *-commutative 78.6%

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

   4. associate-*l* 72.6%

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

   5. distribute-lft-out 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in u around 0 72.3%

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

   1. *-commutative 72.3%

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

7. Simplified 72.3%

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

8. Taylor expanded in u around inf 73.3%

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

   1. +-commutative 73.3%

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

   2. associate-/l* 78.2%

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

   3. sub-neg 78.2%

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

   4. +-commutative 78.2%

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

   5. neg-mul-1 78.2%

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

   6. fma-undefine 78.2%

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

   7. *-commutative 78.2%

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

   8. *-commutative 78.2%

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

   9. times-frac 98.8%

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

   10. *-commutative 98.8%

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

   11. fma-undefine 98.8%

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

   12. neg-mul-1 98.8%

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

   13. +-commutative 98.8%

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

   14. sub-neg 98.8%

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

10. Simplified 98.8%

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

Alternative 2: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(n1\_i + \frac{n0\_i \cdot \left(1 - u\right)}{u}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot \frac{n0\_i \cdot {\left(1 - u\right)}^{3}}{u} + \left(-0.16666666666666666 \cdot \frac{n0\_i \cdot \left(u + -1\right)}{u} - n1\_i \cdot -0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (u * (n1_i + ((n0_i * (1.0f - u)) / u))) + (powf(normAngle, 2.0f) * (u * ((-0.16666666666666666f * ((n0_i * powf((1.0f - u), 3.0f)) / u)) + ((-0.16666666666666666f * ((n0_i * (u + -1.0f)) / u)) - (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 = (u * (n1_i + ((n0_i * (1.0e0 - u)) / u))) + ((normangle ** 2.0e0) * (u * (((-0.16666666666666666e0) * ((n0_i * ((1.0e0 - u) ** 3.0e0)) / u)) + (((-0.16666666666666666e0) * ((n0_i * (u + (-1.0e0))) / u)) - (n1_i * (-0.16666666666666666e0))))))
end function

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (u * (n1_i + ((n0_i * (single(1.0) - u)) / u))) + ((normAngle ^ single(2.0)) * (u * ((single(-0.16666666666666666) * ((n0_i * ((single(1.0) - u) ^ single(3.0))) / u)) + ((single(-0.16666666666666666) * ((n0_i * (u + single(-1.0))) / u)) - (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. *-commutative 97.5%

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

   2. associate-*l* 78.6%

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

   3. *-commutative 78.6%

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

   4. associate-*l* 72.6%

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

   5. distribute-lft-out 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in u around 0 72.3%

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

   1. *-commutative 72.3%

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

7. Simplified 72.3%

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

8. Taylor expanded in u around inf 73.3%

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

   1. +-commutative 73.3%

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

   2. associate-/l* 78.2%

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

   3. sub-neg 78.2%

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

   4. +-commutative 78.2%

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

   5. neg-mul-1 78.2%

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

   6. fma-undefine 78.2%

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

   7. *-commutative 78.2%

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

   8. *-commutative 78.2%

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

   9. times-frac 98.8%

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

   10. *-commutative 98.8%

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

   11. fma-undefine 98.8%

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

   12. neg-mul-1 98.8%

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

   13. +-commutative 98.8%

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

   14. sub-neg 98.8%

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

10. Simplified 98.8%

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

11. Taylor expanded in normAngle around 0 98.5%

    \[\leadsto \color{blue}{u \cdot \left(n1\_i + \frac{n0\_i \cdot \left(1 - u\right)}{u}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot \frac{n0\_i \cdot {\left(1 - u\right)}^{3}}{u} - \left(-0.16666666666666666 \cdot n1\_i + -0.16666666666666666 \cdot \frac{n0\_i \cdot \left(1 - u\right)}{u}\right)\right)\right)} \]

12. Final simplification 98.5%

    \[\leadsto u \cdot \left(n1\_i + \frac{n0\_i \cdot \left(1 - u\right)}{u}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot \frac{n0\_i \cdot {\left(1 - u\right)}^{3}}{u} + \left(-0.16666666666666666 \cdot \frac{n0\_i \cdot \left(u + -1\right)}{u} - n1\_i \cdot -0.16666666666666666\right)\right)\right) \]
13. Add Preprocessing

Alternative 3: 98.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(n1\_i + \frac{n0\_i \cdot \left(1 - u\right)}{u}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.5 \cdot \left(u \cdot n0\_i\right) + n0\_i \cdot 0.5\right) - \left(n1\_i \cdot -0.16666666666666666 + n0\_i \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* u (+ n1_i (/ (* n0_i (- 1.0 u)) u)))
  (*
   (pow normAngle 2.0)
   (*
    u
    (-
     (+ (* -0.5 (* u n0_i)) (* n0_i 0.5))
     (+ (* n1_i -0.16666666666666666) (* n0_i 0.16666666666666666)))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (u * (n1_i + ((n0_i * (1.0f - u)) / u))) + (powf(normAngle, 2.0f) * (u * (((-0.5f * (u * n0_i)) + (n0_i * 0.5f)) - ((n1_i * -0.16666666666666666f) + (n0_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 = (u * (n1_i + ((n0_i * (1.0e0 - u)) / u))) + ((normangle ** 2.0e0) * (u * ((((-0.5e0) * (u * n0_i)) + (n0_i * 0.5e0)) - ((n1_i * (-0.16666666666666666e0)) + (n0_i * 0.16666666666666666e0)))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(u * Float32(n1_i + Float32(Float32(n0_i * Float32(Float32(1.0) - u)) / u))) + Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(Float32(Float32(Float32(-0.5) * Float32(u * n0_i)) + Float32(n0_i * Float32(0.5))) - Float32(Float32(n1_i * Float32(-0.16666666666666666)) + Float32(n0_i * Float32(0.16666666666666666)))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (u * (n1_i + ((n0_i * (single(1.0) - u)) / u))) + ((normAngle ^ single(2.0)) * (u * (((single(-0.5) * (u * n0_i)) + (n0_i * single(0.5))) - ((n1_i * single(-0.16666666666666666)) + (n0_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. *-commutative 97.5%

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

   2. associate-*l* 78.6%

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

   3. *-commutative 78.6%

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

   4. associate-*l* 72.6%

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

   5. distribute-lft-out 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in u around 0 72.3%

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

   1. *-commutative 72.3%

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

7. Simplified 72.3%

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

8. Taylor expanded in u around inf 73.3%

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

   1. +-commutative 73.3%

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

   2. associate-/l* 78.2%

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

   3. sub-neg 78.2%

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

   4. +-commutative 78.2%

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

   5. neg-mul-1 78.2%

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

   6. fma-undefine 78.2%

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

   7. *-commutative 78.2%

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

   8. *-commutative 78.2%

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

   9. times-frac 98.8%

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

   10. *-commutative 98.8%

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

   11. fma-undefine 98.8%

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

   12. neg-mul-1 98.8%

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

   13. +-commutative 98.8%

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

   14. sub-neg 98.8%

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

10. Simplified 98.8%

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

11. Taylor expanded in normAngle around 0 98.5%

    \[\leadsto \color{blue}{u \cdot \left(n1\_i + \frac{n0\_i \cdot \left(1 - u\right)}{u}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot \frac{n0\_i \cdot {\left(1 - u\right)}^{3}}{u} - \left(-0.16666666666666666 \cdot n1\_i + -0.16666666666666666 \cdot \frac{n0\_i \cdot \left(1 - u\right)}{u}\right)\right)\right)} \]

12. Taylor expanded in u around 0 98.5%

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

13. Final simplification 98.5%

    \[\leadsto u \cdot \left(n1\_i + \frac{n0\_i \cdot \left(1 - u\right)}{u}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.5 \cdot \left(u \cdot n0\_i\right) + n0\_i \cdot 0.5\right) - \left(n1\_i \cdot -0.16666666666666666 + n0\_i \cdot 0.16666666666666666\right)\right)\right) \]
14. Add Preprocessing

Alternative 4: 98.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(n1\_i + \frac{n0\_i \cdot \left(1 - u\right)}{u}\right) - {normAngle}^{2} \cdot \left(u \cdot \left(\left(n1\_i \cdot -0.16666666666666666 + n0\_i \cdot 0.16666666666666666\right) - n0\_i \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (-
  (* u (+ n1_i (/ (* n0_i (- 1.0 u)) u)))
  (*
   (pow normAngle 2.0)
   (*
    u
    (-
     (+ (* n1_i -0.16666666666666666) (* n0_i 0.16666666666666666))
     (* n0_i 0.5))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (u * (n1_i + ((n0_i * (1.0f - u)) / u))) - (powf(normAngle, 2.0f) * (u * (((n1_i * -0.16666666666666666f) + (n0_i * 0.16666666666666666f)) - (n0_i * 0.5f))));
}

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 = (u * (n1_i + ((n0_i * (1.0e0 - u)) / u))) - ((normangle ** 2.0e0) * (u * (((n1_i * (-0.16666666666666666e0)) + (n0_i * 0.16666666666666666e0)) - (n0_i * 0.5e0))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(u * Float32(n1_i + Float32(Float32(n0_i * Float32(Float32(1.0) - u)) / u))) - Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(Float32(Float32(n1_i * Float32(-0.16666666666666666)) + Float32(n0_i * Float32(0.16666666666666666))) - Float32(n0_i * Float32(0.5))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (u * (n1_i + ((n0_i * (single(1.0) - u)) / u))) - ((normAngle ^ single(2.0)) * (u * (((n1_i * single(-0.16666666666666666)) + (n0_i * single(0.16666666666666666))) - (n0_i * single(0.5)))));
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. *-commutative 97.5%

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

   2. associate-*l* 78.6%

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

   3. *-commutative 78.6%

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

   4. associate-*l* 72.6%

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

   5. distribute-lft-out 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in u around 0 72.3%

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

   1. *-commutative 72.3%

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

7. Simplified 72.3%

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

8. Taylor expanded in u around inf 73.3%

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

   1. +-commutative 73.3%

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

   2. associate-/l* 78.2%

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

   3. sub-neg 78.2%

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

   4. +-commutative 78.2%

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

   5. neg-mul-1 78.2%

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

   6. fma-undefine 78.2%

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

   7. *-commutative 78.2%

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

   8. *-commutative 78.2%

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

   9. times-frac 98.8%

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

   10. *-commutative 98.8%

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

   11. fma-undefine 98.8%

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

   12. neg-mul-1 98.8%

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

   13. +-commutative 98.8%

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

   14. sub-neg 98.8%

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

10. Simplified 98.8%

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

11. Taylor expanded in normAngle around 0 98.5%

    \[\leadsto \color{blue}{u \cdot \left(n1\_i + \frac{n0\_i \cdot \left(1 - u\right)}{u}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot \frac{n0\_i \cdot {\left(1 - u\right)}^{3}}{u} - \left(-0.16666666666666666 \cdot n1\_i + -0.16666666666666666 \cdot \frac{n0\_i \cdot \left(1 - u\right)}{u}\right)\right)\right)} \]

12. Taylor expanded in u around 0 98.4%

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

13. Final simplification 98.4%

    \[\leadsto u \cdot \left(n1\_i + \frac{n0\_i \cdot \left(1 - u\right)}{u}\right) - {normAngle}^{2} \cdot \left(u \cdot \left(\left(n1\_i \cdot -0.16666666666666666 + n0\_i \cdot 0.16666666666666666\right) - n0\_i \cdot 0.5\right)\right) \]
14. Add Preprocessing

Alternative 5: 98.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(n1\_i + \frac{n0\_i \cdot \left(1 - u\right)}{u}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n1\_i \cdot 0.16666666666666666\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (u * (n1_i + ((n0_i * (single(1.0) - u)) / u))) + ((normAngle ^ single(2.0)) * (u * (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. *-commutative 97.5%

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

   2. associate-*l* 78.6%

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

   3. *-commutative 78.6%

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

   4. associate-*l* 72.6%

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

   5. distribute-lft-out 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in u around 0 72.3%

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

   1. *-commutative 72.3%

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

7. Simplified 72.3%

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

8. Taylor expanded in u around inf 73.3%

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

   1. +-commutative 73.3%

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

   2. associate-/l* 78.2%

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

   3. sub-neg 78.2%

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

   4. +-commutative 78.2%

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

   5. neg-mul-1 78.2%

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

   6. fma-undefine 78.2%

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

   7. *-commutative 78.2%

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

   8. *-commutative 78.2%

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

   9. times-frac 98.8%

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

   10. *-commutative 98.8%

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

   11. fma-undefine 98.8%

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

   12. neg-mul-1 98.8%

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

   13. +-commutative 98.8%

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

   14. sub-neg 98.8%

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

10. Simplified 98.8%

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

11. Taylor expanded in normAngle around 0 98.5%

    \[\leadsto \color{blue}{u \cdot \left(n1\_i + \frac{n0\_i \cdot \left(1 - u\right)}{u}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot \frac{n0\_i \cdot {\left(1 - u\right)}^{3}}{u} - \left(-0.16666666666666666 \cdot n1\_i + -0.16666666666666666 \cdot \frac{n0\_i \cdot \left(1 - u\right)}{u}\right)\right)\right)} \]

12. Taylor expanded in n0_i around 0 98.2%

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

    1. *-commutative 98.2%

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

14. Simplified 98.2%

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

Alternative 6: 68.7% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.5000000170217692 \cdot 10^{-18} \lor \neg \left(n0\_i \leq 4.0000000126843074 \cdot 10^{-29}\right):\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -1.5000000170217692e-18)
         (not (<= n0_i 4.0000000126843074e-29)))
   (- 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 <= -1.5000000170217692e-18f) || !(n0_i <= 4.0000000126843074e-29f)) {
		tmp = n0_i - (u * n0_i);
	} 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 <= (-1.5000000170217692e-18)) .or. (.not. (n0_i <= 4.0000000126843074e-29))) then
        tmp = n0_i - (u * n0_i)
    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(-1.5000000170217692e-18)) || !(n0_i <= Float32(4.0000000126843074e-29)))
		tmp = Float32(n0_i - Float32(u * n0_i));
	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(-1.5000000170217692e-18)) || ~((n0_i <= single(4.0000000126843074e-29))))
		tmp = n0_i - (u * n0_i);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.50000002e-18 or 4.00000001e-29 < n0_i

   1. Initial program 98.2%

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

   3. Taylor expanded in u around 0 94.1%

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

      1. mul-1-neg 94.1%

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

      2. unsub-neg 94.1%

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

      3. associate-/l* 99.0%

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

      4. associate-/l* 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in normAngle around 0 98.8%

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

   7. Taylor expanded in n1_i around 0 82.1%

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

   if -1.50000002e-18 < n0_i < 4.00000001e-29

   1. Initial program 96.4%

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

      1. associate-*l* 96.0%

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

      2. cancel-sign-sub 96.0%

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

      3. *-commutative 96.0%

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

      4. associate-*r* 74.7%

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

      5. associate-*r/ 74.7%

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

      6. *-rgt-identity 74.7%

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

      7. sin-neg 74.7%

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

      8. distribute-lft-neg-out 74.7%

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

      9. associate-*l* 75.1%

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

      10. *-commutative 75.1%

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

      11. distribute-lft-neg-out 75.1%

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

      12. distribute-rgt-neg-out 75.1%

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

      13. associate-*r/ 75.5%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in n0_i around 0 53.4%

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

      1. *-commutative 53.4%

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

   7. Simplified 53.4%

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

   8. Taylor expanded in normAngle around 0 66.1%

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

      1. *-commutative 66.1%

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

   10. Simplified 66.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.5000000170217692 \cdot 10^{-18} \lor \neg \left(n0\_i \leq 4.0000000126843074 \cdot 10^{-29}\right):\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.6% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.5000000170217692 \cdot 10^{-18} \lor \neg \left(n0\_i \leq 4.0000000126843074 \cdot 10^{-29}\right):\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -1.5000000170217692e-18)
         (not (<= n0_i 4.0000000126843074e-29)))
   (* 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 <= -1.5000000170217692e-18f) || !(n0_i <= 4.0000000126843074e-29f)) {
		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 <= (-1.5000000170217692e-18)) .or. (.not. (n0_i <= 4.0000000126843074e-29))) 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(-1.5000000170217692e-18)) || !(n0_i <= Float32(4.0000000126843074e-29)))
		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(-1.5000000170217692e-18)) || ~((n0_i <= single(4.0000000126843074e-29))))
		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 < -1.50000002e-18 or 4.00000001e-29 < n0_i

   1. Initial program 98.2%

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

   3. Taylor expanded in u around 0 94.1%

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

      1. mul-1-neg 94.1%

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

      2. unsub-neg 94.1%

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

      3. associate-/l* 99.0%

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

      4. associate-/l* 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in normAngle around 0 98.8%

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

   7. Taylor expanded in n0_i around inf 81.8%

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

   if -1.50000002e-18 < n0_i < 4.00000001e-29

   1. Initial program 96.4%

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

      1. associate-*l* 96.0%

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

      2. cancel-sign-sub 96.0%

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

      3. *-commutative 96.0%

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

      4. associate-*r* 74.7%

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

      5. associate-*r/ 74.7%

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

      6. *-rgt-identity 74.7%

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

      7. sin-neg 74.7%

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

      8. distribute-lft-neg-out 74.7%

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

      9. associate-*l* 75.1%

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

      10. *-commutative 75.1%

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

      11. distribute-lft-neg-out 75.1%

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

      12. distribute-rgt-neg-out 75.1%

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

      13. associate-*r/ 75.5%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in n0_i around 0 53.4%

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

      1. *-commutative 53.4%

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

   7. Simplified 53.4%

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

   8. Taylor expanded in normAngle around 0 66.1%

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

      1. *-commutative 66.1%

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

   10. Simplified 66.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.5000000170217692 \cdot 10^{-18} \lor \neg \left(n0\_i \leq 4.0000000126843074 \cdot 10^{-29}\right):\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.6% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.5000000170217692 \cdot 10^{-18}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 5.000000097707407 \cdot 10^{-25}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.5000000170217692e-18)
   n0_i
   (if (<= n0_i 5.000000097707407e-25) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.5000000170217692e-18f) {
		tmp = n0_i;
	} else if (n0_i <= 5.000000097707407e-25f) {
		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 <= (-1.5000000170217692e-18)) then
        tmp = n0_i
    else if (n0_i <= 5.000000097707407e-25) 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(-1.5000000170217692e-18))
		tmp = n0_i;
	elseif (n0_i <= Float32(5.000000097707407e-25))
		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(-1.5000000170217692e-18))
		tmp = n0_i;
	elseif (n0_i <= single(5.000000097707407e-25))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.50000002e-18 or 5.0000001e-25 < n0_i

   1. Initial program 98.5%

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

      1. fma-define 98.5%

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

      2. associate-*r/ 99.0%

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

      3. *-rgt-identity 99.0%

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

      4. associate-*r/ 99.0%

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

      5. *-rgt-identity 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in u around 0 68.4%

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

   if -1.50000002e-18 < n0_i < 5.0000001e-25

   1. Initial program 96.2%

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

      1. associate-*l* 95.8%

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

      2. cancel-sign-sub 95.8%

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

      3. *-commutative 95.8%

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

      4. associate-*r* 72.5%

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

      5. associate-*r/ 72.4%

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

      6. *-rgt-identity 72.4%

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

      7. sin-neg 72.4%

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

      8. distribute-lft-neg-out 72.4%

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

      9. associate-*l* 72.9%

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

      10. *-commutative 72.9%

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

      11. distribute-lft-neg-out 72.9%

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

      12. distribute-rgt-neg-out 72.9%

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

      13. associate-*r/ 73.2%

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

   3. Simplified 60.3%

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

   5. Taylor expanded in n0_i around 0 50.4%

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

      1. *-commutative 50.4%

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

   7. Simplified 50.4%

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

   8. Taylor expanded in normAngle around 0 62.9%

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

      1. *-commutative 62.9%

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

   10. Simplified 62.9%

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

Alternative 9: 98.1% accurate, 46.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + ((u * n1_i) - (u * 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. Add Preprocessing

3. Taylor expanded in u around 0 93.2%

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

   1. mul-1-neg 93.2%

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

   2. unsub-neg 93.2%

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

   3. associate-/l* 98.1%

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

   4. associate-/l* 98.1%

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

5. Simplified 98.1%

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

6. Taylor expanded in normAngle around 0 98.1%

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

   1. associate--l+ 98.1%

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

   2. *-commutative 98.1%

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

8. Simplified 98.1%

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

9. Final simplification 98.1%

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

Alternative 10: 98.1% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in u around 0 93.2%

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

   1. mul-1-neg 93.2%

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

   2. unsub-neg 93.2%

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

   3. associate-/l* 98.1%

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

   4. associate-/l* 98.1%

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

5. Simplified 98.1%

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

6. Taylor expanded in normAngle around 0 98.1%

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

   1. associate--l+ 98.1%

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

   2. distribute-rgt-out-- 98.1%

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

8. Simplified 98.1%

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

Alternative 11: 47.6% accurate, 421.0× speedup

Math

\[\begin{array}{l} \\ n0\_i \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return n0_i
end

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. fma-define 97.6%

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

   2. associate-*r/ 97.8%

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

   3. *-rgt-identity 97.8%

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

   4. associate-*r/ 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 50.7%

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

Reproduce

herbie shell --seed 2024139 
(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)))