Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.1%

Time: 15.1s

Alternatives: 9

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.3% 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: 99.1% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(n1\_i + \left(\left(\left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) - n0\_i\right) + u \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.5 + u \cdot \left(-0.16666666666666666 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    n1_i
    (+
     (-
      (*
       (* normAngle normAngle)
       (+ (* n1_i 0.16666666666666666) (* n0_i 0.3333333333333333)))
      n0_i)
     (*
      u
      (*
       (* normAngle normAngle)
       (+ (* n0_i -0.5) (* u (* -0.16666666666666666 (- n1_i n0_i)))))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * (n1_i + ((((normAngle * normAngle) * ((n1_i * 0.16666666666666666f) + (n0_i * 0.3333333333333333f))) - n0_i) + (u * ((normAngle * normAngle) * ((n0_i * -0.5f) + (u * (-0.16666666666666666f * (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 + ((((normangle * normangle) * ((n1_i * 0.16666666666666666e0) + (n0_i * 0.3333333333333333e0))) - n0_i) + (u * ((normangle * normangle) * ((n0_i * (-0.5e0)) + (u * ((-0.16666666666666666e0) * (n1_i - n0_i)))))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(n1_i + Float32(Float32(Float32(Float32(normAngle * normAngle) * Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.3333333333333333)))) - n0_i) + Float32(u * Float32(Float32(normAngle * normAngle) * Float32(Float32(n0_i * Float32(-0.5)) + Float32(u * Float32(Float32(-0.16666666666666666) * Float32(n1_i - n0_i))))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * (n1_i + ((((normAngle * normAngle) * ((n1_i * single(0.16666666666666666)) + (n0_i * single(0.3333333333333333)))) - n0_i) + (u * ((normAngle * normAngle) * ((n0_i * single(-0.5)) + (u * (single(-0.16666666666666666) * (n1_i - n0_i)))))))));
end

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - \left(\frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)} \]

5. Simplified 98.7%

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

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\frac{-1}{6} \cdot n1\_i + \frac{1}{6} \cdot n0\_i\right)\right)\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)} \]

8. Simplified 99.0%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(\left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) \cdot \left(normAngle \cdot normAngle\right) - n0\_i\right) + u \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.5 + u \cdot \left(-0.16666666666666666 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)} \]

9. Final simplification 99.0%

   \[\leadsto n0\_i + u \cdot \left(n1\_i + \left(\left(\left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) - n0\_i\right) + u \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.5 + u \cdot \left(-0.16666666666666666 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 2: 99.1% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(n1\_i + \left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) + u \cdot \left(n0\_i \cdot -0.5 + -0.16666666666666666 \cdot \left(u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right) - n0\_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    n1_i
    (-
     (*
      (* normAngle normAngle)
      (+
       (+ (* n1_i 0.16666666666666666) (* n0_i 0.3333333333333333))
       (* u (+ (* n0_i -0.5) (* -0.16666666666666666 (* u (- n1_i n0_i)))))))
     n0_i)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * (n1_i + (((normAngle * normAngle) * (((n1_i * 0.16666666666666666f) + (n0_i * 0.3333333333333333f)) + (u * ((n0_i * -0.5f) + (-0.16666666666666666f * (u * (n1_i - n0_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 + (((normangle * normangle) * (((n1_i * 0.16666666666666666e0) + (n0_i * 0.3333333333333333e0)) + (u * ((n0_i * (-0.5e0)) + ((-0.16666666666666666e0) * (u * (n1_i - n0_i))))))) - n0_i)))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(n1_i + Float32(Float32(Float32(normAngle * normAngle) * Float32(Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.3333333333333333))) + Float32(u * Float32(Float32(n0_i * Float32(-0.5)) + Float32(Float32(-0.16666666666666666) * Float32(u * Float32(n1_i - n0_i))))))) - n0_i))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * (n1_i + (((normAngle * normAngle) * (((n1_i * single(0.16666666666666666)) + (n0_i * single(0.3333333333333333))) + (u * ((n0_i * single(-0.5)) + (single(-0.16666666666666666) * (u * (n1_i - n0_i))))))) - n0_i)));
end

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - \left(\frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)} \]

5. Simplified 98.7%

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

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\frac{-1}{6} \cdot n1\_i + \frac{1}{6} \cdot n0\_i\right)\right)\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)} \]

8. Simplified 99.0%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(\left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) \cdot \left(normAngle \cdot normAngle\right) - n0\_i\right) + u \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.5 + u \cdot \left(-0.16666666666666666 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)} \]

9. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \color{blue}{\left({normAngle}^{2} \cdot \left(\frac{1}{6} \cdot n1\_i + \left(\frac{1}{3} \cdot n0\_i + u \cdot \left(\frac{-1}{2} \cdot n0\_i + \frac{-1}{6} \cdot \left(u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - n0\_i\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. --lowering--.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\left({normAngle}^{2} \cdot \left(\frac{1}{6} \cdot n1\_i + \left(\frac{1}{3} \cdot n0\_i + u \cdot \left(\frac{-1}{2} \cdot n0\_i + \frac{-1}{6} \cdot \left(u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right), \color{blue}{n0\_i}\right)\right)\right)\right) \]

11. Simplified 99.0%

    \[\leadsto n0\_i + u \cdot \left(n1\_i + \color{blue}{\left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(0.16666666666666666 \cdot n1\_i + 0.3333333333333333 \cdot n0\_i\right) + u \cdot \left(n0\_i \cdot -0.5 + -0.16666666666666666 \cdot \left(u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right) - n0\_i\right)}\right) \]

12. Final simplification 99.0%

    \[\leadsto n0\_i + u \cdot \left(n1\_i + \left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) + u \cdot \left(n0\_i \cdot -0.5 + -0.16666666666666666 \cdot \left(u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right) - n0\_i\right)\right) \]
13. Add Preprocessing

Alternative 3: 99.1% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(n1\_i + \left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) + \left(n0\_i \cdot u\right) \cdot \left(-0.5 + u \cdot 0.16666666666666666\right)\right) - n0\_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    n1_i
    (-
     (*
      (* normAngle normAngle)
      (+
       (+ (* n1_i 0.16666666666666666) (* n0_i 0.3333333333333333))
       (* (* n0_i u) (+ -0.5 (* u 0.16666666666666666)))))
     n0_i)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * (n1_i + (((normAngle * normAngle) * (((n1_i * 0.16666666666666666f) + (n0_i * 0.3333333333333333f)) + ((n0_i * u) * (-0.5f + (u * 0.16666666666666666f))))) - 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 + (((normangle * normangle) * (((n1_i * 0.16666666666666666e0) + (n0_i * 0.3333333333333333e0)) + ((n0_i * u) * ((-0.5e0) + (u * 0.16666666666666666e0))))) - n0_i)))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(n1_i + Float32(Float32(Float32(normAngle * normAngle) * Float32(Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.3333333333333333))) + Float32(Float32(n0_i * u) * Float32(Float32(-0.5) + Float32(u * Float32(0.16666666666666666)))))) - n0_i))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * (n1_i + (((normAngle * normAngle) * (((n1_i * single(0.16666666666666666)) + (n0_i * single(0.3333333333333333))) + ((n0_i * u) * (single(-0.5) + (u * single(0.16666666666666666)))))) - n0_i)));
end

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - \left(\frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)} \]

5. Simplified 98.7%

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

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\frac{-1}{6} \cdot n1\_i + \frac{1}{6} \cdot n0\_i\right)\right)\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)} \]

8. Simplified 99.0%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(\left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) \cdot \left(normAngle \cdot normAngle\right) - n0\_i\right) + u \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.5 + u \cdot \left(-0.16666666666666666 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)} \]

9. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \color{blue}{\left({normAngle}^{2} \cdot \left(\frac{1}{6} \cdot n1\_i + \left(\frac{1}{3} \cdot n0\_i + u \cdot \left(\frac{-1}{2} \cdot n0\_i + \frac{-1}{6} \cdot \left(u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - n0\_i\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. --lowering--.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\left({normAngle}^{2} \cdot \left(\frac{1}{6} \cdot n1\_i + \left(\frac{1}{3} \cdot n0\_i + u \cdot \left(\frac{-1}{2} \cdot n0\_i + \frac{-1}{6} \cdot \left(u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right), \color{blue}{n0\_i}\right)\right)\right)\right) \]

11. Simplified 99.0%

    \[\leadsto n0\_i + u \cdot \left(n1\_i + \color{blue}{\left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(0.16666666666666666 \cdot n1\_i + 0.3333333333333333 \cdot n0\_i\right) + u \cdot \left(n0\_i \cdot -0.5 + -0.16666666666666666 \cdot \left(u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right) - n0\_i\right)}\right) \]

12. Taylor expanded in n0_i around inf

    \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \color{blue}{\left(n0\_i \cdot \left(u \cdot \left(\frac{1}{6} \cdot u - \frac{1}{2}\right)\right)\right)}\right)\right), n0\_i\right)\right)\right)\right) \]
13. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \left(\left(n0\_i \cdot u\right) \cdot \left(\frac{1}{6} \cdot u - \frac{1}{2}\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \mathsf{*.f32}\left(\left(n0\_i \cdot u\right), \left(\frac{1}{6} \cdot u - \frac{1}{2}\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \mathsf{*.f32}\left(\left(u \cdot n0\_i\right), \left(\frac{1}{6} \cdot u - \frac{1}{2}\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, n0\_i\right), \left(\frac{1}{6} \cdot u - \frac{1}{2}\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

    5. sub-neg N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, n0\_i\right), \left(\frac{1}{6} \cdot u + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, n0\_i\right), \left(\frac{1}{6} \cdot u + \frac{-1}{2}\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

    7. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, n0\_i\right), \mathsf{+.f32}\left(\left(\frac{1}{6} \cdot u\right), \frac{-1}{2}\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, n0\_i\right), \mathsf{+.f32}\left(\left(u \cdot \frac{1}{6}\right), \frac{-1}{2}\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

    9. *-lowering-*.f32 99.0%

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, n0\_i\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \frac{1}{6}\right), \frac{-1}{2}\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

14. Simplified 99.0%

    \[\leadsto n0\_i + u \cdot \left(n1\_i + \left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(0.16666666666666666 \cdot n1\_i + 0.3333333333333333 \cdot n0\_i\right) + \color{blue}{\left(u \cdot n0\_i\right) \cdot \left(u \cdot 0.16666666666666666 + -0.5\right)}\right) - n0\_i\right)\right) \]

15. Final simplification 99.0%

    \[\leadsto n0\_i + u \cdot \left(n1\_i + \left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) + \left(n0\_i \cdot u\right) \cdot \left(-0.5 + u \cdot 0.16666666666666666\right)\right) - n0\_i\right)\right) \]
16. Add Preprocessing

Alternative 4: 99.0% accurate, 16.8× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(n1\_i + \left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) + -0.5 \cdot \left(n0\_i \cdot u\right)\right) - n0\_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    n1_i
    (-
     (*
      (* normAngle normAngle)
      (+
       (+ (* n1_i 0.16666666666666666) (* n0_i 0.3333333333333333))
       (* -0.5 (* n0_i u))))
     n0_i)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * (n1_i + (((normAngle * normAngle) * (((n1_i * 0.16666666666666666f) + (n0_i * 0.3333333333333333f)) + (-0.5f * (n0_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 + (((normangle * normangle) * (((n1_i * 0.16666666666666666e0) + (n0_i * 0.3333333333333333e0)) + ((-0.5e0) * (n0_i * u)))) - n0_i)))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(n1_i + Float32(Float32(Float32(normAngle * normAngle) * Float32(Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.3333333333333333))) + Float32(Float32(-0.5) * Float32(n0_i * u)))) - n0_i))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * (n1_i + (((normAngle * normAngle) * (((n1_i * single(0.16666666666666666)) + (n0_i * single(0.3333333333333333))) + (single(-0.5) * (n0_i * u)))) - n0_i)));
end

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - \left(\frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)} \]

5. Simplified 98.7%

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

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\frac{-1}{6} \cdot n1\_i + \frac{1}{6} \cdot n0\_i\right)\right)\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)} \]

8. Simplified 99.0%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(\left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) \cdot \left(normAngle \cdot normAngle\right) - n0\_i\right) + u \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.5 + u \cdot \left(-0.16666666666666666 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)} \]

9. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \color{blue}{\left({normAngle}^{2} \cdot \left(\frac{1}{6} \cdot n1\_i + \left(\frac{1}{3} \cdot n0\_i + u \cdot \left(\frac{-1}{2} \cdot n0\_i + \frac{-1}{6} \cdot \left(u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right) - n0\_i\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. --lowering--.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\left({normAngle}^{2} \cdot \left(\frac{1}{6} \cdot n1\_i + \left(\frac{1}{3} \cdot n0\_i + u \cdot \left(\frac{-1}{2} \cdot n0\_i + \frac{-1}{6} \cdot \left(u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right), \color{blue}{n0\_i}\right)\right)\right)\right) \]

11. Simplified 99.0%

    \[\leadsto n0\_i + u \cdot \left(n1\_i + \color{blue}{\left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(0.16666666666666666 \cdot n1\_i + 0.3333333333333333 \cdot n0\_i\right) + u \cdot \left(n0\_i \cdot -0.5 + -0.16666666666666666 \cdot \left(u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right) - n0\_i\right)}\right) \]

12. Taylor expanded in u around 0

    \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \color{blue}{\left(\frac{-1}{2} \cdot \left(n0\_i \cdot u\right)\right)}\right)\right), n0\_i\right)\right)\right)\right) \]
13. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \mathsf{*.f32}\left(\frac{-1}{2}, \left(n0\_i \cdot u\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \mathsf{*.f32}\left(\frac{-1}{2}, \left(u \cdot n0\_i\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

    3. *-lowering-*.f32 98.8%

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(n1\_i, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{6}, n1\_i\right), \mathsf{*.f32}\left(\frac{1}{3}, n0\_i\right)\right), \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u, n0\_i\right)\right)\right)\right), n0\_i\right)\right)\right)\right) \]

14. Simplified 98.8%

    \[\leadsto n0\_i + u \cdot \left(n1\_i + \left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(0.16666666666666666 \cdot n1\_i + 0.3333333333333333 \cdot n0\_i\right) + \color{blue}{-0.5 \cdot \left(u \cdot n0\_i\right)}\right) - n0\_i\right)\right) \]

15. Final simplification 98.8%

    \[\leadsto n0\_i + u \cdot \left(n1\_i + \left(\left(normAngle \cdot normAngle\right) \cdot \left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) + -0.5 \cdot \left(n0\_i \cdot u\right)\right) - n0\_i\right)\right) \]
16. Add Preprocessing

Alternative 5: 99.0% accurate, 22.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - \left(\frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)} \]

5. Simplified 98.7%

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

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \color{blue}{\left(u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)}\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)}\right)\right) \]

   3. associate-+r+ N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\left(n1\_i + -1 \cdot n0\_i\right) + \color{blue}{{normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i + -1 \cdot n0\_i\right), \color{blue}{\left({normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)}\right)\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i + \left(\mathsf{neg}\left(n0\_i\right)\right)\right), \left({normAngle}^{\color{blue}{2}} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i - n0\_i\right), \left(\color{blue}{{normAngle}^{2}} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   7. --lowering--.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \left(\color{blue}{{normAngle}^{2}} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \left(\left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right) \cdot \color{blue}{{normAngle}^{2}}\right)\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right), \color{blue}{\left({normAngle}^{2}\right)}\right)\right)\right)\right) \]

   10. associate--l+ N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\left(\frac{1}{6} \cdot n1\_i + \left(\frac{1}{2} \cdot n0\_i - \frac{1}{6} \cdot n0\_i\right)\right), \left({\color{blue}{normAngle}}^{2}\right)\right)\right)\right)\right) \]

   11. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{6} \cdot n1\_i\right), \left(\frac{1}{2} \cdot n0\_i - \frac{1}{6} \cdot n0\_i\right)\right), \left({\color{blue}{normAngle}}^{2}\right)\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(n1\_i \cdot \frac{1}{6}\right), \left(\frac{1}{2} \cdot n0\_i - \frac{1}{6} \cdot n0\_i\right)\right), \left({normAngle}^{2}\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \left(\frac{1}{2} \cdot n0\_i - \frac{1}{6} \cdot n0\_i\right)\right), \left({normAngle}^{2}\right)\right)\right)\right)\right) \]

   14. distribute-rgt-out-- N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \left(n0\_i \cdot \left(\frac{1}{2} - \frac{1}{6}\right)\right)\right), \left({normAngle}^{2}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{*.f32}\left(n0\_i, \left(\frac{1}{2} - \frac{1}{6}\right)\right)\right), \left({normAngle}^{2}\right)\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right)\right), \left({normAngle}^{2}\right)\right)\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right)\right), \left(normAngle \cdot \color{blue}{normAngle}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right)\right), \mathsf{*.f32}\left(normAngle, \color{blue}{normAngle}\right)\right)\right)\right)\right) \]

8. Simplified 98.6%

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

9. Final simplification 98.6%

   \[\leadsto n0\_i + u \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) + \left(n1\_i - n0\_i\right)\right) \]
10. Add Preprocessing

Alternative 6: 98.8% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - \left(\frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)} \]

5. Simplified 98.7%

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

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \color{blue}{\left(u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)}\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)}\right)\right) \]

   3. associate-+r+ N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\left(n1\_i + -1 \cdot n0\_i\right) + \color{blue}{{normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i + -1 \cdot n0\_i\right), \color{blue}{\left({normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)}\right)\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i + \left(\mathsf{neg}\left(n0\_i\right)\right)\right), \left({normAngle}^{\color{blue}{2}} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i - n0\_i\right), \left(\color{blue}{{normAngle}^{2}} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   7. --lowering--.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \left(\color{blue}{{normAngle}^{2}} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \left(\left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right) \cdot \color{blue}{{normAngle}^{2}}\right)\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right), \color{blue}{\left({normAngle}^{2}\right)}\right)\right)\right)\right) \]

   10. associate--l+ N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\left(\frac{1}{6} \cdot n1\_i + \left(\frac{1}{2} \cdot n0\_i - \frac{1}{6} \cdot n0\_i\right)\right), \left({\color{blue}{normAngle}}^{2}\right)\right)\right)\right)\right) \]

   11. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{6} \cdot n1\_i\right), \left(\frac{1}{2} \cdot n0\_i - \frac{1}{6} \cdot n0\_i\right)\right), \left({\color{blue}{normAngle}}^{2}\right)\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(n1\_i \cdot \frac{1}{6}\right), \left(\frac{1}{2} \cdot n0\_i - \frac{1}{6} \cdot n0\_i\right)\right), \left({normAngle}^{2}\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \left(\frac{1}{2} \cdot n0\_i - \frac{1}{6} \cdot n0\_i\right)\right), \left({normAngle}^{2}\right)\right)\right)\right)\right) \]

   14. distribute-rgt-out-- N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \left(n0\_i \cdot \left(\frac{1}{2} - \frac{1}{6}\right)\right)\right), \left({normAngle}^{2}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{*.f32}\left(n0\_i, \left(\frac{1}{2} - \frac{1}{6}\right)\right)\right), \left({normAngle}^{2}\right)\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right)\right), \left({normAngle}^{2}\right)\right)\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right)\right), \left(normAngle \cdot \color{blue}{normAngle}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{*.f32}\left(n0\_i, \frac{1}{3}\right)\right), \mathsf{*.f32}\left(normAngle, \color{blue}{normAngle}\right)\right)\right)\right)\right) \]

8. Simplified 98.6%

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

9. Taylor expanded in n1_i around inf

   \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \color{blue}{\left(\frac{1}{6} \cdot \left(n1\_i \cdot {normAngle}^{2}\right)\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\frac{1}{6}, \color{blue}{\left(n1\_i \cdot {normAngle}^{2}\right)}\right)\right)\right)\right) \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(n1\_i, \color{blue}{\left({normAngle}^{2}\right)}\right)\right)\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(n1\_i, \left(normAngle \cdot \color{blue}{normAngle}\right)\right)\right)\right)\right)\right) \]

    4. *-lowering-*.f32 98.4%

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(n1\_i, \mathsf{*.f32}\left(normAngle, \color{blue}{normAngle}\right)\right)\right)\right)\right)\right) \]

11. Simplified 98.4%

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

Alternative 7: 98.2% 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 97.6%

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

3. Taylor expanded in normAngle around 0

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - \left(\frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)} \]

5. Simplified 98.7%

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

6. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + \left(u \cdot \left(\frac{-1}{2} \cdot \left(n0\_i \cdot {normAngle}^{2}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\frac{-1}{6} \cdot n1\_i + \frac{1}{6} \cdot n0\_i\right)\right)\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)} \]

8. Simplified 99.0%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + \left(\left(\left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right) \cdot \left(normAngle \cdot normAngle\right) - n0\_i\right) + u \cdot \left(\left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot -0.5 + u \cdot \left(-0.16666666666666666 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\right)\right)} \]

9. Taylor expanded in normAngle around 0

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

    1. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \color{blue}{\left(u \cdot \left(n1\_i - n0\_i\right)\right)}\right) \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(n1\_i - n0\_i\right)}\right)\right) \]

    3. --lowering--.f32 98.0%

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(n1\_i, \color{blue}{n0\_i}\right)\right)\right) \]

11. Simplified 98.0%

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

Alternative 8: 59.1% accurate, 84.2× speedup

Math

\[\begin{array}{l} \\ n0\_i \cdot \left(1 - u\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in n0_i around inf

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\sin \left(normAngle \cdot \left(1 - u\right)\right), \color{blue}{\left(\frac{n0\_i}{\sin normAngle}\right)}\right) \]

   4. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sin.f32}\left(\left(normAngle \cdot \left(1 - u\right)\right)\right), \left(\frac{\color{blue}{n0\_i}}{\sin normAngle}\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(normAngle, \left(1 - u\right)\right)\right), \left(\frac{n0\_i}{\sin normAngle}\right)\right) \]

   6. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{\_.f32}\left(1, u\right)\right)\right), \left(\frac{n0\_i}{\sin normAngle}\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{\_.f32}\left(1, u\right)\right)\right), \mathsf{/.f32}\left(n0\_i, \color{blue}{\sin normAngle}\right)\right) \]

   8. sin-lowering-sin.f32 56.8%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{\_.f32}\left(1, u\right)\right)\right), \mathsf{/.f32}\left(n0\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right) \]

5. Simplified 56.8%

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

6. Taylor expanded in normAngle around 0

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

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(n0\_i, \color{blue}{\left(1 - u\right)}\right) \]

   2. --lowering--.f32 56.4%

      \[\leadsto \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(1, \color{blue}{u}\right)\right) \]

8. Simplified 56.4%

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

Alternative 9: 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 97.6%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i} \]
4. Step-by-step derivation

5. Simplified 45.3%

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

Reproduce

herbie shell --seed 2024161 
(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)))