Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.0%

Time: 14.6s

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: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t_0\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot t_0\right) \cdot n1_i \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(n0_i - n0_i \cdot u\right) - \left({normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot \left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) - \left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right)\right) - u \cdot n1_i\right) \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n0_i - (n0_i * u)) - ((powf(normAngle, 2.0f) * ((-0.16666666666666666f * ((u * n1_i) + (n0_i * (1.0f - u)))) - ((-0.16666666666666666f * (n0_i * powf((1.0f - u), 3.0f))) + (-0.16666666666666666f * (n1_i * powf(u, 3.0f)))))) - (u * n1_i));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = (n0_i - (n0_i * u)) - (((normangle ** 2.0e0) * (((-0.16666666666666666e0) * ((u * n1_i) + (n0_i * (1.0e0 - u)))) - (((-0.16666666666666666e0) * (n0_i * ((1.0e0 - u) ** 3.0e0))) + ((-0.16666666666666666e0) * (n1_i * (u ** 3.0e0)))))) - (u * n1_i))
end function

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n0_i - (n0_i * u)) - (((normAngle ^ single(2.0)) * ((single(-0.16666666666666666) * ((u * n1_i) + (n0_i * (single(1.0) - u)))) - ((single(-0.16666666666666666) * (n0_i * ((single(1.0) - u) ^ single(3.0)))) + (single(-0.16666666666666666) * (n1_i * (u ^ single(3.0))))))) - (u * n1_i));
end

Derivation

1. Initial program 96.9%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. *-commutative 96.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. associate-*l* 78.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. *-commutative 78.3%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

   4. associate-*l* 72.4%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   5. distribute-lft-out 72.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Simplified 72.3%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

4. Taylor expanded in normAngle around 0 99.1%

   \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right)} \]

5. Taylor expanded in u around 0 99.3%

   \[\leadsto \color{blue}{\left(n0_i + -1 \cdot \left(n0_i \cdot u\right)\right)} + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg 99.3%

      \[\leadsto \left(n0_i + \color{blue}{\left(-n0_i \cdot u\right)}\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

   2. unsub-neg 99.3%

      \[\leadsto \color{blue}{\left(n0_i - n0_i \cdot u\right)} + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

   3. *-commutative 99.3%

      \[\leadsto \left(n0_i - \color{blue}{u \cdot n0_i}\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

7. Simplified 99.3%

   \[\leadsto \color{blue}{\left(n0_i - u \cdot n0_i\right)} + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

8. Final simplification 99.3%

   \[\leadsto \left(n0_i - n0_i \cdot u\right) - \left({normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot \left(u \cdot n1_i + n0_i \cdot \left(1 - u\right)\right) - \left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right)\right) - u \cdot n1_i\right) \]

Alternative 2: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ n0_i \cdot \left(1 - u\right) + \left(u \cdot n1_i + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) + -0.16666666666666666 \cdot \left(n0_i \cdot \left(u + -1\right) - u \cdot n1_i\right)\right)\right) \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n0_i * (1.0f - u)) + ((u * n1_i) + (powf(normAngle, 2.0f) * (((-0.16666666666666666f * (n0_i * powf((1.0f - u), 3.0f))) + (-0.16666666666666666f * (n1_i * powf(u, 3.0f)))) + (-0.16666666666666666f * ((n0_i * (u + -1.0f)) - (u * n1_i))))));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = (n0_i * (1.0e0 - u)) + ((u * n1_i) + ((normangle ** 2.0e0) * ((((-0.16666666666666666e0) * (n0_i * ((1.0e0 - u) ** 3.0e0))) + ((-0.16666666666666666e0) * (n1_i * (u ** 3.0e0)))) + ((-0.16666666666666666e0) * ((n0_i * (u + (-1.0e0))) - (u * n1_i))))))
end function

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n0_i * (single(1.0) - u)) + ((u * n1_i) + ((normAngle ^ single(2.0)) * (((single(-0.16666666666666666) * (n0_i * ((single(1.0) - u) ^ single(3.0)))) + (single(-0.16666666666666666) * (n1_i * (u ^ single(3.0))))) + (single(-0.16666666666666666) * ((n0_i * (u + single(-1.0))) - (u * n1_i))))));
end

Derivation

1. Initial program 96.9%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. *-commutative 96.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. associate-*l* 78.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. *-commutative 78.3%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

   4. associate-*l* 72.4%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   5. distribute-lft-out 72.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Simplified 72.3%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

4. Taylor expanded in normAngle around 0 99.1%

   \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right)} \]

5. Final simplification 99.1%

   \[\leadsto n0_i \cdot \left(1 - u\right) + \left(u \cdot n1_i + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) + -0.16666666666666666 \cdot \left(n0_i \cdot \left(u + -1\right) - u \cdot n1_i\right)\right)\right) \]

Alternative 3: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(n0_i - n0_i \cdot u\right) + \left(u \cdot n1_i - {normAngle}^{2} \cdot \left(n0_i \cdot \left(-0.16666666666666666 \cdot \left(\left(1 - u\right) - {\left(1 - u\right)}^{3}\right)\right)\right)\right) \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n0_i - (n0_i * u)) + ((u * n1_i) - (powf(normAngle, 2.0f) * (n0_i * (-0.16666666666666666f * ((1.0f - u) - powf((1.0f - u), 3.0f))))));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = (n0_i - (n0_i * u)) + ((u * n1_i) - ((normangle ** 2.0e0) * (n0_i * ((-0.16666666666666666e0) * ((1.0e0 - u) - ((1.0e0 - u) ** 3.0e0))))))
end function

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n0_i - (n0_i * u)) + ((u * n1_i) - ((normAngle ^ single(2.0)) * (n0_i * (single(-0.16666666666666666) * ((single(1.0) - u) - ((single(1.0) - u) ^ single(3.0)))))));
end

Derivation

1. Initial program 96.9%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. *-commutative 96.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. associate-*l* 78.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. *-commutative 78.3%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

   4. associate-*l* 72.4%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   5. distribute-lft-out 72.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Simplified 72.3%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

4. Taylor expanded in normAngle around 0 99.1%

   \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right)} \]

5. Taylor expanded in u around 0 99.3%

   \[\leadsto \color{blue}{\left(n0_i + -1 \cdot \left(n0_i \cdot u\right)\right)} + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg 99.3%

      \[\leadsto \left(n0_i + \color{blue}{\left(-n0_i \cdot u\right)}\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

   2. unsub-neg 99.3%

      \[\leadsto \color{blue}{\left(n0_i - n0_i \cdot u\right)} + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

   3. *-commutative 99.3%

      \[\leadsto \left(n0_i - \color{blue}{u \cdot n0_i}\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

7. Simplified 99.3%

   \[\leadsto \color{blue}{\left(n0_i - u \cdot n0_i\right)} + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

8. Taylor expanded in n0_i around inf 98.8%

   \[\leadsto \left(n0_i - u \cdot n0_i\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(n0_i \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. distribute-lft-out-- 98.8%

      \[\leadsto \left(n0_i - u \cdot n0_i\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(n0_i \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right)\right)}\right)\right) \]

10. Simplified 98.8%

    \[\leadsto \left(n0_i - u \cdot n0_i\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(n0_i \cdot \left(-0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right)\right)\right)}\right) \]

11. Final simplification 98.8%

    \[\leadsto \left(n0_i - n0_i \cdot u\right) + \left(u \cdot n1_i - {normAngle}^{2} \cdot \left(n0_i \cdot \left(-0.16666666666666666 \cdot \left(\left(1 - u\right) - {\left(1 - u\right)}^{3}\right)\right)\right)\right) \]

Alternative 4: 98.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ n0_i \cdot \left(1 - u\right) + \left(u \cdot n1_i + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.5 - 0.16666666666666666 \cdot \left(n0_i - n1_i\right)\right)\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = (n0_i * (1.0e0 - u)) + ((u * n1_i) + ((normangle ** 2.0e0) * (u * ((n0_i * 0.5e0) - (0.16666666666666666e0 * (n0_i - n1_i))))))
end function

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n0_i * (single(1.0) - u)) + ((u * n1_i) + ((normAngle ^ single(2.0)) * (u * ((n0_i * single(0.5)) - (single(0.16666666666666666) * (n0_i - n1_i))))));
end

Derivation

1. Initial program 96.9%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. *-commutative 96.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. associate-*l* 78.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. *-commutative 78.3%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

   4. associate-*l* 72.4%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   5. distribute-lft-out 72.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Simplified 72.3%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

4. Taylor expanded in normAngle around 0 99.1%

   \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right)} \]

5. Taylor expanded in u around 0 98.8%

   \[\leadsto n0_i \cdot \left(1 - u\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(0.5 \cdot n0_i - -0.16666666666666666 \cdot \left(n1_i + -1 \cdot n0_i\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 98.8%

      \[\leadsto n0_i \cdot \left(1 - u\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \color{blue}{\left(0.5 \cdot n0_i + \left(--0.16666666666666666\right) \cdot \left(n1_i + -1 \cdot n0_i\right)\right)}\right)\right) \]

   2. *-commutative 98.8%

      \[\leadsto n0_i \cdot \left(1 - u\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\color{blue}{n0_i \cdot 0.5} + \left(--0.16666666666666666\right) \cdot \left(n1_i + -1 \cdot n0_i\right)\right)\right)\right) \]

   3. metadata-eval 98.8%

      \[\leadsto n0_i \cdot \left(1 - u\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.5 + \color{blue}{0.16666666666666666} \cdot \left(n1_i + -1 \cdot n0_i\right)\right)\right)\right) \]

   4. mul-1-neg 98.8%

      \[\leadsto n0_i \cdot \left(1 - u\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1_i + \color{blue}{\left(-n0_i\right)}\right)\right)\right)\right) \]

7. Simplified 98.8%

   \[\leadsto n0_i \cdot \left(1 - u\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(n0_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1_i + \left(-n0_i\right)\right)\right)\right)}\right) \]

8. Final simplification 98.8%

   \[\leadsto n0_i \cdot \left(1 - u\right) + \left(u \cdot n1_i + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.5 - 0.16666666666666666 \cdot \left(n0_i - n1_i\right)\right)\right)\right) \]

Alternative 5: 98.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.9%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. *-commutative 96.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. associate-*l* 78.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. *-commutative 78.3%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

   4. associate-*l* 72.4%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   5. distribute-lft-out 72.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Simplified 72.3%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

4. Taylor expanded in normAngle around 0 98.3%

   \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

5. Taylor expanded in u around 0 98.6%

   \[\leadsto \color{blue}{n0_i + u \cdot \left(n1_i + -1 \cdot n0_i\right)} \]
6. Step-by-step derivation

   1. +-commutative 98.6%

      \[\leadsto \color{blue}{u \cdot \left(n1_i + -1 \cdot n0_i\right) + n0_i} \]

   2. fma-def 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1_i + -1 \cdot n0_i, n0_i\right)} \]

   3. mul-1-neg 98.7%

      \[\leadsto \mathsf{fma}\left(u, n1_i + \color{blue}{\left(-n0_i\right)}, n0_i\right) \]

   4. unsub-neg 98.7%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1_i - n0_i}, n0_i\right) \]

7. Simplified 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right)} \]

8. Final simplification 98.7%

   \[\leadsto \mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \]

Alternative 6: 70.6% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -2.499999990010493 \cdot 10^{-13} \lor \neg \left(n1_i \leq 5.000000018137469 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -2.499999990010493e-13)
         (not (<= n1_i 5.000000018137469e-16)))
   (* u n1_i)
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -2.499999990010493e-13f) || !(n1_i <= 5.000000018137469e-16f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-2.499999990010493e-13)) .or. (.not. (n1_i <= 5.000000018137469e-16))) then
        tmp = u * n1_i
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-2.499999990010493e-13)) || !(n1_i <= Float32(5.000000018137469e-16)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-2.499999990010493e-13)) || ~((n1_i <= single(5.000000018137469e-16))))
		tmp = u * n1_i;
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -2.49999999e-13 or 5.00000002e-16 < n1_i

   1. Initial program 94.9%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. *-commutative 94.9%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      2. associate-*l* 91.9%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      3. *-commutative 91.9%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

      4. associate-*l* 81.1%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

      5. distribute-lft-out 81.1%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   3. Simplified 81.1%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 98.3%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

   5. Taylor expanded in n0_i around 0 65.4%

      \[\leadsto \color{blue}{n1_i \cdot u} \]

   if -2.49999999e-13 < n1_i < 5.00000002e-16

   1. Initial program 98.1%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. *-commutative 98.1%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      2. associate-*l* 70.7%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      3. *-commutative 70.7%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

      4. associate-*l* 67.5%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

      5. distribute-lft-out 67.4%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 98.4%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

   5. Taylor expanded in n0_i around inf 77.6%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -2.499999990010493 \cdot 10^{-13} \lor \neg \left(n1_i \leq 5.000000018137469 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 7: 86.5% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999943633011 \cdot 10^{-27} \lor \neg \left(n1_i \leq 4.0000000126843074 \cdot 10^{-28}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -4.999999943633011e-27)
         (not (<= n1_i 4.0000000126843074e-28)))
   (+ n0_i (* u n1_i))
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -4.999999943633011e-27f) || !(n1_i <= 4.0000000126843074e-28f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-4.999999943633011e-27)) .or. (.not. (n1_i <= 4.0000000126843074e-28))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-4.999999943633011e-27)) || !(n1_i <= Float32(4.0000000126843074e-28)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-4.999999943633011e-27)) || ~((n1_i <= single(4.0000000126843074e-28))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.99999994e-27 or 4.00000001e-28 < n1_i

   1. Initial program 96.3%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. *-commutative 96.3%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      2. associate-*l* 85.6%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      3. *-commutative 85.6%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

      4. associate-*l* 77.1%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

      5. distribute-lft-out 77.1%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   3. Simplified 77.1%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 98.6%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

   5. Taylor expanded in u around 0 84.7%

      \[\leadsto \color{blue}{n0_i} + n1_i \cdot u \]

   if -4.99999994e-27 < n1_i < 4.00000001e-28

   1. Initial program 98.4%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      2. associate-*l* 61.9%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      3. *-commutative 61.9%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

      4. associate-*l* 61.6%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

      5. distribute-lft-out 61.6%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   3. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 97.8%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

   5. Taylor expanded in n0_i around inf 90.5%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999943633011 \cdot 10^{-27} \lor \neg \left(n1_i \leq 4.0000000126843074 \cdot 10^{-28}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 8: 86.6% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999943633011 \cdot 10^{-27} \lor \neg \left(n1_i \leq 4.0000000126843074 \cdot 10^{-28}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -4.999999943633011e-27)
         (not (<= n1_i 4.0000000126843074e-28)))
   (+ n0_i (* u n1_i))
   (- n0_i (* n0_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -4.999999943633011e-27f) || !(n1_i <= 4.0000000126843074e-28f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i - (n0_i * u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-4.999999943633011e-27)) .or. (.not. (n1_i <= 4.0000000126843074e-28))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i - (n0_i * u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-4.999999943633011e-27)) || !(n1_i <= Float32(4.0000000126843074e-28)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i - Float32(n0_i * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-4.999999943633011e-27)) || ~((n1_i <= single(4.0000000126843074e-28))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i - (n0_i * u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.99999994e-27 or 4.00000001e-28 < n1_i

   1. Initial program 96.3%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. *-commutative 96.3%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      2. associate-*l* 85.6%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      3. *-commutative 85.6%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

      4. associate-*l* 77.1%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

      5. distribute-lft-out 77.1%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   3. Simplified 77.1%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 98.6%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

   5. Taylor expanded in u around 0 84.7%

      \[\leadsto \color{blue}{n0_i} + n1_i \cdot u \]

   if -4.99999994e-27 < n1_i < 4.00000001e-28

   1. Initial program 98.4%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      2. associate-*l* 61.9%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      3. *-commutative 61.9%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

      4. associate-*l* 61.6%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

      5. distribute-lft-out 61.6%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   3. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 97.8%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

   5. Taylor expanded in u around 0 98.2%

      \[\leadsto \color{blue}{\left(n0_i + -1 \cdot \left(n0_i \cdot u\right)\right)} + n1_i \cdot u \]
   6. Step-by-step derivation

      1. mul-1-neg 98.9%

         \[\leadsto \left(n0_i + \color{blue}{\left(-n0_i \cdot u\right)}\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

      2. unsub-neg 98.9%

         \[\leadsto \color{blue}{\left(n0_i - n0_i \cdot u\right)} + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

      3. *-commutative 98.9%

         \[\leadsto \left(n0_i - \color{blue}{u \cdot n0_i}\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

   7. Simplified 98.2%

      \[\leadsto \color{blue}{\left(n0_i - u \cdot n0_i\right)} + n1_i \cdot u \]

   8. Taylor expanded in n1_i around 0 90.8%

      \[\leadsto \color{blue}{n0_i - n0_i \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999943633011 \cdot 10^{-27} \lor \neg \left(n1_i \leq 4.0000000126843074 \cdot 10^{-28}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \]

Alternative 9: 60.1% accurate, 57.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -2.499999990010493 \cdot 10^{-13} \lor \neg \left(n1_i \leq 5.000000018137469 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -2.499999990010493e-13)
         (not (<= n1_i 5.000000018137469e-16)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -2.499999990010493e-13f) || !(n1_i <= 5.000000018137469e-16f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-2.499999990010493e-13)) .or. (.not. (n1_i <= 5.000000018137469e-16))) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-2.499999990010493e-13)) || !(n1_i <= Float32(5.000000018137469e-16)))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-2.499999990010493e-13)) || ~((n1_i <= single(5.000000018137469e-16))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -2.49999999e-13 or 5.00000002e-16 < n1_i

   1. Initial program 94.9%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. *-commutative 94.9%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      2. associate-*l* 91.9%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      3. *-commutative 91.9%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

      4. associate-*l* 81.1%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

      5. distribute-lft-out 81.1%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   3. Simplified 81.1%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 98.3%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

   5. Taylor expanded in n0_i around 0 65.4%

      \[\leadsto \color{blue}{n1_i \cdot u} \]

   if -2.49999999e-13 < n1_i < 5.00000002e-16

   1. Initial program 98.1%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. *-commutative 98.1%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      2. associate-*l* 70.7%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

      3. *-commutative 70.7%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

      4. associate-*l* 67.5%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

      5. distribute-lft-out 67.4%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   4. Taylor expanded in normAngle around 0 98.4%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

   5. Taylor expanded in u around 0 98.7%

      \[\leadsto \color{blue}{\left(n0_i + -1 \cdot \left(n0_i \cdot u\right)\right)} + n1_i \cdot u \]
   6. Step-by-step derivation

      1. mul-1-neg 99.3%

         \[\leadsto \left(n0_i + \color{blue}{\left(-n0_i \cdot u\right)}\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

      2. unsub-neg 99.3%

         \[\leadsto \color{blue}{\left(n0_i - n0_i \cdot u\right)} + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

      3. *-commutative 99.3%

         \[\leadsto \left(n0_i - \color{blue}{u \cdot n0_i}\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

   7. Simplified 98.7%

      \[\leadsto \color{blue}{\left(n0_i - u \cdot n0_i\right)} + n1_i \cdot u \]

   8. Taylor expanded in u around 0 57.6%

      \[\leadsto \color{blue}{n0_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -2.499999990010493 \cdot 10^{-13} \lor \neg \left(n1_i \leq 5.000000018137469 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 10: 98.2% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ n0_i - u \cdot \left(n0_i - n1_i\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. *-commutative 96.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. associate-*l* 78.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. *-commutative 78.3%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

   4. associate-*l* 72.4%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   5. distribute-lft-out 72.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Simplified 72.3%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

4. Taylor expanded in normAngle around 0 98.3%

   \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

5. Taylor expanded in u around -inf 98.6%

   \[\leadsto \color{blue}{n0_i + -1 \cdot \left(u \cdot \left(n0_i + -1 \cdot n1_i\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 98.6%

      \[\leadsto n0_i + \color{blue}{\left(-u \cdot \left(n0_i + -1 \cdot n1_i\right)\right)} \]

   2. unsub-neg 98.6%

      \[\leadsto \color{blue}{n0_i - u \cdot \left(n0_i + -1 \cdot n1_i\right)} \]

   3. mul-1-neg 98.6%

      \[\leadsto n0_i - u \cdot \left(n0_i + \color{blue}{\left(-n1_i\right)}\right) \]

   4. unsub-neg 98.6%

      \[\leadsto n0_i - u \cdot \color{blue}{\left(n0_i - n1_i\right)} \]

7. Simplified 98.6%

   \[\leadsto \color{blue}{n0_i - u \cdot \left(n0_i - n1_i\right)} \]

8. Final simplification 98.6%

   \[\leadsto n0_i - u \cdot \left(n0_i - n1_i\right) \]

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 96.9%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. *-commutative 96.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. associate-*l* 78.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. *-commutative 78.3%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

   4. associate-*l* 72.4%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   5. distribute-lft-out 72.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Simplified 72.3%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

4. Taylor expanded in normAngle around 0 98.3%

   \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

5. Taylor expanded in u around 0 98.5%

   \[\leadsto \color{blue}{\left(n0_i + -1 \cdot \left(n0_i \cdot u\right)\right)} + n1_i \cdot u \]
6. Step-by-step derivation

   1. mul-1-neg 99.3%

      \[\leadsto \left(n0_i + \color{blue}{\left(-n0_i \cdot u\right)}\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

   2. unsub-neg 99.3%

      \[\leadsto \color{blue}{\left(n0_i - n0_i \cdot u\right)} + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

   3. *-commutative 99.3%

      \[\leadsto \left(n0_i - \color{blue}{u \cdot n0_i}\right) + \left(n1_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0_i \cdot \left(1 - u\right) + n1_i \cdot u\right)\right)\right) \]

7. Simplified 98.5%

   \[\leadsto \color{blue}{\left(n0_i - u \cdot n0_i\right)} + n1_i \cdot u \]

8. Taylor expanded in u around 0 46.4%

   \[\leadsto \color{blue}{n0_i} \]

9. Final simplification 46.4%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023336 
(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)))