Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 98.8%

Time: 16.3s

Alternatives: 10

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.2% 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: 98.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (-
  (* n0_i (- 1.0 u))
  (-
   (*
    (pow normAngle 2.0)
    (-
     (* -0.16666666666666666 (* n1_i (- u (pow u 3.0))))
     (* -0.16666666666666666 (* n0_i (+ (pow (- 1.0 u) 3.0) (+ 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)) - ((powf(normAngle, 2.0f) * ((-0.16666666666666666f * (n1_i * (u - powf(u, 3.0f)))) - (-0.16666666666666666f * (n0_i * (powf((1.0f - u), 3.0f) + (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)) - (((normangle ** 2.0e0) * (((-0.16666666666666666e0) * (n1_i * (u - (u ** 3.0e0)))) - ((-0.16666666666666666e0) * (n0_i * (((1.0e0 - u) ** 3.0e0) + (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((normAngle ^ Float32(2.0)) * Float32(Float32(Float32(-0.16666666666666666) * Float32(n1_i * Float32(u - (u ^ Float32(3.0))))) - Float32(Float32(-0.16666666666666666) * Float32(n0_i * Float32((Float32(Float32(1.0) - u) ^ Float32(3.0)) + 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)) - (((normAngle ^ single(2.0)) * ((single(-0.16666666666666666) * (n1_i * (u - (u ^ single(3.0))))) - (single(-0.16666666666666666) * (n0_i * (((single(1.0) - u) ^ single(3.0)) + (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* 74.4%

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

   3. *-commutative 74.4%

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

   4. associate-*l* 67.8%

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

   5. distribute-lft-out 67.8%

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

3. Simplified 67.8%

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

5. Taylor expanded in normAngle around 0 98.9%

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

6. Taylor expanded in n0_i around 0 98.9%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot \left(n1\_i \cdot {u}^{3}\right) + n0\_i \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right)\right) - -0.16666666666666666 \cdot \left(n1\_i \cdot u\right)\right)}\right) \]
7. Step-by-step derivation

   1. +-commutative 98.9%

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

   2. associate--l+ 98.9%

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

   3. *-commutative 98.9%

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

   4. distribute-lft-out-- 98.9%

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

   5. associate-*l* 98.9%

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

   6. distribute-lft-out-- 98.9%

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

   7. distribute-lft-out-- 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 2: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n0_i * (1.0f - u)) - ((powf(normAngle, 2.0f) * ((-0.16666666666666666f * (n0_i * ((1.0f - u) - powf((1.0f - u), 3.0f)))) - (n1_i * (u * 0.16666666666666666f)))) - (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)) - (((normangle ** 2.0e0) * (((-0.16666666666666666e0) * (n0_i * ((1.0e0 - u) - ((1.0e0 - u) ** 3.0e0)))) - (n1_i * (u * 0.16666666666666666e0)))) - (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((normAngle ^ Float32(2.0)) * Float32(Float32(Float32(-0.16666666666666666) * Float32(n0_i * Float32(Float32(Float32(1.0) - u) - (Float32(Float32(1.0) - u) ^ Float32(3.0))))) - Float32(n1_i * Float32(u * Float32(0.16666666666666666))))) - Float32(u * n1_i)))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n0_i * (single(1.0) - u)) - (((normAngle ^ single(2.0)) * ((single(-0.16666666666666666) * (n0_i * ((single(1.0) - u) - ((single(1.0) - u) ^ single(3.0))))) - (n1_i * (u * single(0.16666666666666666))))) - (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* 74.4%

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

   3. *-commutative 74.4%

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

   4. associate-*l* 67.8%

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

   5. distribute-lft-out 67.8%

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

3. Simplified 67.8%

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

5. Taylor expanded in normAngle around 0 98.9%

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

6. Taylor expanded in n0_i around 0 98.9%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot \left(n1\_i \cdot {u}^{3}\right) + n0\_i \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right)\right) - -0.16666666666666666 \cdot \left(n1\_i \cdot u\right)\right)}\right) \]
7. Step-by-step derivation

   1. +-commutative 98.9%

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

   2. associate--l+ 98.9%

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

   3. *-commutative 98.9%

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

   4. distribute-lft-out-- 98.9%

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

   5. associate-*l* 98.9%

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

   6. distribute-lft-out-- 98.9%

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

   7. distribute-lft-out-- 98.9%

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

8. Simplified 98.9%

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

9. Taylor expanded in u around 0 98.8%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot \left(\left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right) \cdot n0\_i\right) + \color{blue}{0.16666666666666666 \cdot \left(n1\_i \cdot u\right)}\right)\right) \]
10. Step-by-step derivation

    1. associate-*r* 98.8%

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

    2. *-commutative 98.8%

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

    3. associate-*l* 98.8%

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

11. Simplified 98.8%

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

12. Final simplification 98.8%

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

Alternative 3: 98.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ n0\_i \cdot \left(1 - u\right) + \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(u \cdot \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* n0_i (- 1.0 u))
  (+
   (* u n1_i)
   (*
    (pow normAngle 2.0)
    (* u (+ (* n1_i 0.16666666666666666) (* n0_i 0.3333333333333333)))))))

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 * ((n1_i * 0.16666666666666666f) + (n0_i * 0.3333333333333333f)))));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = (n0_i * (1.0e0 - u)) + ((u * n1_i) + ((normangle ** 2.0e0) * (u * ((n1_i * 0.16666666666666666e0) + (n0_i * 0.3333333333333333e0)))))
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(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.3333333333333333)))))))
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 * ((n1_i * single(0.16666666666666666)) + (n0_i * single(0.3333333333333333))))));
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* 74.4%

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

   3. *-commutative 74.4%

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

   4. associate-*l* 67.8%

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

   5. distribute-lft-out 67.8%

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

3. Simplified 67.8%

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

5. Taylor expanded in normAngle around 0 98.9%

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

6. Taylor expanded in u around 0 98.6%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. *-commutative 98.6%

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

   2. mul-1-neg 98.6%

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

8. Simplified 98.6%

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

9. Taylor expanded in n0_i around 0 98.6%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \color{blue}{\left(0.16666666666666666 \cdot n1\_i + 0.3333333333333333 \cdot n0\_i\right)}\right)\right) \]

10. Final simplification 98.6%

    \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(u \cdot \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right)\right)\right) \]
11. Add Preprocessing

Alternative 4: 98.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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* 74.4%

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

   3. *-commutative 74.4%

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

   4. associate-*l* 67.8%

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

   5. distribute-lft-out 67.8%

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

3. Simplified 67.8%

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

5. Taylor expanded in normAngle around 0 98.1%

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

6. Taylor expanded in u around 0 98.4%

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

   1. +-commutative 98.4%

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

   2. fma-def 98.5%

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

   3. mul-1-neg 98.5%

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

   4. unsub-neg 98.5%

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

8. Simplified 98.5%

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

9. Final simplification 98.5%

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

Alternative 5: 71.3% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.999999967550318 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 1.100000014578155 \cdot 10^{-14}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.999999967550318e-17)
         (not (<= n1_i 1.100000014578155e-14)))
   (* u n1_i)
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.999999967550318e-17f) || !(n1_i <= 1.100000014578155e-14f)) {
		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 <= (-1.999999967550318e-17)) .or. (.not. (n1_i <= 1.100000014578155e-14))) 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(-1.999999967550318e-17)) || !(n1_i <= Float32(1.100000014578155e-14)))
		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(-1.999999967550318e-17)) || ~((n1_i <= single(1.100000014578155e-14))))
		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 < -1.99999997e-17 or 1.10000001e-14 < n1_i

   1. Initial program 94.6%

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

      1. *-commutative 94.6%

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

      2. associate-*l* 90.9%

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

      3. *-commutative 90.9%

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

      4. associate-*l* 79.7%

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

      5. distribute-lft-out 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in normAngle around 0 97.7%

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

   6. Taylor expanded in n0_i around 0 66.9%

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

   if -1.99999997e-17 < n1_i < 1.10000001e-14

   1. Initial program 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 64.3%

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

      3. *-commutative 64.3%

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

      4. associate-*l* 60.6%

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

      5. distribute-lft-out 60.6%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in normAngle around 0 98.4%

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

   6. Taylor expanded in u around 0 98.7%

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

      1. +-commutative 98.7%

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

      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%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in n1_i around 0 82.4%

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

       1. *-rgt-identity 82.4%

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

       2. mul-1-neg 82.4%

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

       3. distribute-rgt-neg-in 82.4%

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

       4. distribute-lft-in 82.2%

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

       5. sub-neg 82.2%

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

   11. Simplified 82.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.999999967550318 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 1.100000014578155 \cdot 10^{-14}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.999999682655225 \cdot 10^{-21} \lor \neg \left(n1\_i \leq 5.000000097707407 \cdot 10^{-25}\right):\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -9.999999682655225e-21)
         (not (<= n1_i 5.000000097707407e-25)))
   (+ 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 <= -9.999999682655225e-21f) || !(n1_i <= 5.000000097707407e-25f)) {
		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 <= (-9.999999682655225e-21)) .or. (.not. (n1_i <= 5.000000097707407e-25))) 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(-9.999999682655225e-21)) || !(n1_i <= Float32(5.000000097707407e-25)))
		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(-9.999999682655225e-21)) || ~((n1_i <= single(5.000000097707407e-25))))
		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 < -9.99999968e-21 or 5.0000001e-25 < n1_i

   1. Initial program 95.1%

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

   3. Taylor expanded in normAngle around 0 97.6%

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

   4. Taylor expanded in u around 0 87.3%

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

   if -9.99999968e-21 < n1_i < 5.0000001e-25

   1. Initial program 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l* 62.6%

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

      3. *-commutative 62.6%

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

      4. associate-*l* 61.0%

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

      5. distribute-lft-out 61.0%

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

   3. Simplified 61.0%

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

   5. Taylor expanded in normAngle around 0 98.9%

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

   6. Taylor expanded in u around 0 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-def 99.2%

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

      3. mul-1-neg 99.2%

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

      4. unsub-neg 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in n1_i around 0 89.2%

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

       1. *-rgt-identity 89.2%

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

       2. mul-1-neg 89.2%

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

       3. distribute-rgt-neg-in 89.2%

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

       4. distribute-lft-in 89.0%

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

       5. sub-neg 89.0%

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

   11. Simplified 89.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.999999682655225 \cdot 10^{-21} \lor \neg \left(n1\_i \leq 5.000000097707407 \cdot 10^{-25}\right):\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.999999682655225 \cdot 10^{-21} \lor \neg \left(n1\_i \leq 5.000000097707407 \cdot 10^{-25}\right):\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -9.999999682655225e-21)
         (not (<= n1_i 5.000000097707407e-25)))
   (+ 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 <= -9.999999682655225e-21f) || !(n1_i <= 5.000000097707407e-25f)) {
		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 <= (-9.999999682655225e-21)) .or. (.not. (n1_i <= 5.000000097707407e-25))) 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(-9.999999682655225e-21)) || !(n1_i <= Float32(5.000000097707407e-25)))
		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(-9.999999682655225e-21)) || ~((n1_i <= single(5.000000097707407e-25))))
		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 < -9.99999968e-21 or 5.0000001e-25 < n1_i

   1. Initial program 95.1%

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

   3. Taylor expanded in normAngle around 0 97.6%

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

   4. Taylor expanded in u around 0 87.3%

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

   if -9.99999968e-21 < n1_i < 5.0000001e-25

   1. Initial program 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l* 62.6%

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

      3. *-commutative 62.6%

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

      4. associate-*l* 61.0%

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

      5. distribute-lft-out 61.0%

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

   3. Simplified 61.0%

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

   5. Taylor expanded in normAngle around 0 98.9%

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

   6. Taylor expanded in u around 0 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-def 99.2%

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

      3. mul-1-neg 99.2%

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

      4. unsub-neg 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in n1_i around 0 89.2%

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

       1. mul-1-neg 89.2%

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

       2. *-commutative 89.2%

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

   11. Simplified 89.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.999999682655225 \cdot 10^{-21} \lor \neg \left(n1\_i \leq 5.000000097707407 \cdot 10^{-25}\right):\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.2% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.999999967550318 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 1.100000014578155 \cdot 10^{-14}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.999999967550318e-17)
         (not (<= n1_i 1.100000014578155e-14)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.999999967550318e-17f) || !(n1_i <= 1.100000014578155e-14f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-1.999999967550318e-17)) .or. (.not. (n1_i <= 1.100000014578155e-14))) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-1.999999967550318e-17)) || !(n1_i <= Float32(1.100000014578155e-14)))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-1.999999967550318e-17)) || ~((n1_i <= single(1.100000014578155e-14))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.99999997e-17 or 1.10000001e-14 < n1_i

   1. Initial program 94.6%

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

      1. *-commutative 94.6%

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

      2. associate-*l* 90.9%

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

      3. *-commutative 90.9%

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

      4. associate-*l* 79.7%

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

      5. distribute-lft-out 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in normAngle around 0 97.7%

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

   6. Taylor expanded in n0_i around 0 66.9%

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

   if -1.99999997e-17 < n1_i < 1.10000001e-14

   1. Initial program 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 64.3%

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

      3. *-commutative 64.3%

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

      4. associate-*l* 60.6%

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

      5. distribute-lft-out 60.6%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in normAngle around 0 98.4%

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

   6. Taylor expanded in u around 0 98.7%

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

      1. +-commutative 98.7%

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

      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%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in u around 0 60.8%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.999999967550318 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 1.100000014578155 \cdot 10^{-14}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.0% accurate, 60.1× speedup

Math

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

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* 74.4%

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

   3. *-commutative 74.4%

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

   4. associate-*l* 67.8%

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

   5. distribute-lft-out 67.8%

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

3. Simplified 67.8%

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

5. Taylor expanded in normAngle around 0 98.1%

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

6. Taylor expanded in u around 0 98.4%

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

   1. +-commutative 98.4%

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

   2. fma-def 98.5%

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

   3. mul-1-neg 98.5%

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

   4. unsub-neg 98.5%

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

8. Simplified 98.5%

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

9. Taylor expanded in u around 0 98.4%

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

10. Final simplification 98.4%

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

Alternative 10: 47.8% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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* 74.4%

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

   3. *-commutative 74.4%

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

   4. associate-*l* 67.8%

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

   5. distribute-lft-out 67.8%

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

3. Simplified 67.8%

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

5. Taylor expanded in normAngle around 0 98.1%

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

6. Taylor expanded in u around 0 98.4%

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

   1. +-commutative 98.4%

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

   2. fma-def 98.5%

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

   3. mul-1-neg 98.5%

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

   4. unsub-neg 98.5%

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

8. Simplified 98.5%

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

9. Taylor expanded in u around 0 47.6%

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

10. Final simplification 47.6%

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

Reproduce

herbie shell --seed 2024027 
(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)))