Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 98.8%

Time: 38.4s

Alternatives: 11

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 3.0× speedup

Math

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

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)
       (*
        u
        (+
         (* n0_i -0.5)
         (*
          u
          (+ (* n1_i -0.16666666666666666) (* n0_i 0.16666666666666666))))))
      (* -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) + (u * ((n0_i * -0.5f) + (u * ((n1_i * -0.16666666666666666f) + (n0_i * 0.16666666666666666f)))))) + (-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) + (u * ((n0_i * (-0.5e0)) + (u * ((n1_i * (-0.16666666666666666e0)) + (n0_i * 0.16666666666666666e0)))))) + ((-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(Float32(n0_i * Float32(0.5)) + Float32(u * Float32(Float32(n0_i * Float32(-0.5)) + Float32(u * Float32(Float32(n1_i * Float32(-0.16666666666666666)) + Float32(n0_i * Float32(0.16666666666666666))))))) + 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)) + (u * ((n0_i * single(-0.5)) + (u * ((n1_i * single(-0.16666666666666666)) + (n0_i * single(0.16666666666666666))))))) + (single(-0.16666666666666666) * (n0_i - n1_i))))));
end

Derivation

1. Initial program 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-*l* 79.5%

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

   3. *-commutative 79.5%

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

   4. associate-*l* 72.6%

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

   5. distribute-lft-out 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in normAngle around 0 99.0%

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

6. Taylor expanded in u around 0 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 98.6% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (-
  (* n0_i (- 1.0 u))
  (-
   (*
    (pow normAngle 2.0)
    (*
     u
     (-
      (* n1_i -0.16666666666666666)
      (+
       (* n0_i 0.5)
       (* u (+ (* n0_i -0.5) (* u (* n1_i -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) * (u * ((n1_i * -0.16666666666666666f) - ((n0_i * 0.5f) + (u * ((n0_i * -0.5f) + (u * (n1_i * -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) * (u * ((n1_i * (-0.16666666666666666e0)) - ((n0_i * 0.5e0) + (u * ((n0_i * (-0.5e0)) + (u * (n1_i * (-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(u * Float32(Float32(n1_i * Float32(-0.16666666666666666)) - Float32(Float32(n0_i * Float32(0.5)) + Float32(u * Float32(Float32(n0_i * Float32(-0.5)) + Float32(u * Float32(n1_i * 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)) * (u * ((n1_i * single(-0.16666666666666666)) - ((n0_i * single(0.5)) + (u * ((n0_i * single(-0.5)) + (u * (n1_i * single(-0.16666666666666666))))))))) - (u * n1_i));
end

Derivation

1. Initial program 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-*l* 79.5%

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

   3. *-commutative 79.5%

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

   4. associate-*l* 72.6%

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

   5. distribute-lft-out 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in normAngle around 0 99.0%

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

6. Taylor expanded in u around 0 99.0%

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

7. Taylor expanded in n1_i around inf 98.9%

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

8. Taylor expanded in n1_i around inf 98.9%

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

   1. associate-*r* 98.9%

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

   2. *-commutative 98.9%

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

10. Simplified 98.9%

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

11. Final simplification 98.9%

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

Alternative 3: 98.7% accurate, 3.2× speedup

Math

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

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) (* -0.5 (* n0_i u)))
      (* -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) + (-0.5f * (n0_i * u))) + (-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) + ((-0.5e0) * (n0_i * u))) + ((-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(Float32(n0_i * Float32(0.5)) + Float32(Float32(-0.5) * Float32(n0_i * u))) + 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(-0.5) * (n0_i * u))) + (single(-0.16666666666666666) * (n0_i - n1_i))))));
end

Derivation

1. Initial program 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-*l* 79.5%

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

   3. *-commutative 79.5%

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

   4. associate-*l* 72.6%

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

   5. distribute-lft-out 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in normAngle around 0 99.0%

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

6. Taylor expanded in u around 0 98.8%

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

7. Final simplification 98.8%

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

Alternative 4: 98.6% 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 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-*l* 79.5%

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

   3. *-commutative 79.5%

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

   4. associate-*l* 72.6%

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

   5. distribute-lft-out 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in normAngle around 0 99.0%

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

6. Taylor expanded in u around 0 98.8%

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

   1. *-commutative 98.8%

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

   2. mul-1-neg 98.8%

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 5: 98.5% accurate, 3.5× speedup

Math

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

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 0.5)))))))

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 * 0.5f)))));
}

Fortran

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

Derivation

1. Initial program 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-*l* 79.5%

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

   3. *-commutative 79.5%

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

   4. associate-*l* 72.6%

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

   5. distribute-lft-out 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in normAngle around 0 99.0%

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

6. Taylor expanded in u around 0 99.0%

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

7. Taylor expanded in n1_i around inf 98.9%

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

8. Taylor expanded in u around 0 98.7%

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

9. Final simplification 98.7%

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

Alternative 6: 97.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.2%

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

   1. fma-define 97.2%

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

   2. associate-*r/ 97.3%

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

   3. *-rgt-identity 97.3%

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

   4. associate-*r/ 97.9%

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

   5. *-rgt-identity 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in normAngle around 0 97.8%

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

   1. fma-define 97.9%

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

   2. *-commutative 97.9%

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

7. Simplified 97.9%

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

8. Final simplification 97.9%

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right) \]
9. Add Preprocessing

Alternative 7: 70.1% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.5000000170217692 \cdot 10^{-18} \lor \neg \left(n1\_i \leq 9.99999983775159 \cdot 10^{-18}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.5000000170217692e-18)
         (not (<= n1_i 9.99999983775159e-18)))
   (* 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 <= -1.5000000170217692e-18f) || !(n1_i <= 9.99999983775159e-18f)) {
		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 <= (-1.5000000170217692e-18)) .or. (.not. (n1_i <= 9.99999983775159e-18))) 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(-1.5000000170217692e-18)) || !(n1_i <= Float32(9.99999983775159e-18)))
		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(-1.5000000170217692e-18)) || ~((n1_i <= single(9.99999983775159e-18))))
		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 < -1.50000002e-18 or 9.99999984e-18 < n1_i

   1. Initial program 97.3%

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

      1. fma-define 97.3%

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

      2. associate-*r/ 97.3%

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

      3. *-rgt-identity 97.3%

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

      4. associate-*r/ 97.4%

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

      5. *-rgt-identity 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in n0_i around 0 58.8%

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

      1. *-commutative 58.8%

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

      2. *-commutative 58.8%

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

      3. associate-*r/ 66.1%

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

      4. *-commutative 66.1%

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

      5. *-commutative 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in normAngle around 0 66.6%

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

   if -1.50000002e-18 < n1_i < 9.99999984e-18

   1. Initial program 97.0%

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

      1. fma-define 97.1%

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

      2. associate-*r/ 97.2%

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

      3. *-rgt-identity 97.2%

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

      4. associate-*r/ 98.5%

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

      5. *-rgt-identity 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in normAngle around 0 98.8%

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

      1. fma-define 99.0%

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

      2. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in n0_i around inf 80.9%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.5000000170217692 \cdot 10^{-18} \lor \neg \left(n1\_i \leq 9.99999983775159 \cdot 10^{-18}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.3% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.1999999840067324 \cdot 10^{-23} \lor \neg \left(n1\_i \leq 9.99999983775159 \cdot 10^{-18}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.1999999840067324e-23)
         (not (<= n1_i 9.99999983775159e-18)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.1999999840067324e-23f) || !(n1_i <= 9.99999983775159e-18f)) {
		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 <= (-1.1999999840067324e-23)) .or. (.not. (n1_i <= 9.99999983775159e-18))) 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(-1.1999999840067324e-23)) || !(n1_i <= Float32(9.99999983775159e-18)))
		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(-1.1999999840067324e-23)) || ~((n1_i <= single(9.99999983775159e-18))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.19999998e-23 or 9.99999984e-18 < n1_i

   1. Initial program 97.3%

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

      1. fma-define 97.4%

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

      2. associate-*r/ 97.3%

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

      3. *-rgt-identity 97.3%

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

      4. associate-*r/ 97.4%

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

      5. *-rgt-identity 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in n0_i around 0 55.4%

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

      1. *-commutative 55.4%

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

      2. *-commutative 55.4%

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

      3. associate-*r/ 63.8%

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

      4. *-commutative 63.8%

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

      5. *-commutative 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in normAngle around 0 64.2%

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

   if -1.19999998e-23 < n1_i < 9.99999984e-18

   1. Initial program 97.0%

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

      1. fma-define 97.0%

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

      2. associate-*r/ 97.2%

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

      3. *-rgt-identity 97.2%

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

      4. associate-*r/ 98.6%

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

      5. *-rgt-identity 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in u around 0 63.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.1999999840067324 \cdot 10^{-23} \lor \neg \left(n1\_i \leq 9.99999983775159 \cdot 10^{-18}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.0% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

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

   1. associate-*l* 97.0%

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

   2. cancel-sign-sub 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 79.4%

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

   5. associate-*r/ 79.5%

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

   6. *-rgt-identity 79.5%

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

   7. sin-neg 79.5%

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

   8. distribute-lft-neg-out 79.5%

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

   9. associate-*l* 79.7%

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

   10. *-commutative 79.7%

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

   11. distribute-lft-neg-out 79.7%

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

   12. distribute-rgt-neg-out 79.7%

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

   13. associate-*r/ 80.2%

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

3. Simplified 72.8%

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

5. Taylor expanded in normAngle around 0 71.0%

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

6. Taylor expanded in normAngle around 0 97.8%

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

   1. +-commutative 97.8%

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

   2. sub-neg 97.8%

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

   3. distribute-lft-in 97.8%

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

   4. *-rgt-identity 97.8%

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

   5. distribute-rgt-neg-in 97.8%

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

   6. mul-1-neg 97.8%

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

   7. associate-*r* 97.8%

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

   8. +-commutative 97.8%

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

   9. associate-+l+ 97.9%

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

   10. distribute-rgt-in 97.9%

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

   11. +-commutative 97.9%

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

   12. neg-mul-1 97.9%

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

   13. unsub-neg 97.9%

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

8. Simplified 97.9%

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

9. Final simplification 97.9%

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

Alternative 10: 81.5% accurate, 84.2× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot n1\_i \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

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

   1. fma-define 97.2%

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

   2. associate-*l* 97.1%

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

3. Simplified 97.1%

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

5. Taylor expanded in normAngle around 0 97.0%

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

   1. *-commutative 97.0%

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

7. Simplified 97.0%

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

8. Taylor expanded in u around inf 78.8%

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

   1. *-commutative 78.8%

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

   2. sub-neg 78.8%

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

   3. neg-mul-1 78.8%

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

   4. *-commutative 78.8%

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

   5. associate-/l* 94.5%

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

   6. *-commutative 94.5%

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

   7. neg-mul-1 94.5%

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

   8. sub-neg 94.5%

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

10. Simplified 94.5%

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

11. Taylor expanded in u around 0 82.3%

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

12. Taylor expanded in u around 0 82.4%

    \[\leadsto \color{blue}{n0\_i + n1\_i \cdot u} \]
13. Step-by-step derivation

    1. *-commutative 82.4%

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

14. Simplified 82.4%

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

15. Final simplification 82.4%

    \[\leadsto n0\_i + u \cdot n1\_i \]
16. Add Preprocessing

Alternative 11: 47.3% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

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

   1. fma-define 97.2%

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

   2. associate-*r/ 97.3%

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

   3. *-rgt-identity 97.3%

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

   4. associate-*r/ 97.9%

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

   5. *-rgt-identity 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in u around 0 41.7%

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

6. Final simplification 41.7%

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

Reproduce

herbie shell --seed 2024076 
(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)))