Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 91.6% → 93.4%

Time: 14.9s

Alternatives: 9

Speedup: 1.3×

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: 91.6% 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: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot \left(n1\_i - n0\_i\right)\\ \mathbf{if}\;normAngle \leq 0.0020000000949949026:\\ \;\;\;\;n0\_i + \left(t\_0 + {normAngle}^{2} \cdot \left(\left(n0\_i \cdot -0.16666666666666666 + 0.5 \cdot \left(u \cdot n0\_i\right)\right) - -0.16666666666666666 \cdot \left(n0\_i + t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right), n0\_i, \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)}{\sin normAngle}\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* u (- n1_i n0_i))))
   (if (<= normAngle 0.0020000000949949026)
     (+
      n0_i
      (+
       t_0
       (*
        (pow normAngle 2.0)
        (-
         (+ (* n0_i -0.16666666666666666) (* 0.5 (* u n0_i)))
         (* -0.16666666666666666 (+ n0_i t_0))))))
     (/
      (fma (sin (* (- 1.0 u) normAngle)) n0_i (* (sin (* u normAngle)) n1_i))
      (sin normAngle)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = u * (n1_i - n0_i);
	float tmp;
	if (normAngle <= 0.0020000000949949026f) {
		tmp = n0_i + (t_0 + (powf(normAngle, 2.0f) * (((n0_i * -0.16666666666666666f) + (0.5f * (u * n0_i))) - (-0.16666666666666666f * (n0_i + t_0)))));
	} else {
		tmp = fmaf(sinf(((1.0f - u) * normAngle)), n0_i, (sinf((u * normAngle)) * n1_i)) / sinf(normAngle);
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(u * Float32(n1_i - n0_i))
	tmp = Float32(0.0)
	if (normAngle <= Float32(0.0020000000949949026))
		tmp = Float32(n0_i + Float32(t_0 + Float32((normAngle ^ Float32(2.0)) * Float32(Float32(Float32(n0_i * Float32(-0.16666666666666666)) + Float32(Float32(0.5) * Float32(u * n0_i))) - Float32(Float32(-0.16666666666666666) * Float32(n0_i + t_0))))));
	else
		tmp = Float32(fma(sin(Float32(Float32(Float32(1.0) - u) * normAngle)), n0_i, Float32(sin(Float32(u * normAngle)) * n1_i)) / sin(normAngle));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if normAngle < 0.00200000009

   1. Initial program 97.4%

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

      1. associate-*l* 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* 77.0%

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

      5. associate-*r/ 77.2%

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

      6. *-rgt-identity 77.2%

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

      7. sin-neg 77.2%

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

      8. distribute-lft-neg-out 77.2%

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

      9. associate-*l* 77.5%

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

      10. *-commutative 77.5%

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

      11. distribute-lft-neg-out 77.5%

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

      12. distribute-rgt-neg-out 77.5%

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

      13. associate-*r/ 77.6%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in u around 0 70.0%

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

      1. fma-define 70.1%

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

      2. +-commutative 70.1%

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

      3. mul-1-neg 70.1%

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

      4. unsub-neg 70.1%

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

      5. *-commutative 70.1%

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

      6. associate-*r* 70.1%

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

   7. Simplified 70.1%

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

   8. Taylor expanded in normAngle around 0 99.9%

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

   if 0.00200000009 < normAngle

   1. Initial program 74.7%

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

      1. associate-*l* 74.7%

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

      2. cancel-sign-sub 74.7%

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

      3. *-commutative 74.7%

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

      4. associate-*r* 74.7%

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

      5. associate-*r/ 74.8%

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

      6. *-rgt-identity 74.8%

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

      7. sin-neg 74.8%

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

      8. distribute-lft-neg-out 74.8%

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

      9. associate-*l* 74.8%

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

      10. *-commutative 74.8%

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

      11. distribute-lft-neg-out 74.8%

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

      12. distribute-rgt-neg-out 74.8%

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

      13. associate-*r/ 74.9%

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

   3. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right), n0\_i, \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)}{\sin normAngle}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;normAngle \leq 0.0020000000949949026:\\ \;\;\;\;n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(\left(n0\_i \cdot -0.16666666666666666 + 0.5 \cdot \left(u \cdot n0\_i\right)\right) - -0.16666666666666666 \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right), n0\_i, \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)}{\sin normAngle}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 92.2%

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

   1. +-commutative 92.2%

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

   2. fma-define 92.2%

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

   3. associate-*r/ 92.3%

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

   4. *-rgt-identity 92.3%

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

   5. *-commutative 92.3%

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

   6. associate-*r* 76.8%

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

   7. associate-*r/ 77.0%

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

   8. *-rgt-identity 77.0%

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

3. Simplified 77.0%

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

   1. associate-/l* 92.8%

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

   2. un-div-inv 92.3%

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

   3. *-commutative 92.3%

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

   4. fma-define 92.3%

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

   5. div-inv 92.2%

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

   6. +-commutative 92.2%

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

   7. associate-*l* 92.0%

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

   8. associate-*l/ 92.2%

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

   9. *-un-lft-identity 92.2%

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

   10. associate-*r* 91.9%

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

   11. associate-*l/ 92.4%

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

6. Applied egg-rr 92.4%

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

Alternative 3: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot \left(n1\_i - n0\_i\right)\\ \mathbf{if}\;normAngle \leq 0.0020000000949949026:\\ \;\;\;\;n0\_i + \left(t\_0 + {normAngle}^{2} \cdot \left(\left(n0\_i \cdot -0.16666666666666666 + 0.5 \cdot \left(u \cdot n0\_i\right)\right) - -0.16666666666666666 \cdot \left(n0\_i + t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u \cdot normAngle\right) \cdot n1\_i + \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i}{\sin normAngle}\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* u (- n1_i n0_i))))
   (if (<= normAngle 0.0020000000949949026)
     (+
      n0_i
      (+
       t_0
       (*
        (pow normAngle 2.0)
        (-
         (+ (* n0_i -0.16666666666666666) (* 0.5 (* u n0_i)))
         (* -0.16666666666666666 (+ n0_i t_0))))))
     (/
      (+ (* (sin (* u normAngle)) n1_i) (* (sin (* (- 1.0 u) normAngle)) n0_i))
      (sin normAngle)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = u * (n1_i - n0_i);
	float tmp;
	if (normAngle <= 0.0020000000949949026f) {
		tmp = n0_i + (t_0 + (powf(normAngle, 2.0f) * (((n0_i * -0.16666666666666666f) + (0.5f * (u * n0_i))) - (-0.16666666666666666f * (n0_i + t_0)))));
	} else {
		tmp = ((sinf((u * normAngle)) * n1_i) + (sinf(((1.0f - u) * normAngle)) * n0_i)) / sinf(normAngle);
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = u * (n1_i - n0_i)
    if (normangle <= 0.0020000000949949026e0) then
        tmp = n0_i + (t_0 + ((normangle ** 2.0e0) * (((n0_i * (-0.16666666666666666e0)) + (0.5e0 * (u * n0_i))) - ((-0.16666666666666666e0) * (n0_i + t_0)))))
    else
        tmp = ((sin((u * normangle)) * n1_i) + (sin(((1.0e0 - u) * normangle)) * n0_i)) / sin(normangle)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(u * Float32(n1_i - n0_i))
	tmp = Float32(0.0)
	if (normAngle <= Float32(0.0020000000949949026))
		tmp = Float32(n0_i + Float32(t_0 + Float32((normAngle ^ Float32(2.0)) * Float32(Float32(Float32(n0_i * Float32(-0.16666666666666666)) + Float32(Float32(0.5) * Float32(u * n0_i))) - Float32(Float32(-0.16666666666666666) * Float32(n0_i + t_0))))));
	else
		tmp = Float32(Float32(Float32(sin(Float32(u * normAngle)) * n1_i) + Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * n0_i)) / sin(normAngle));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	t_0 = u * (n1_i - n0_i);
	tmp = single(0.0);
	if (normAngle <= single(0.0020000000949949026))
		tmp = n0_i + (t_0 + ((normAngle ^ single(2.0)) * (((n0_i * single(-0.16666666666666666)) + (single(0.5) * (u * n0_i))) - (single(-0.16666666666666666) * (n0_i + t_0)))));
	else
		tmp = ((sin((u * normAngle)) * n1_i) + (sin(((single(1.0) - u) * normAngle)) * n0_i)) / sin(normAngle);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if normAngle < 0.00200000009

   1. Initial program 97.4%

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

      1. associate-*l* 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* 77.0%

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

      5. associate-*r/ 77.2%

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

      6. *-rgt-identity 77.2%

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

      7. sin-neg 77.2%

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

      8. distribute-lft-neg-out 77.2%

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

      9. associate-*l* 77.5%

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

      10. *-commutative 77.5%

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

      11. distribute-lft-neg-out 77.5%

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

      12. distribute-rgt-neg-out 77.5%

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

      13. associate-*r/ 77.6%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in u around 0 70.0%

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

      1. fma-define 70.1%

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

      2. +-commutative 70.1%

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

      3. mul-1-neg 70.1%

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

      4. unsub-neg 70.1%

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

      5. *-commutative 70.1%

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

      6. associate-*r* 70.1%

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

   7. Simplified 70.1%

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

   8. Taylor expanded in normAngle around 0 99.9%

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

   if 0.00200000009 < normAngle

   1. Initial program 74.7%

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

      1. associate-*l* 74.7%

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

      2. cancel-sign-sub 74.7%

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

      3. *-commutative 74.7%

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

      4. associate-*r* 74.7%

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

      5. associate-*r/ 74.8%

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

      6. *-rgt-identity 74.8%

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

      7. sin-neg 74.8%

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

      8. distribute-lft-neg-out 74.8%

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

      9. associate-*l* 74.8%

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

      10. *-commutative 74.8%

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

      11. distribute-lft-neg-out 74.8%

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

      12. distribute-rgt-neg-out 74.8%

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

      13. associate-*r/ 74.9%

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

   3. Simplified 75.0%

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

      1. fma-define 75.0%

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

   6. Applied egg-rr 75.0%

      \[\leadsto \frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i}}{\sin normAngle} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;normAngle \leq 0.0020000000949949026:\\ \;\;\;\;n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(\left(n0\_i \cdot -0.16666666666666666 + 0.5 \cdot \left(u \cdot n0\_i\right)\right) - -0.16666666666666666 \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u \cdot normAngle\right) \cdot n1\_i + \sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i}{\sin normAngle}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot \left(n1\_i - n0\_i\right)\\ \mathbf{if}\;normAngle \leq 0.4000000059604645:\\ \;\;\;\;n0\_i + \left(t\_0 + {normAngle}^{2} \cdot \left(\left(n0\_i \cdot -0.16666666666666666 + 0.5 \cdot \left(u \cdot n0\_i\right)\right) - -0.16666666666666666 \cdot \left(n0\_i + t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n0\_i + n1\_i \cdot \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* u (- n1_i n0_i))))
   (if (<= normAngle 0.4000000059604645)
     (+
      n0_i
      (+
       t_0
       (*
        (pow normAngle 2.0)
        (-
         (+ (* n0_i -0.16666666666666666) (* 0.5 (* u n0_i)))
         (* -0.16666666666666666 (+ n0_i t_0))))))
     (+ n0_i (* n1_i (* (sin (* u normAngle)) (/ 1.0 (sin normAngle))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = u * (n1_i - n0_i);
	float tmp;
	if (normAngle <= 0.4000000059604645f) {
		tmp = n0_i + (t_0 + (powf(normAngle, 2.0f) * (((n0_i * -0.16666666666666666f) + (0.5f * (u * n0_i))) - (-0.16666666666666666f * (n0_i + t_0)))));
	} else {
		tmp = n0_i + (n1_i * (sinf((u * normAngle)) * (1.0f / sinf(normAngle))));
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = u * (n1_i - n0_i)
    if (normangle <= 0.4000000059604645e0) then
        tmp = n0_i + (t_0 + ((normangle ** 2.0e0) * (((n0_i * (-0.16666666666666666e0)) + (0.5e0 * (u * n0_i))) - ((-0.16666666666666666e0) * (n0_i + t_0)))))
    else
        tmp = n0_i + (n1_i * (sin((u * normangle)) * (1.0e0 / sin(normangle))))
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(u * Float32(n1_i - n0_i))
	tmp = Float32(0.0)
	if (normAngle <= Float32(0.4000000059604645))
		tmp = Float32(n0_i + Float32(t_0 + Float32((normAngle ^ Float32(2.0)) * Float32(Float32(Float32(n0_i * Float32(-0.16666666666666666)) + Float32(Float32(0.5) * Float32(u * n0_i))) - Float32(Float32(-0.16666666666666666) * Float32(n0_i + t_0))))));
	else
		tmp = Float32(n0_i + Float32(n1_i * Float32(sin(Float32(u * normAngle)) * Float32(Float32(1.0) / sin(normAngle)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	t_0 = u * (n1_i - n0_i);
	tmp = single(0.0);
	if (normAngle <= single(0.4000000059604645))
		tmp = n0_i + (t_0 + ((normAngle ^ single(2.0)) * (((n0_i * single(-0.16666666666666666)) + (single(0.5) * (u * n0_i))) - (single(-0.16666666666666666) * (n0_i + t_0)))));
	else
		tmp = n0_i + (n1_i * (sin((u * normAngle)) * (single(1.0) / sin(normAngle))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if normAngle < 0.400000006

   1. Initial program 97.5%

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

      1. associate-*l* 97.1%

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

      2. cancel-sign-sub 97.1%

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

      3. *-commutative 97.1%

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

      4. associate-*r* 78.1%

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

      5. associate-*r/ 78.2%

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

      6. *-rgt-identity 78.2%

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

      7. sin-neg 78.2%

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

      8. distribute-lft-neg-out 78.2%

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

      9. associate-*l* 78.5%

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

      10. *-commutative 78.5%

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

      11. distribute-lft-neg-out 78.5%

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

      12. distribute-rgt-neg-out 78.5%

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

      13. associate-*r/ 78.7%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in u around 0 71.0%

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

      1. fma-define 71.1%

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

      2. +-commutative 71.1%

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

      3. mul-1-neg 71.1%

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

      4. unsub-neg 71.1%

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

      5. *-commutative 71.1%

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

      6. associate-*r* 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in normAngle around 0 99.5%

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

   if 0.400000006 < normAngle

   1. Initial program 69.8%

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

   3. Taylor expanded in u around 0 59.3%

      \[\leadsto \color{blue}{1} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;normAngle \leq 0.4000000059604645:\\ \;\;\;\;n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(\left(n0\_i \cdot -0.16666666666666666 + 0.5 \cdot \left(u \cdot n0\_i\right)\right) - -0.16666666666666666 \cdot \left(n0\_i + u \cdot \left(n1\_i - n0\_i\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;n0\_i + n1\_i \cdot \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 92.2%

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

   1. +-commutative 92.2%

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

   2. fma-define 92.2%

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

   3. associate-*r/ 92.3%

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

   4. *-rgt-identity 92.3%

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

   5. *-commutative 92.3%

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

   6. associate-*r* 76.8%

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

   7. associate-*r/ 77.0%

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

   8. *-rgt-identity 77.0%

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

3. Simplified 77.0%

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

   1. associate-/l* 92.8%

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

   2. un-div-inv 92.3%

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

   3. *-commutative 92.3%

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

   4. fma-define 92.3%

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

   5. div-inv 92.2%

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

   6. +-commutative 92.2%

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

   7. associate-*l* 92.0%

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

   8. associate-*l/ 92.2%

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

   9. *-un-lft-identity 92.2%

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

   10. associate-*r* 91.9%

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

   11. associate-*l/ 92.4%

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

6. Applied egg-rr 92.4%

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

7. Taylor expanded in normAngle around 0 90.4%

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

8. Final simplification 90.4%

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

Alternative 6: 60.7% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -3.0000001167615996 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 2.000000033724767 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -3.0000001167615996e-17)
         (not (<= n1_i 2.000000033724767e-16)))
   (* u n1_i)
   (* (- 1.0 u) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -3.0000001167615996e-17f) || !(n1_i <= 2.000000033724767e-16f)) {
		tmp = u * n1_i;
	} else {
		tmp = (1.0f - u) * 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 <= (-3.0000001167615996e-17)) .or. (.not. (n1_i <= 2.000000033724767e-16))) then
        tmp = u * n1_i
    else
        tmp = (1.0e0 - u) * 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(-3.0000001167615996e-17)) || !(n1_i <= Float32(2.000000033724767e-16)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(Float32(Float32(1.0) - u) * 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(-3.0000001167615996e-17)) || ~((n1_i <= single(2.000000033724767e-16))))
		tmp = u * n1_i;
	else
		tmp = (single(1.0) - u) * n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -3.0000001e-17 or 2.00000003e-16 < n1_i

   1. Initial program 94.5%

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

   3. Taylor expanded in normAngle around 0 85.4%

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

   4. Taylor expanded in n0_i around 0 59.8%

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

      1. *-commutative 59.8%

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

   6. Simplified 59.8%

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

   if -3.0000001e-17 < n1_i < 2.00000003e-16

   1. Initial program 90.6%

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

   3. Taylor expanded in u around inf 90.0%

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

   4. Taylor expanded in n0_i around inf 56.6%

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

      1. associate-/l* 75.2%

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

      2. associate-*r* 73.3%

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

      3. *-commutative 73.3%

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

      4. sub-neg 73.3%

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

      5. metadata-eval 73.3%

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

      6. +-commutative 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in normAngle around 0 73.3%

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

      1. sub-neg 73.3%

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

      2. metadata-eval 73.3%

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

      3. distribute-rgt-in 73.3%

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

      4. lft-mult-inverse 73.5%

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

      5. mul-1-neg 73.5%

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

      6. sub-neg 73.5%

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

   9. Simplified 73.5%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -3.0000001167615996 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 2.000000033724767 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.8% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -3.0000001167615996 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 2.000000033724767 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -3.0000001167615996e-17)
         (not (<= n1_i 2.000000033724767e-16)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -3.0000001167615996e-17f) || !(n1_i <= 2.000000033724767e-16f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-3.0000001167615996e-17)) .or. (.not. (n1_i <= 2.000000033724767e-16))) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-3.0000001167615996e-17)) || !(n1_i <= Float32(2.000000033724767e-16)))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-3.0000001167615996e-17)) || ~((n1_i <= single(2.000000033724767e-16))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -3.0000001e-17 or 2.00000003e-16 < n1_i

   1. Initial program 94.5%

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

   3. Taylor expanded in normAngle around 0 85.4%

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

   4. Taylor expanded in n0_i around 0 59.8%

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

      1. *-commutative 59.8%

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

   6. Simplified 59.8%

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

   if -3.0000001e-17 < n1_i < 2.00000003e-16

   1. Initial program 90.6%

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

   3. Taylor expanded in u around 0 55.4%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -3.0000001167615996 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 2.000000033724767 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.8% 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 92.2%

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

3. Taylor expanded in normAngle around 0 86.1%

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

4. Taylor expanded in u around 0 86.2%

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

   1. mul-1-neg 86.2%

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

   2. sub-neg 86.2%

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

6. Simplified 86.2%

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

Alternative 9: 42.5% 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 92.2%

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

3. Taylor expanded in u around 0 42.3%

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

Reproduce

herbie shell --seed 2024169 
(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)))