Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 98.9%

Time: 13.6s

Alternatives: 10

Speedup: 60.1×

Specification

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.1% 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: 98.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

   1. +-commutative 96.9%

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

   2. *-commutative 96.9%

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

   3. associate-*r* 89.1%

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

   4. *-commutative 89.1%

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

   5. associate-*r* 77.4%

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

   6. distribute-rgt-out 77.3%

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

   7. *-commutative 77.3%

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

   8. associate-*r/ 77.6%

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

   9. associate-/l* 77.6%

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

   10. *-commutative 77.6%

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

   11. fma-def 77.6%

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

   12. *-commutative 77.6%

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

   13. /-rgt-identity 77.6%

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

3. Simplified 77.6%

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

4. Taylor expanded in u around 0 77.1%

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

5. Taylor expanded in normAngle around 0 99.1%

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

6. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(n1_i - n0_i\right) \cdot u + \left(n0_i + \left(u \cdot \left(normAngle \cdot normAngle\right)\right) \cdot \mathsf{fma}\left(0.16666666666666666, n1_i, n0_i \cdot 0.3333333333333333\right)\right) \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((n1_i - n0_i) * u) + (n0_i + ((u * (normAngle * normAngle)) * fmaf(0.16666666666666666f, n1_i, (n0_i * 0.3333333333333333f))));
}

Julia

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

Derivation

1. Initial program 96.9%

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

   1. +-commutative 96.9%

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

   2. *-commutative 96.9%

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

   3. associate-*r* 89.1%

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

   4. *-commutative 89.1%

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

   5. associate-*r* 77.4%

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

   6. distribute-rgt-out 77.3%

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

   7. *-commutative 77.3%

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

   8. associate-*r/ 77.6%

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

   9. associate-/l* 77.6%

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

   10. *-commutative 77.6%

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

   11. fma-def 77.6%

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

   12. *-commutative 77.6%

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

   13. /-rgt-identity 77.6%

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

3. Simplified 77.6%

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

4. Taylor expanded in u around 0 77.1%

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

5. Taylor expanded in normAngle around 0 99.1%

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

6. Taylor expanded in u around 0 99.1%

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

   1. *-commutative 99.1%

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

   2. cancel-sign-sub-inv 99.1%

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

   3. metadata-eval 99.1%

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

   4. neg-mul-1 99.1%

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

   5. unsub-neg 99.1%

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

8. Simplified 99.1%

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

9. Taylor expanded in n0_i around 0 99.1%

   \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\color{blue}{\left(0.3333333333333333 \cdot \left(n0_i \cdot \left(u \cdot {normAngle}^{2}\right)\right) + 0.16666666666666666 \cdot \left(n1_i \cdot \left(u \cdot {normAngle}^{2}\right)\right)\right)} + n0_i\right) \]
10. Step-by-step derivation

    1. +-commutative 99.1%

       \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\color{blue}{\left(0.16666666666666666 \cdot \left(n1_i \cdot \left(u \cdot {normAngle}^{2}\right)\right) + 0.3333333333333333 \cdot \left(n0_i \cdot \left(u \cdot {normAngle}^{2}\right)\right)\right)} + n0_i\right) \]

    2. associate-*r* 99.1%

       \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(\color{blue}{\left(0.16666666666666666 \cdot n1_i\right) \cdot \left(u \cdot {normAngle}^{2}\right)} + 0.3333333333333333 \cdot \left(n0_i \cdot \left(u \cdot {normAngle}^{2}\right)\right)\right) + n0_i\right) \]

    3. associate-*r* 99.1%

       \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(\left(0.16666666666666666 \cdot n1_i\right) \cdot \left(u \cdot {normAngle}^{2}\right) + \color{blue}{\left(0.3333333333333333 \cdot n0_i\right) \cdot \left(u \cdot {normAngle}^{2}\right)}\right) + n0_i\right) \]

    4. *-commutative 99.1%

       \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(\left(0.16666666666666666 \cdot n1_i\right) \cdot \left(u \cdot {normAngle}^{2}\right) + \color{blue}{\left(n0_i \cdot 0.3333333333333333\right)} \cdot \left(u \cdot {normAngle}^{2}\right)\right) + n0_i\right) \]

    5. metadata-eval 99.1%

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

    6. distribute-rgt-out 99.1%

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

    7. distribute-rgt-out 99.1%

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

    8. unpow2 99.1%

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

    9. fma-def 99.1%

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

    10. distribute-rgt-out 99.1%

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

    11. metadata-eval 99.1%

        \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(u \cdot \left(normAngle \cdot normAngle\right)\right) \cdot \mathsf{fma}\left(0.16666666666666666, n1_i, n0_i \cdot \color{blue}{0.3333333333333333}\right) + n0_i\right) \]

    12. *-commutative 99.1%

        \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\left(u \cdot \left(normAngle \cdot normAngle\right)\right) \cdot \mathsf{fma}\left(0.16666666666666666, n1_i, \color{blue}{0.3333333333333333 \cdot n0_i}\right) + n0_i\right) \]

11. Simplified 99.1%

    \[\leadsto \left(n1_i + -1 \cdot n0_i\right) \cdot u + \left(\color{blue}{\left(u \cdot \left(normAngle \cdot normAngle\right)\right) \cdot \mathsf{fma}\left(0.16666666666666666, n1_i, 0.3333333333333333 \cdot n0_i\right)} + n0_i\right) \]

12. Final simplification 99.1%

    \[\leadsto \left(n1_i - n0_i\right) \cdot u + \left(n0_i + \left(u \cdot \left(normAngle \cdot normAngle\right)\right) \cdot \mathsf{fma}\left(0.16666666666666666, n1_i, n0_i \cdot 0.3333333333333333\right)\right) \]

Alternative 3: 98.7% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

   1. +-commutative 96.9%

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

   2. *-commutative 96.9%

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

   3. associate-*r* 89.1%

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

   4. *-commutative 89.1%

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

   5. associate-*r* 77.4%

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

   6. distribute-rgt-out 77.3%

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

   7. *-commutative 77.3%

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

   8. associate-*r/ 77.6%

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

   9. associate-/l* 77.6%

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

   10. *-commutative 77.6%

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

   11. fma-def 77.6%

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

   12. *-commutative 77.6%

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

   13. /-rgt-identity 77.6%

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

3. Simplified 77.6%

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

4. Taylor expanded in u around 0 77.1%

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

5. Taylor expanded in normAngle around 0 99.1%

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

6. Taylor expanded in u around 0 99.1%

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

   1. *-commutative 99.1%

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

   2. cancel-sign-sub-inv 99.1%

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

   3. metadata-eval 99.1%

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

   4. neg-mul-1 99.1%

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

   5. unsub-neg 99.1%

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

8. Simplified 99.1%

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

9. Taylor expanded in n0_i around 0 98.8%

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

    1. associate-*r* 98.8%

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

    2. unpow2 98.8%

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

    3. associate-*r* 98.8%

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

    4. *-commutative 98.8%

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

    5. associate-*l* 98.8%

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

11. Simplified 98.8%

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

12. Final simplification 98.8%

    \[\leadsto \left(n1_i - n0_i\right) \cdot u + \left(n0_i + 0.16666666666666666 \cdot \left(normAngle \cdot \left(u \cdot \left(n1_i \cdot normAngle\right)\right)\right)\right) \]

Alternative 4: 61.5% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -8.199999739777285 \cdot 10^{-21} \lor \neg \left(n1_i \leq 1.0000000168623835 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot \left(n1_i + n0_i\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -8.199999739777285e-21)
         (not (<= n1_i 1.0000000168623835e-16)))
   (* u (+ n1_i n0_i))
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -8.199999739777285e-21f) || !(n1_i <= 1.0000000168623835e-16f)) {
		tmp = u * (n1_i + n0_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 <= (-8.199999739777285e-21)) .or. (.not. (n1_i <= 1.0000000168623835e-16))) then
        tmp = u * (n1_i + n0_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(-8.199999739777285e-21)) || !(n1_i <= Float32(1.0000000168623835e-16)))
		tmp = Float32(u * Float32(n1_i + n0_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(-8.199999739777285e-21)) || ~((n1_i <= single(1.0000000168623835e-16))))
		tmp = u * (n1_i + n0_i);
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -8.19999974e-21 or 1.00000002e-16 < n1_i

   1. Initial program 96.1%

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

      1. fma-def 96.2%

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

      2. associate-*r/ 96.2%

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

      3. *-rgt-identity 96.2%

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

      4. associate-*r/ 97.1%

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

      5. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in normAngle around 0 97.4%

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

      1. flip-+ 91.3%

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

   6. Applied egg-rr 91.3%

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

   7. Taylor expanded in u around inf 58.2%

      \[\leadsto \color{blue}{\frac{\left({n1_i}^{2} - {n0_i}^{2}\right) \cdot u}{n1_i - -1 \cdot n0_i}} \]
   8. Step-by-step derivation

      1. associate-/l* 58.9%

         \[\leadsto \color{blue}{\frac{{n1_i}^{2} - {n0_i}^{2}}{\frac{n1_i - -1 \cdot n0_i}{u}}} \]

      2. unpow2 58.9%

         \[\leadsto \frac{\color{blue}{n1_i \cdot n1_i} - {n0_i}^{2}}{\frac{n1_i - -1 \cdot n0_i}{u}} \]

      3. unpow2 58.9%

         \[\leadsto \frac{n1_i \cdot n1_i - \color{blue}{n0_i \cdot n0_i}}{\frac{n1_i - -1 \cdot n0_i}{u}} \]

      4. neg-mul-1 58.9%

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

   9. Simplified 58.9%

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

       1. associate-/r/ 59.2%

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

       2. sqr-neg 59.2%

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

       3. flip-+ 59.4%

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

       4. +-commutative 59.4%

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

       5. add-sqr-sqrt 33.0%

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

       6. sqrt-unprod 60.6%

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

       7. sqr-neg 60.6%

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

       8. sqrt-prod 27.7%

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

       9. add-sqr-sqrt 62.1%

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

   11. Applied egg-rr 62.1%

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

   if -8.19999974e-21 < n1_i < 1.00000002e-16

   1. Initial program 97.7%

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

      1. fma-def 97.7%

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

      2. associate-*r/ 97.9%

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

      3. *-rgt-identity 97.9%

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

      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. Taylor expanded in u around 0 64.1%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -8.199999739777285 \cdot 10^{-21} \lor \neg \left(n1_i \leq 1.0000000168623835 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot \left(n1_i + n0_i\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 5: 71.8% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -4.99999991225835 \cdot 10^{-15} \lor \neg \left(n1_i \leq 1.0000000168623835 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot \left(n1_i + n0_i\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -4.99999991225835e-15)
         (not (<= n1_i 1.0000000168623835e-16)))
   (* u (+ n1_i n0_i))
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -4.99999991225835e-15f) || !(n1_i <= 1.0000000168623835e-16f)) {
		tmp = u * (n1_i + n0_i);
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-4.99999991225835e-15)) .or. (.not. (n1_i <= 1.0000000168623835e-16))) then
        tmp = u * (n1_i + n0_i)
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-4.99999991225835e-15)) || !(n1_i <= Float32(1.0000000168623835e-16)))
		tmp = Float32(u * Float32(n1_i + n0_i));
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-4.99999991225835e-15)) || ~((n1_i <= single(1.0000000168623835e-16))))
		tmp = u * (n1_i + n0_i);
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.99999991e-15 or 1.00000002e-16 < n1_i

   1. Initial program 96.7%

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

      1. fma-def 96.7%

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

      2. associate-*r/ 96.8%

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

      3. *-rgt-identity 96.8%

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

      4. associate-*r/ 97.2%

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

      5. *-rgt-identity 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in normAngle around 0 97.4%

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

      1. flip-+ 95.0%

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

   6. Applied egg-rr 95.0%

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

   7. Taylor expanded in u around inf 63.3%

      \[\leadsto \color{blue}{\frac{\left({n1_i}^{2} - {n0_i}^{2}\right) \cdot u}{n1_i - -1 \cdot n0_i}} \]
   8. Step-by-step derivation

      1. associate-/l* 63.1%

         \[\leadsto \color{blue}{\frac{{n1_i}^{2} - {n0_i}^{2}}{\frac{n1_i - -1 \cdot n0_i}{u}}} \]

      2. unpow2 63.1%

         \[\leadsto \frac{\color{blue}{n1_i \cdot n1_i} - {n0_i}^{2}}{\frac{n1_i - -1 \cdot n0_i}{u}} \]

      3. unpow2 63.1%

         \[\leadsto \frac{n1_i \cdot n1_i - \color{blue}{n0_i \cdot n0_i}}{\frac{n1_i - -1 \cdot n0_i}{u}} \]

      4. neg-mul-1 63.1%

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

   9. Simplified 63.1%

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

       1. associate-/r/ 63.5%

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

       2. sqr-neg 63.5%

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

       3. flip-+ 63.6%

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

       4. +-commutative 63.6%

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

       5. add-sqr-sqrt 35.6%

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

       6. sqrt-unprod 64.8%

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

       7. sqr-neg 64.8%

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

       8. sqrt-prod 29.2%

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

       9. add-sqr-sqrt 66.0%

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

   11. Applied egg-rr 66.0%

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

   if -4.99999991e-15 < n1_i < 1.00000002e-16

   1. Initial program 97.1%

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

      1. fma-def 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/ 97.7%

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

      5. *-rgt-identity 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in normAngle around 0 97.5%

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

   5. Taylor expanded in n1_i around 0 80.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -4.99999991225835 \cdot 10^{-15} \lor \neg \left(n1_i \leq 1.0000000168623835 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot \left(n1_i + n0_i\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 6: 86.2% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -5.000000097707407 \cdot 10^{-26} \lor \neg \left(n1_i \leq 5.000000156871975 \cdot 10^{-23}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -5.000000097707407e-26)
         (not (<= n1_i 5.000000156871975e-23)))
   (+ n0_i (* n1_i u))
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -5.000000097707407e-26f) || !(n1_i <= 5.000000156871975e-23f)) {
		tmp = n0_i + (n1_i * u);
	} 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 <= (-5.000000097707407e-26)) .or. (.not. (n1_i <= 5.000000156871975e-23))) then
        tmp = n0_i + (n1_i * u)
    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(-5.000000097707407e-26)) || !(n1_i <= Float32(5.000000156871975e-23)))
		tmp = Float32(n0_i + Float32(n1_i * u));
	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(-5.000000097707407e-26)) || ~((n1_i <= single(5.000000156871975e-23))))
		tmp = n0_i + (n1_i * u);
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -5.0000001e-26 or 5.00000016e-23 < n1_i

   1. Initial program 95.8%

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

      1. fma-def 95.9%

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

      2. associate-*r/ 96.0%

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

      3. *-rgt-identity 96.0%

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

      4. associate-*r/ 96.6%

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

      5. *-rgt-identity 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in normAngle around 0 97.7%

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

   5. Taylor expanded in u around 0 86.1%

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

   if -5.0000001e-26 < n1_i < 5.00000016e-23

   1. Initial program 98.9%

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

      1. fma-def 98.9%

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

      2. associate-*r/ 99.0%

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

      3. *-rgt-identity 99.0%

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

      4. associate-*r/ 99.1%

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

      5. *-rgt-identity 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in normAngle around 0 97.0%

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

   5. Taylor expanded in n1_i around 0 90.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -5.000000097707407 \cdot 10^{-26} \lor \neg \left(n1_i \leq 5.000000156871975 \cdot 10^{-23}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 7: 86.3% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -5.000000097707407 \cdot 10^{-26} \lor \neg \left(n1_i \leq 5.000000156871975 \cdot 10^{-23}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -5.000000097707407e-26)
         (not (<= n1_i 5.000000156871975e-23)))
   (+ n0_i (* n1_i u))
   (- n0_i (* n0_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -5.000000097707407e-26f) || !(n1_i <= 5.000000156871975e-23f)) {
		tmp = n0_i + (n1_i * u);
	} else {
		tmp = n0_i - (n0_i * u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-5.000000097707407e-26)) .or. (.not. (n1_i <= 5.000000156871975e-23))) then
        tmp = n0_i + (n1_i * u)
    else
        tmp = n0_i - (n0_i * u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-5.000000097707407e-26)) || !(n1_i <= Float32(5.000000156871975e-23)))
		tmp = Float32(n0_i + Float32(n1_i * u));
	else
		tmp = Float32(n0_i - Float32(n0_i * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-5.000000097707407e-26)) || ~((n1_i <= single(5.000000156871975e-23))))
		tmp = n0_i + (n1_i * u);
	else
		tmp = n0_i - (n0_i * u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -5.0000001e-26 or 5.00000016e-23 < n1_i

   1. Initial program 95.8%

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

      1. fma-def 95.9%

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

      2. associate-*r/ 96.0%

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

      3. *-rgt-identity 96.0%

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

      4. associate-*r/ 96.6%

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

      5. *-rgt-identity 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in normAngle around 0 97.7%

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

   5. Taylor expanded in u around 0 86.1%

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

   if -5.0000001e-26 < n1_i < 5.00000016e-23

   1. Initial program 98.9%

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

      1. fma-def 98.9%

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

      2. associate-*r/ 99.0%

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

      3. *-rgt-identity 99.0%

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

      4. associate-*r/ 99.1%

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

      5. *-rgt-identity 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in normAngle around 0 97.0%

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

      1. flip-+ 64.7%

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

   6. Applied egg-rr 64.7%

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

   7. Taylor expanded in n1_i around 0 90.8%

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

      1. *-commutative 90.8%

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

      2. sub-neg 90.8%

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

      3. mul-1-neg 90.8%

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

      4. distribute-lft-in 91.0%

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

      5. *-rgt-identity 91.0%

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

      6. mul-1-neg 91.0%

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

      7. distribute-rgt-neg-in 91.0%

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

      8. unsub-neg 91.0%

         \[\leadsto \color{blue}{n0_i - n0_i \cdot u} \]

      9. *-commutative 91.0%

         \[\leadsto n0_i - \color{blue}{u \cdot n0_i} \]

   9. Simplified 91.0%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -5.000000097707407 \cdot 10^{-26} \lor \neg \left(n1_i \leq 5.000000156871975 \cdot 10^{-23}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \]

Alternative 8: 60.6% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -8.199999739777285 \cdot 10^{-21}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -8.199999739777285e-21)
   (* n1_i u)
   (if (<= n1_i 1.0000000168623835e-16) n0_i (* n1_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -8.199999739777285e-21f) {
		tmp = n1_i * u;
	} else if (n1_i <= 1.0000000168623835e-16f) {
		tmp = n0_i;
	} else {
		tmp = n1_i * u;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n1_i <= (-8.199999739777285e-21)) then
        tmp = n1_i * u
    else if (n1_i <= 1.0000000168623835e-16) then
        tmp = n0_i
    else
        tmp = n1_i * u
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-8.199999739777285e-21))
		tmp = Float32(n1_i * u);
	elseif (n1_i <= Float32(1.0000000168623835e-16))
		tmp = n0_i;
	else
		tmp = Float32(n1_i * u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n1_i <= single(-8.199999739777285e-21))
		tmp = n1_i * u;
	elseif (n1_i <= single(1.0000000168623835e-16))
		tmp = n0_i;
	else
		tmp = n1_i * u;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -8.19999974e-21 or 1.00000002e-16 < n1_i

   1. Initial program 96.1%

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

      1. fma-def 96.2%

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

      2. associate-*r/ 96.2%

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

      3. *-rgt-identity 96.2%

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

      4. associate-*r/ 97.1%

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

      5. *-rgt-identity 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in normAngle around 0 97.4%

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

   5. Taylor expanded in n1_i around inf 60.4%

      \[\leadsto \color{blue}{n1_i \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 60.4%

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

   7. Simplified 60.4%

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

   if -8.19999974e-21 < n1_i < 1.00000002e-16

   1. Initial program 97.7%

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

      1. fma-def 97.7%

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

      2. associate-*r/ 97.9%

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

      3. *-rgt-identity 97.9%

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

      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. Taylor expanded in u around 0 64.1%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -8.199999739777285 \cdot 10^{-21}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 9: 98.0% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

   1. fma-def 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/ 97.1%

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

   3. *-rgt-identity 97.1%

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

   4. associate-*r/ 97.5%

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

   5. *-rgt-identity 97.5%

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

3. Simplified 97.5%

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

4. Taylor expanded in normAngle around 0 97.5%

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

5. Taylor expanded in u around -inf 97.7%

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

   1. +-commutative 97.7%

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

   2. mul-1-neg 97.7%

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

   3. unsub-neg 97.7%

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

   4. +-commutative 97.7%

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

   5. mul-1-neg 97.7%

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

   6. unsub-neg 97.7%

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

7. Simplified 97.7%

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

8. Final simplification 97.7%

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

Alternative 10: 46.6% accurate, 421.0× speedup

Math

\[\begin{array}{l} \\ n0_i \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

   1. fma-def 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/ 97.1%

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

   3. *-rgt-identity 97.1%

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

   4. associate-*r/ 97.5%

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

   5. *-rgt-identity 97.5%

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

3. Simplified 97.5%

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

4. Taylor expanded in u around 0 48.2%

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

5. Final simplification 48.2%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023194 
(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)))