Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 99.3%

Time: 13.4s

Alternatives: 11

Speedup: 60.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.1% 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: 99.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Taylor expanded in normAngle around 0 96.6%

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

3. Taylor expanded in u around 0 89.2%

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

   1. fma-def 89.3%

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

   2. associate-/l* 99.1%

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

   3. mul-1-neg 99.1%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 2: 99.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Taylor expanded in normAngle around 0 96.6%

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

3. Taylor expanded in u around 0 89.2%

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

   1. *-commutative 89.2%

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

   2. fma-def 89.3%

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

   3. mul-1-neg 89.3%

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

   4. unsub-neg 89.3%

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

   5. associate-/l* 99.1%

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

   6. associate-/r/ 99.0%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 3: 99.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 97.1%

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

   2. associate-*r/ 97.2%

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

   3. *-rgt-identity 97.2%

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

   4. associate-*r/ 97.3%

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

   5. *-rgt-identity 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in u around 0 86.4%

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

   1. *-commutative 86.4%

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

   2. associate-*r/ 86.4%

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

   3. mul-1-neg 86.4%

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

   4. *-commutative 86.4%

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

6. Simplified 86.4%

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

7. Taylor expanded in normAngle around 0 98.7%

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

   1. *-commutative 98.7%

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

   2. fma-def 98.7%

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

   3. mul-1-neg 98.7%

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

   4. unsub-neg 98.7%

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

   5. cancel-sign-sub-inv 98.7%

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

   6. metadata-eval 98.7%

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

   7. +-commutative 98.7%

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

   8. *-commutative 98.7%

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

   9. mul-1-neg 98.7%

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

   10. distribute-rgt-out-- 98.7%

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

   11. distribute-rgt-neg-in 98.7%

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

   12. metadata-eval 98.7%

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

   13. metadata-eval 98.7%

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

   14. unpow2 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

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

Alternative 4: 99.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ n0_i + u \cdot \left(\left(n1_i + {normAngle}^{2} \cdot \left(n0_i \cdot 0.3333333333333333 + n1_i \cdot 0.16666666666666666\right)\right) - n0_i\right) \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((n1_i + (powf(normAngle, 2.0f) * ((n0_i * 0.3333333333333333f) + (n1_i * 0.16666666666666666f)))) - 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 + ((normangle ** 2.0e0) * ((n0_i * 0.3333333333333333e0) + (n1_i * 0.16666666666666666e0)))) - n0_i))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(Float32(n1_i + Float32((normAngle ^ Float32(2.0)) * Float32(Float32(n0_i * Float32(0.3333333333333333)) + Float32(n1_i * Float32(0.16666666666666666))))) - n0_i)))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * ((n1_i + ((normAngle ^ single(2.0)) * ((n0_i * single(0.3333333333333333)) + (n1_i * single(0.16666666666666666))))) - n0_i));
end

Derivation

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 97.1%

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

   2. associate-*r/ 97.2%

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

   3. *-rgt-identity 97.2%

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

   4. associate-*r/ 97.3%

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

   5. *-rgt-identity 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in u around 0 86.4%

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

   1. *-commutative 86.4%

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

   2. associate-*r/ 86.4%

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

   3. mul-1-neg 86.4%

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

   4. *-commutative 86.4%

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

6. Simplified 86.4%

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

7. Taylor expanded in normAngle around 0 98.7%

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

   1. *-commutative 98.7%

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

   2. fma-def 98.7%

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

   3. mul-1-neg 98.7%

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

   4. unsub-neg 98.7%

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

   5. cancel-sign-sub-inv 98.7%

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

   6. metadata-eval 98.7%

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

   7. +-commutative 98.7%

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

   8. *-commutative 98.7%

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

   9. mul-1-neg 98.7%

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

   10. distribute-rgt-out-- 98.7%

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

   11. distribute-rgt-neg-in 98.7%

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

   12. metadata-eval 98.7%

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

   13. metadata-eval 98.7%

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

   14. unpow2 98.7%

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

9. Simplified 98.7%

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

10. Taylor expanded in u around 0 98.6%

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

11. Final simplification 98.6%

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

Alternative 5: 98.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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 97.1%

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

   2. associate-*r/ 97.2%

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

   3. *-rgt-identity 97.2%

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

   4. associate-*r/ 97.3%

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

   5. *-rgt-identity 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in normAngle around 0 97.3%

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

5. Taylor expanded in u around 0 97.5%

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

   1. *-commutative 97.5%

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

   2. fma-def 97.8%

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

   3. mul-1-neg 97.8%

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

   4. unsub-neg 97.8%

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

7. Simplified 97.8%

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

8. Final simplification 97.8%

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

Alternative 6: 69.4% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -3.0000000340435383 \cdot 10^{-19} \lor \neg \left(n0_i \leq 9.999999682655225 \cdot 10^{-22}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -3.0000000340435383e-19)
         (not (<= n0_i 9.999999682655225e-22)))
   (* n0_i (- 1.0 u))
   (* n1_i u)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -3.0000000340435383e-19f) || !(n0_i <= 9.999999682655225e-22f)) {
		tmp = n0_i * (1.0f - u);
	} 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 ((n0_i <= (-3.0000000340435383e-19)) .or. (.not. (n0_i <= 9.999999682655225e-22))) then
        tmp = n0_i * (1.0e0 - u)
    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 ((n0_i <= Float32(-3.0000000340435383e-19)) || !(n0_i <= Float32(9.999999682655225e-22)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	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 ((n0_i <= single(-3.0000000340435383e-19)) || ~((n0_i <= single(9.999999682655225e-22))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = n1_i * u;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -3.00000003e-19 or 9.9999997e-22 < n0_i

   1. Initial program 97.2%

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

      1. fma-def 97.3%

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

      2. associate-*r/ 97.5%

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

      3. *-rgt-identity 97.5%

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

      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 98.0%

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

   5. Taylor expanded in n1_i around 0 77.3%

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

   if -3.00000003e-19 < n0_i < 9.9999997e-22

   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 96.9%

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

      2. associate-*r/ 96.9%

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

      3. *-rgt-identity 96.9%

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

      4. associate-*r/ 97.1%

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

      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 96.6%

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

   5. Taylor expanded in n1_i around inf 67.5%

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

      1. *-commutative 67.5%

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

   7. Simplified 67.5%

      \[\leadsto \color{blue}{u \cdot n1_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -3.0000000340435383 \cdot 10^{-19} \lor \neg \left(n0_i \leq 9.999999682655225 \cdot 10^{-22}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 7: 60.1% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -3.0000000340435383 \cdot 10^{-19}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -3.0000000340435383e-19)
   n0_i
   (if (<= n0_i 9.999999682655225e-22) (* n1_i u) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -3.0000000340435383e-19f) {
		tmp = n0_i;
	} else if (n0_i <= 9.999999682655225e-22f) {
		tmp = n1_i * u;
	} 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 (n0_i <= (-3.0000000340435383e-19)) then
        tmp = n0_i
    else if (n0_i <= 9.999999682655225e-22) then
        tmp = n1_i * u
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-3.0000000340435383e-19))
		tmp = n0_i;
	elseif (n0_i <= Float32(9.999999682655225e-22))
		tmp = Float32(n1_i * u);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-3.0000000340435383e-19))
		tmp = n0_i;
	elseif (n0_i <= single(9.999999682655225e-22))
		tmp = n1_i * u;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -3.00000003e-19 or 9.9999997e-22 < n0_i

   1. Initial program 97.2%

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

      1. fma-def 97.3%

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

      2. associate-*r/ 97.5%

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

      3. *-rgt-identity 97.5%

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

      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 59.0%

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

   if -3.00000003e-19 < n0_i < 9.9999997e-22

   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 96.9%

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

      2. associate-*r/ 96.9%

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

      3. *-rgt-identity 96.9%

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

      4. associate-*r/ 97.1%

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

      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 96.6%

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

   5. Taylor expanded in n1_i around inf 67.5%

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

      1. *-commutative 67.5%

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

   7. Simplified 67.5%

      \[\leadsto \color{blue}{u \cdot n1_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -3.0000000340435383 \cdot 10^{-19}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 8: 83.7% accurate, 59.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i + n1_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -9.9999998245167e-14) (* n0_i (- 1.0 u)) (+ n0_i (* n1_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -9.9999998245167e-14f) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = n0_i + (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 (n0_i <= (-9.9999998245167e-14)) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = n0_i + (n1_i * u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-9.9999998245167e-14))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(n0_i + Float32(n1_i * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-9.9999998245167e-14))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = n0_i + (n1_i * u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -9.99999982e-14

   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/ 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. associate-*r/ 98.8%

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

      5. *-rgt-identity 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in normAngle around 0 98.3%

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

   5. Taylor expanded in n1_i around 0 94.0%

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

   if -9.99999982e-14 < n0_i

   1. Initial program 96.6%

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

      1. 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/ 96.9%

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

      5. *-rgt-identity 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in normAngle around 0 97.1%

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

   5. Taylor expanded in u around 0 82.9%

      \[\leadsto n1_i \cdot u + \color{blue}{n0_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i + n1_i \cdot u\\ \end{array} \]

Alternative 9: 83.8% accurate, 59.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i + n1_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -9.9999998245167e-14) (- n0_i (* n0_i u)) (+ n0_i (* n1_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -9.9999998245167e-14f) {
		tmp = n0_i - (n0_i * u);
	} else {
		tmp = n0_i + (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 (n0_i <= (-9.9999998245167e-14)) then
        tmp = n0_i - (n0_i * u)
    else
        tmp = n0_i + (n1_i * u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-9.9999998245167e-14))
		tmp = Float32(n0_i - Float32(n0_i * u));
	else
		tmp = Float32(n0_i + Float32(n1_i * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-9.9999998245167e-14))
		tmp = n0_i - (n0_i * u);
	else
		tmp = n0_i + (n1_i * u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -9.99999982e-14

   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/ 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. associate-*r/ 98.8%

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

      5. *-rgt-identity 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in normAngle around 0 98.3%

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

      1. flip-+ 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in n1_i around 0 94.0%

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

      1. sub-neg 94.0%

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

      2. mul-1-neg 94.0%

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

      3. +-commutative 94.0%

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

      4. distribute-rgt1-in 94.4%

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

      5. associate-*r* 94.4%

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

      6. mul-1-neg 94.4%

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

      7. *-commutative 94.4%

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

      8. unsub-neg 94.4%

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

      9. *-commutative 94.4%

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

   9. Simplified 94.4%

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

   if -9.99999982e-14 < n0_i

   1. Initial program 96.6%

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

      1. 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/ 96.9%

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

      5. *-rgt-identity 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in normAngle around 0 97.1%

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

   5. Taylor expanded in u around 0 82.9%

      \[\leadsto n1_i \cdot u + \color{blue}{n0_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i + n1_i \cdot u\\ \end{array} \]

Alternative 10: 98.3% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.1%

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

   1. fma-def 97.1%

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

   2. associate-*r/ 97.2%

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

   3. *-rgt-identity 97.2%

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

   4. associate-*r/ 97.3%

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

   5. *-rgt-identity 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in normAngle around 0 97.3%

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

5. Taylor expanded in u around -inf 97.5%

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

   1. +-commutative 97.5%

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

   2. mul-1-neg 97.5%

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

   3. unsub-neg 97.5%

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

   4. +-commutative 97.5%

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

   5. mul-1-neg 97.5%

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

   6. unsub-neg 97.5%

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

7. Simplified 97.5%

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

8. Final simplification 97.5%

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

Alternative 11: 47.5% 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 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 97.1%

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

   2. associate-*r/ 97.2%

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

   3. *-rgt-identity 97.2%

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

   4. associate-*r/ 97.3%

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

   5. *-rgt-identity 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in u around 0 43.7%

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

5. Final simplification 43.7%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023199 
(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)))