Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 98.8%

Time: 13.3s

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: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. fma-def 97.6%

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

   2. associate-*r/ 97.8%

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

   3. *-rgt-identity 97.8%

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

   4. associate-*r/ 97.9%

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

   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 88.3%

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

   1. *-commutative 88.3%

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

   2. associate-*r/ 88.3%

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

   3. mul-1-neg 88.3%

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

   4. *-commutative 88.3%

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

6. Simplified 88.3%

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

7. Taylor expanded in normAngle around 0 99.0%

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

   1. fma-def 99.0%

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

   2. mul-1-neg 99.0%

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

   3. cancel-sign-sub-inv 99.0%

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

   4. metadata-eval 99.0%

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

   5. mul-1-neg 99.0%

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

   6. distribute-rgt-out-- 99.0%

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

   7. metadata-eval 99.0%

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

   8. *-commutative 99.0%

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

   9. unpow2 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. fma-def 97.6%

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

   2. associate-*r/ 97.8%

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

   3. *-rgt-identity 97.8%

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

   4. associate-*r/ 97.9%

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

   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 88.3%

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

   1. *-commutative 88.3%

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

   2. associate-*r/ 88.3%

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

   3. mul-1-neg 88.3%

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

   4. *-commutative 88.3%

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

6. Simplified 88.3%

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

7. Taylor expanded in normAngle around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. fma-def 99.0%

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

   3. cancel-sign-sub-inv 99.0%

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

   4. mul-1-neg 99.0%

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

   5. distribute-rgt-out-- 99.0%

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

   6. distribute-rgt-neg-in 99.0%

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

   7. metadata-eval 99.0%

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

   8. metadata-eval 99.0%

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

   9. metadata-eval 99.0%

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

   10. *-commutative 99.0%

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

   11. *-commutative 99.0%

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

   12. unpow2 99.0%

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

   13. associate-*l* 99.0%

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

   14. *-commutative 99.0%

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

   15. *-commutative 99.0%

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

   16. mul-1-neg 99.0%

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

   17. unsub-neg 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

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

Alternative 3: 98.8% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (-
  n0_i
  (*
   u
   (-
    n0_i
    (+
     n1_i
     (*
      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 n0_i - (u * (n0_i - (n1_i + (normAngle * (normAngle * fmaf(0.16666666666666666f, n1_i, (n0_i * 0.3333333333333333f)))))));
}

Julia

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

Derivation

1. Initial program 97.6%

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

   1. fma-def 97.6%

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

   2. associate-*r/ 97.8%

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

   3. *-rgt-identity 97.8%

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

   4. associate-*r/ 97.9%

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

   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 88.3%

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

   1. *-commutative 88.3%

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

   2. associate-*r/ 88.3%

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

   3. mul-1-neg 88.3%

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

   4. *-commutative 88.3%

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

6. Simplified 88.3%

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

7. Taylor expanded in normAngle around 0 99.0%

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

   1. fma-def 99.0%

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

   2. mul-1-neg 99.0%

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

   3. cancel-sign-sub-inv 99.0%

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

   4. metadata-eval 99.0%

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

   5. mul-1-neg 99.0%

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

   6. distribute-rgt-out-- 99.0%

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

   7. metadata-eval 99.0%

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

   8. *-commutative 99.0%

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

   9. unpow2 99.0%

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

9. Simplified 99.0%

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

10. Taylor expanded in u around 0 99.0%

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

11. Taylor expanded in n1_i around 0 99.0%

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

    1. +-commutative 99.0%

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

    2. associate-*r* 99.0%

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

    3. associate-*r* 99.0%

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

    4. metadata-eval 99.0%

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

    5. distribute-rgt-in 99.0%

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

    6. cancel-sign-sub-inv 99.0%

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

    7. unpow2 99.0%

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

    8. associate-*l* 99.0%

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

    9. cancel-sign-sub-inv 99.0%

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

    10. fma-def 99.0%

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

    11. metadata-eval 99.0%

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

13. Simplified 99.0%

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

14. Final simplification 99.0%

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

Alternative 4: 98.1% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i - (u * (n0_i - (n1_i + (n0_i * ((normangle * normangle) * 0.3333333333333333e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. fma-def 97.6%

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

   2. associate-*r/ 97.8%

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

   3. *-rgt-identity 97.8%

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

   4. associate-*r/ 97.9%

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

   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 88.3%

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

   1. *-commutative 88.3%

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

   2. associate-*r/ 88.3%

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

   3. mul-1-neg 88.3%

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

   4. *-commutative 88.3%

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

6. Simplified 88.3%

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

7. Taylor expanded in normAngle around 0 99.0%

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

   1. fma-def 99.0%

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

   2. mul-1-neg 99.0%

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

   3. cancel-sign-sub-inv 99.0%

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

   4. metadata-eval 99.0%

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

   5. mul-1-neg 99.0%

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

   6. distribute-rgt-out-- 99.0%

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

   7. metadata-eval 99.0%

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

   8. *-commutative 99.0%

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

   9. unpow2 99.0%

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

9. Simplified 99.0%

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

10. Taylor expanded in u around 0 99.0%

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

11. Taylor expanded in n1_i around 0 98.3%

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

    1. *-commutative 98.3%

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

    2. associate-*l* 98.3%

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

    3. *-commutative 98.3%

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

    4. unpow2 98.3%

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

13. Simplified 98.3%

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

14. Final simplification 98.3%

    \[\leadsto n0_i - u \cdot \left(n0_i - \left(n1_i + n0_i \cdot \left(\left(normAngle \cdot normAngle\right) \cdot 0.3333333333333333\right)\right)\right) \]

Alternative 5: 98.6% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. fma-def 97.6%

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

   2. associate-*r/ 97.8%

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

   3. *-rgt-identity 97.8%

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

   4. associate-*r/ 97.9%

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

   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 88.3%

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

   1. *-commutative 88.3%

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

   2. associate-*r/ 88.3%

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

   3. mul-1-neg 88.3%

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

   4. *-commutative 88.3%

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

6. Simplified 88.3%

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

7. Taylor expanded in normAngle around 0 99.0%

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

   1. fma-def 99.0%

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

   2. mul-1-neg 99.0%

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

   3. cancel-sign-sub-inv 99.0%

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

   4. metadata-eval 99.0%

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

   5. mul-1-neg 99.0%

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

   6. distribute-rgt-out-- 99.0%

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

   7. metadata-eval 99.0%

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

   8. *-commutative 99.0%

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

   9. unpow2 99.0%

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

9. Simplified 99.0%

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

10. Taylor expanded in u around 0 99.0%

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

11. Taylor expanded in n1_i around inf 98.9%

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

    1. associate-*r* 98.9%

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

    2. *-commutative 98.9%

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

    3. unpow2 98.9%

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

13. Simplified 98.9%

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

14. Final simplification 98.9%

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

Alternative 6: 85.9% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -5.000000136226006 \cdot 10^{-28} \lor \neg \left(n1_i \leq 1.0000000031710769 \cdot 10^{-28}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -5.000000136226006e-28)
         (not (<= n1_i 1.0000000031710769e-28)))
   (+ 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 <= -5.000000136226006e-28f) || !(n1_i <= 1.0000000031710769e-28f)) {
		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 <= (-5.000000136226006e-28)) .or. (.not. (n1_i <= 1.0000000031710769e-28))) 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(-5.000000136226006e-28)) || !(n1_i <= Float32(1.0000000031710769e-28)))
		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(-5.000000136226006e-28)) || ~((n1_i <= single(1.0000000031710769e-28))))
		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 < -5.00000014e-28 or 1e-28 < n1_i

   1. Initial program 97.0%

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

      1. fma-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/ 97.2%

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

      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/ 97.4%

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

      5. *-rgt-identity 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in normAngle around 0 97.6%

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

   5. Taylor expanded in u around 0 86.5%

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

   if -5.00000014e-28 < n1_i < 1e-28

   1. Initial program 98.8%

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

      1. fma-def 98.8%

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

      2. associate-*r/ 99.1%

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

      3. *-rgt-identity 99.1%

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

      4. associate-*r/ 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 99.4%

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

   5. Taylor expanded in n1_i around 0 93.8%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -5.000000136226006 \cdot 10^{-28} \lor \neg \left(n1_i \leq 1.0000000031710769 \cdot 10^{-28}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 7: 86.0% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -5.000000136226006 \cdot 10^{-28} \lor \neg \left(n1_i \leq 1.0000000031710769 \cdot 10^{-28}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -5.000000136226006e-28)
         (not (<= n1_i 1.0000000031710769e-28)))
   (+ 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 <= -5.000000136226006e-28f) || !(n1_i <= 1.0000000031710769e-28f)) {
		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 <= (-5.000000136226006e-28)) .or. (.not. (n1_i <= 1.0000000031710769e-28))) 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(-5.000000136226006e-28)) || !(n1_i <= Float32(1.0000000031710769e-28)))
		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(-5.000000136226006e-28)) || ~((n1_i <= single(1.0000000031710769e-28))))
		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 < -5.00000014e-28 or 1e-28 < n1_i

   1. Initial program 97.0%

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

      1. fma-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/ 97.2%

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

      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/ 97.4%

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

      5. *-rgt-identity 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in normAngle around 0 97.6%

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

   5. Taylor expanded in u around 0 86.5%

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

   if -5.00000014e-28 < n1_i < 1e-28

   1. Initial program 98.8%

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

      1. fma-def 98.8%

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

      2. associate-*r/ 99.1%

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

      3. *-rgt-identity 99.1%

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

      4. associate-*r/ 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 99.4%

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

   5. Taylor expanded in u around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. fma-def 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in n1_i around 0 93.9%

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

      1. +-commutative 93.9%

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

      2. associate-*r* 93.9%

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

      3. mul-1-neg 93.9%

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

      4. cancel-sign-sub-inv 93.9%

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

   10. Simplified 93.9%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -5.000000136226006 \cdot 10^{-28} \lor \neg \left(n1_i \leq 1.0000000031710769 \cdot 10^{-28}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \]

Alternative 8: 70.0% accurate, 45.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -3.0000000340435383 \cdot 10^{-18}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -3.0000000340435383e-18)
   (* n1_i u)
   (if (<= n1_i 9.999999960041972e-12) (* n0_i (- 1.0 u)) (* n1_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -3.0000000340435383e-18f) {
		tmp = n1_i * u;
	} else if (n1_i <= 9.999999960041972e-12f) {
		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 (n1_i <= (-3.0000000340435383e-18)) then
        tmp = n1_i * u
    else if (n1_i <= 9.999999960041972e-12) 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 (n1_i <= Float32(-3.0000000340435383e-18))
		tmp = Float32(n1_i * u);
	elseif (n1_i <= Float32(9.999999960041972e-12))
		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 (n1_i <= single(-3.0000000340435383e-18))
		tmp = n1_i * u;
	elseif (n1_i <= single(9.999999960041972e-12))
		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 n1_i < -3.00000003e-18 or 9.99999996e-12 < n1_i

   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/ 97.8%

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

      3. *-rgt-identity 97.8%

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

      4. associate-*r/ 98.1%

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

      5. *-rgt-identity 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in normAngle around 0 97.7%

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

   5. Taylor expanded in n1_i around inf 71.6%

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

      1. *-commutative 71.6%

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

   7. Simplified 71.6%

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

   if -3.00000003e-18 < n1_i < 9.99999996e-12

   1. Initial program 97.5%

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

      1. fma-def 97.5%

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

      2. associate-*r/ 97.8%

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

      3. *-rgt-identity 97.8%

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

      4. associate-*r/ 97.8%

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

      5. *-rgt-identity 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in normAngle around 0 98.5%

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

   5. Taylor expanded in n1_i around 0 76.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -3.0000000340435383 \cdot 10^{-18}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 9.999999960041972 \cdot 10^{-12}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 9: 60.5% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -3.0000000340435383 \cdot 10^{-18}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -3.0000000340435383e-18)
   (* n1_i u)
   (if (<= n1_i 5.000000018137469e-16) n0_i (* n1_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -3.0000000340435383e-18f) {
		tmp = n1_i * u;
	} else if (n1_i <= 5.000000018137469e-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 <= (-3.0000000340435383e-18)) then
        tmp = n1_i * u
    else if (n1_i <= 5.000000018137469e-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(-3.0000000340435383e-18))
		tmp = Float32(n1_i * u);
	elseif (n1_i <= Float32(5.000000018137469e-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(-3.0000000340435383e-18))
		tmp = n1_i * u;
	elseif (n1_i <= single(5.000000018137469e-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 < -3.00000003e-18 or 5.00000002e-16 < n1_i

   1. Initial program 97.4%

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

      1. fma-def 97.4%

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

      2. associate-*r/ 97.6%

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

      3. *-rgt-identity 97.6%

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

      4. associate-*r/ 97.8%

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

      5. *-rgt-identity 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in normAngle around 0 97.8%

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

   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%

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

   if -3.00000003e-18 < n1_i < 5.00000002e-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/ 98.0%

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

      5. *-rgt-identity 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in u around 0 61.5%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -3.0000000340435383 \cdot 10^{-18}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 10: 97.9% 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 97.6%

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

   1. fma-def 97.6%

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

   2. associate-*r/ 97.8%

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

   3. *-rgt-identity 97.8%

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

   4. associate-*r/ 97.9%

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

   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 normAngle around 0 98.2%

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

5. Taylor expanded in u around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. +-commutative 98.3%

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

   5. mul-1-neg 98.3%

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

   6. unsub-neg 98.3%

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

7. Simplified 98.3%

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

8. Final simplification 98.3%

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

Alternative 11: 47.6% 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 97.6%

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

   1. fma-def 97.6%

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

   2. associate-*r/ 97.8%

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

   3. *-rgt-identity 97.8%

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

   4. associate-*r/ 97.9%

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

   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 45.3%

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

5. Final simplification 45.3%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023215 
(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)))