Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.4% → 99.3%

Time: 13.0s

Alternatives: 9

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: 97.4% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ n0_i + u \cdot \mathsf{fma}\left(-1, \frac{n0_i}{\frac{\sin normAngle}{normAngle \cdot \cos normAngle}}, \frac{n1_i}{\frac{\sin normAngle}{normAngle}}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

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. *-commutative 97.0%

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

   2. associate-*l* 81.8%

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

   3. *-commutative 81.8%

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

   4. associate-*l* 75.1%

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

   5. distribute-lft-out 75.2%

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

3. Simplified 75.2%

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

4. Taylor expanded in u around 0 88.5%

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

   1. fma-def 88.5%

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

   2. associate-/l* 90.4%

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

   3. associate-/l* 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 98.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i - ((u * (n0_i - n1_i)) - (powf(normAngle, 2.0f) * (u * ((n0_i * 0.3333333333333333f) + (n1_i * 0.16666666666666666f)))));
}

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)) - ((normangle ** 2.0e0) * (u * ((n0_i * 0.3333333333333333e0) + (n1_i * 0.16666666666666666e0)))))
end function

Julia

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

MATLAB

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

Derivation

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. *-commutative 97.0%

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

   2. associate-*l* 81.8%

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

   3. *-commutative 81.8%

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

   4. associate-*l* 75.1%

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

   5. distribute-lft-out 75.2%

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

3. Simplified 75.2%

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

4. Taylor expanded in u around 0 88.5%

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

   1. fma-def 88.5%

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

   2. associate-/l* 90.4%

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

   3. associate-/l* 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in normAngle around 0 99.0%

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

8. Taylor expanded in n0_i around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. associate-*r* 99.0%

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

   3. *-commutative 99.0%

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

   4. metadata-eval 99.0%

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

   5. distribute-rgt-neg-in 99.0%

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

   6. associate-*r* 99.0%

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

   7. *-commutative 99.0%

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

   8. distribute-rgt-out 99.0%

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

   9. distribute-rgt-neg-in 99.0%

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

   10. metadata-eval 99.0%

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

   11. *-commutative 99.0%

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

10. Simplified 99.0%

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

11. Taylor expanded in n1_i around 0 99.0%

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

    1. +-commutative 99.0%

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

    2. associate-*r* 99.0%

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

    3. distribute-rgt-in 99.0%

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

    4. mul-1-neg 99.0%

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

    5. unsub-neg 99.0%

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

13. Simplified 99.0%

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

14. Final simplification 99.0%

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

Alternative 3: 98.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. *-commutative 97.0%

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

   2. associate-*l* 81.8%

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

   3. *-commutative 81.8%

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

   4. associate-*l* 75.1%

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

   5. distribute-lft-out 75.2%

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

3. Simplified 75.2%

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

4. Taylor expanded in u around 0 88.5%

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

   1. fma-def 88.5%

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

   2. associate-/l* 90.4%

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

   3. associate-/l* 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in normAngle around 0 99.0%

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

8. Taylor expanded in n0_i around 0 98.9%

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

   1. associate-*r* 98.9%

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

   2. *-commutative 98.9%

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

   3. associate-*l* 98.9%

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

10. Simplified 98.9%

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

11. Taylor expanded in u around 0 98.9%

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

12. Final simplification 98.9%

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

Alternative 4: 98.3% 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 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*l* 81.8%

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

   3. *-commutative 81.8%

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

   4. associate-*l* 75.1%

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

   5. distribute-lft-out 75.2%

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

3. Simplified 75.2%

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

4. Taylor expanded in u around 0 88.5%

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

   1. fma-def 88.5%

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

   2. associate-/l* 90.4%

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

   3. associate-/l* 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in normAngle around 0 98.0%

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

   1. +-commutative 98.0%

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

   2. fma-def 98.1%

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

   3. mul-1-neg 98.1%

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

   4. unsub-neg 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

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

Alternative 5: 70.3% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -9.999999998199587 \cdot 10^{-24} \lor \neg \left(n0_i \leq 4.00000012549758 \cdot 10^{-22}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -9.999999998199587e-24) (not (<= n0_i 4.00000012549758e-22)))
   (* n0_i (- 1.0 u))
   (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -9.999999998199587e-24f) || !(n0_i <= 4.00000012549758e-22f)) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = u * n1_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 <= (-9.999999998199587e-24)) .or. (.not. (n0_i <= 4.00000012549758e-22))) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-9.999999998199587e-24)) || !(n0_i <= Float32(4.00000012549758e-22)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-9.999999998199587e-24)) || ~((n0_i <= single(4.00000012549758e-22))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1e-23 or 4.00000013e-22 < n0_i

   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. *-commutative 98.8%

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

      2. associate-*l* 87.7%

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

      3. *-commutative 87.7%

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

      4. associate-*l* 87.6%

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

      5. distribute-lft-out 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in u around 0 97.4%

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

      1. fma-def 97.4%

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

      2. associate-/l* 98.9%

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

      3. associate-/l* 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in normAngle around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. fma-def 98.7%

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

      3. mul-1-neg 98.7%

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

      4. unsub-neg 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in n1_i around 0 78.3%

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

       1. mul-1-neg 78.3%

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

       2. *-rgt-identity 78.3%

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

       3. distribute-rgt-neg-in 78.3%

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

       4. distribute-lft-in 78.0%

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

       5. sub-neg 78.0%

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

   12. Simplified 78.0%

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

   if -1e-23 < n0_i < 4.00000013e-22

   1. Initial program 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*l* 73.4%

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

      3. *-commutative 73.4%

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

      4. associate-*l* 57.5%

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

      5. distribute-lft-out 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in normAngle around 0 97.0%

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

   5. Taylor expanded in n0_i around 0 66.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -9.999999998199587 \cdot 10^{-24} \lor \neg \left(n0_i \leq 4.00000012549758 \cdot 10^{-22}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 6: 60.5% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 4.00000012549758 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -9.999999998199587e-24)
   n0_i
   (if (<= n0_i 4.00000012549758e-22) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -9.999999998199587e-24f) {
		tmp = n0_i;
	} else if (n0_i <= 4.00000012549758e-22f) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-9.999999998199587e-24)) then
        tmp = n0_i
    else if (n0_i <= 4.00000012549758e-22) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-9.999999998199587e-24))
		tmp = n0_i;
	elseif (n0_i <= Float32(4.00000012549758e-22))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-9.999999998199587e-24))
		tmp = n0_i;
	elseif (n0_i <= single(4.00000012549758e-22))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1e-23 or 4.00000013e-22 < n0_i

   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. *-commutative 98.8%

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

      2. associate-*l* 87.7%

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

      3. *-commutative 87.7%

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

      4. associate-*l* 87.6%

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

      5. distribute-lft-out 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in u around 0 97.4%

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

      1. fma-def 97.4%

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

      2. associate-/l* 98.9%

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

      3. associate-/l* 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in u around 0 62.6%

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

   if -1e-23 < n0_i < 4.00000013e-22

   1. Initial program 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*l* 73.4%

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

      3. *-commutative 73.4%

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

      4. associate-*l* 57.5%

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

      5. distribute-lft-out 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in normAngle around 0 97.0%

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

   5. Taylor expanded in n0_i around 0 66.3%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 4.00000012549758 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 7: 98.1% 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 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*l* 81.8%

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

   3. *-commutative 81.8%

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

   4. associate-*l* 75.1%

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

   5. distribute-lft-out 75.2%

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

3. Simplified 75.2%

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

4. Taylor expanded in normAngle around 0 97.8%

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

5. Taylor expanded in u around -inf 98.0%

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

   1. mul-1-neg 98.0%

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

   2. unsub-neg 98.0%

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

   3. mul-1-neg 98.0%

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

   4. unsub-neg 98.0%

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

7. Simplified 98.0%

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

8. Final simplification 98.0%

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

Alternative 8: 82.1% accurate, 84.2× speedup

Math

\[\begin{array}{l} \\ n0_i + u \cdot n1_i \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.0%

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

2. Taylor expanded in normAngle around 0 97.6%

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

3. Taylor expanded in u around 0 83.2%

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

4. Final simplification 83.2%

   \[\leadsto n0_i + u \cdot n1_i \]

Alternative 9: 47.2% 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 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*l* 81.8%

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

   3. *-commutative 81.8%

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

   4. associate-*l* 75.1%

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

   5. distribute-lft-out 75.2%

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

3. Simplified 75.2%

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

4. Taylor expanded in u around 0 88.5%

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

   1. fma-def 88.5%

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

   2. associate-/l* 90.4%

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

   3. associate-/l* 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in u around 0 47.4%

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

8. Final simplification 47.4%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023334 
(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)))