Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.3%

Time: 12.1s

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: 97.3% 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 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*l* 85.3%

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

   3. *-commutative 85.3%

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

   4. associate-*l* 71.9%

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

   5. distribute-lft-out 72.0%

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

   6. associate-*l/ 72.2%

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

   7. *-lft-identity 72.2%

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

   8. fma-def 72.2%

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

   9. *-commutative 72.2%

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

   10. distribute-rgt-out-- 72.2%

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

   11. *-lft-identity 72.2%

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

3. Simplified 72.2%

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

4. Taylor expanded in u around 0 87.4%

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

   1. fma-def 87.4%

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

   2. associate-/l* 90.0%

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

   3. associate-/l* 99.1%

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

6. Simplified 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 99.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

2. Taylor expanded in normAngle around 0 96.7%

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

3. Taylor expanded in u around 0 97.2%

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

   1. associate-/l* 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in u around 0 89.7%

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

   1. mul-1-neg 89.7%

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

   2. associate-*r/ 98.8%

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

   3. +-commutative 98.8%

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

   4. sub-neg 98.8%

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 3: 98.8% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

2. Taylor expanded in normAngle around 0 96.7%

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

3. Taylor expanded in u around 0 89.7%

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

4. Taylor expanded in normAngle around 0 98.3%

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

   1. mul-1-neg 98.3%

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

   2. associate-+r+ 98.3%

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

   3. sub-neg 98.3%

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

   4. associate-*r* 98.3%

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

   5. *-commutative 98.3%

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

   6. unpow2 98.3%

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

6. Simplified 98.3%

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

7. Final simplification 98.3%

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

Alternative 4: 85.7% accurate, 37.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -6.000000163471207 \cdot 10^{-27} \lor \neg \left(n1_i \leq 5.000000097707407 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + u \cdot \left(n0_i + n1_i\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -6.000000163471207e-27)
         (not (<= n1_i 5.000000097707407e-26)))
   (+ n0_i (* u (+ n0_i n1_i)))
   (- n0_i (* n0_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -6.000000163471207e-27f) || !(n1_i <= 5.000000097707407e-26f)) {
		tmp = n0_i + (u * (n0_i + n1_i));
	} 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 <= (-6.000000163471207e-27)) .or. (.not. (n1_i <= 5.000000097707407e-26))) then
        tmp = n0_i + (u * (n0_i + n1_i))
    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(-6.000000163471207e-27)) || !(n1_i <= Float32(5.000000097707407e-26)))
		tmp = Float32(n0_i + Float32(u * Float32(n0_i + n1_i)));
	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(-6.000000163471207e-27)) || ~((n1_i <= single(5.000000097707407e-26))))
		tmp = n0_i + (u * (n0_i + n1_i));
	else
		tmp = n0_i - (n0_i * u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -6.00000016e-27 or 5.0000001e-26 < n1_i

   1. Initial program 96.0%

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

      1. *-commutative 96.0%

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

      2. associate-*l* 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-*l* 76.7%

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

      5. distribute-lft-out 76.8%

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

      6. associate-*l/ 77.0%

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

      7. *-lft-identity 77.0%

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

      8. fma-def 77.0%

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

      9. *-commutative 77.0%

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

      10. distribute-rgt-out-- 77.1%

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

      11. *-lft-identity 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in u around 0 87.4%

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

      1. fma-def 87.4%

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

      2. associate-/l* 87.9%

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

      3. associate-/l* 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in normAngle around 0 96.7%

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

      1. +-commutative 96.7%

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

      2. fma-def 96.7%

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

      3. mul-1-neg 96.7%

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

      4. unsub-neg 96.7%

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

   9. Simplified 96.7%

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

       1. fma-udef 96.7%

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

       2. sub-neg 96.7%

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

       3. mul-1-neg 96.7%

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

       4. add-sqr-sqrt 51.8%

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

       5. sqrt-unprod 89.2%

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

       6. mul-1-neg 89.2%

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

       7. mul-1-neg 89.2%

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

       8. sqr-neg 89.2%

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

       9. sqrt-unprod 38.2%

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

       10. add-sqr-sqrt 83.0%

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

   11. Applied egg-rr 83.0%

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

   if -6.00000016e-27 < n1_i < 5.0000001e-26

   1. Initial program 98.5%

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

   2. Taylor expanded in normAngle around 0 98.3%

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

   3. Taylor expanded in u around 0 98.3%

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

      1. associate-/l* 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in u around 0 93.5%

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

      1. mul-1-neg 93.5%

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

      2. associate-*r/ 98.6%

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

      3. +-commutative 98.6%

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

      4. sub-neg 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in n0_i around inf 89.3%

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

       1. distribute-lft-in 89.5%

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

       2. *-rgt-identity 89.5%

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

       3. mul-1-neg 89.5%

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

       4. distribute-rgt-neg-in 89.5%

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

       5. unsub-neg 89.5%

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

       6. *-commutative 89.5%

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

   11. Simplified 89.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -6.000000163471207 \cdot 10^{-27} \lor \neg \left(n1_i \leq 5.000000097707407 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + u \cdot \left(n0_i + n1_i\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \]

Alternative 5: 69.4% accurate, 45.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 7.199999872485958 \cdot 10^{-22}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -4.999999980020986e-13)
   (* u n1_i)
   (if (<= n1_i 7.199999872485958e-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 (n1_i <= -4.999999980020986e-13f) {
		tmp = u * n1_i;
	} else if (n1_i <= 7.199999872485958e-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 (n1_i <= (-4.999999980020986e-13)) then
        tmp = u * n1_i
    else if (n1_i <= 7.199999872485958e-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 (n1_i <= Float32(-4.999999980020986e-13))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(7.199999872485958e-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 (n1_i <= single(-4.999999980020986e-13))
		tmp = u * n1_i;
	elseif (n1_i <= single(7.199999872485958e-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 n1_i < -4.99999998e-13 or 7.19999987e-22 < n1_i

   1. Initial program 95.6%

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

      1. *-commutative 95.6%

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

      2. associate-*l* 84.0%

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

      3. *-commutative 84.0%

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

      4. associate-*l* 80.7%

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

      5. distribute-lft-out 80.9%

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

      6. associate-*l/ 81.0%

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

      7. *-lft-identity 81.0%

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

      8. fma-def 81.1%

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

      9. *-commutative 81.1%

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

      10. distribute-rgt-out-- 81.2%

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

      11. *-lft-identity 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in normAngle around 0 96.7%

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

   5. Taylor expanded in n0_i around 0 64.9%

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

   if -4.99999998e-13 < n1_i < 7.19999987e-22

   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. *-commutative 97.7%

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

      2. associate-*l* 86.1%

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

      3. *-commutative 86.1%

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

      4. associate-*l* 66.1%

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

      5. distribute-lft-out 66.1%

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

      6. associate-*l/ 66.3%

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

      7. *-lft-identity 66.3%

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

      8. fma-def 66.3%

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

      9. *-commutative 66.3%

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

      10. distribute-rgt-out-- 66.2%

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

      11. *-lft-identity 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in normAngle around 0 97.3%

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

   5. Taylor expanded in n0_i around inf 78.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 7.199999872485958 \cdot 10^{-22}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 6: 69.5% accurate, 45.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 7.199999872485958 \cdot 10^{-22}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -4.999999980020986e-13)
   (* u n1_i)
   (if (<= n1_i 7.199999872485958e-22) (- n0_i (* n0_i u)) (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -4.999999980020986e-13f) {
		tmp = u * n1_i;
	} else if (n1_i <= 7.199999872485958e-22f) {
		tmp = n0_i - (n0_i * 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 (n1_i <= (-4.999999980020986e-13)) then
        tmp = u * n1_i
    else if (n1_i <= 7.199999872485958e-22) then
        tmp = n0_i - (n0_i * 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 (n1_i <= Float32(-4.999999980020986e-13))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(7.199999872485958e-22))
		tmp = Float32(n0_i - Float32(n0_i * 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 (n1_i <= single(-4.999999980020986e-13))
		tmp = u * n1_i;
	elseif (n1_i <= single(7.199999872485958e-22))
		tmp = n0_i - (n0_i * u);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.99999998e-13 or 7.19999987e-22 < n1_i

   1. Initial program 95.6%

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

      1. *-commutative 95.6%

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

      2. associate-*l* 84.0%

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

      3. *-commutative 84.0%

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

      4. associate-*l* 80.7%

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

      5. distribute-lft-out 80.9%

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

      6. associate-*l/ 81.0%

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

      7. *-lft-identity 81.0%

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

      8. fma-def 81.1%

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

      9. *-commutative 81.1%

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

      10. distribute-rgt-out-- 81.2%

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

      11. *-lft-identity 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in normAngle around 0 96.7%

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

   5. Taylor expanded in n0_i around 0 64.9%

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

   if -4.99999998e-13 < n1_i < 7.19999987e-22

   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. Taylor expanded in normAngle around 0 97.3%

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

   3. Taylor expanded in u around 0 98.0%

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

      1. associate-/l* 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in u around 0 89.8%

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

      1. mul-1-neg 89.8%

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

      2. associate-*r/ 98.5%

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

      3. +-commutative 98.5%

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

      4. sub-neg 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in n0_i around inf 78.8%

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

       1. distribute-lft-in 79.0%

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

       2. *-rgt-identity 79.0%

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

       3. mul-1-neg 79.0%

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

       4. distribute-rgt-neg-in 79.0%

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

       5. unsub-neg 79.0%

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

       6. *-commutative 79.0%

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

   11. Simplified 79.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 7.199999872485958 \cdot 10^{-22}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 7: 59.8% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 7.199999872485958 \cdot 10^{-22}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -4.999999980020986e-13)
   (* u n1_i)
   (if (<= n1_i 7.199999872485958e-22) n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -4.999999980020986e-13f) {
		tmp = u * n1_i;
	} else if (n1_i <= 7.199999872485958e-22f) {
		tmp = n0_i;
	} 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 (n1_i <= (-4.999999980020986e-13)) then
        tmp = u * n1_i
    else if (n1_i <= 7.199999872485958e-22) then
        tmp = n0_i
    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 (n1_i <= Float32(-4.999999980020986e-13))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(7.199999872485958e-22))
		tmp = n0_i;
	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 (n1_i <= single(-4.999999980020986e-13))
		tmp = u * n1_i;
	elseif (n1_i <= single(7.199999872485958e-22))
		tmp = n0_i;
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.99999998e-13 or 7.19999987e-22 < n1_i

   1. Initial program 95.6%

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

      1. *-commutative 95.6%

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

      2. associate-*l* 84.0%

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

      3. *-commutative 84.0%

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

      4. associate-*l* 80.7%

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

      5. distribute-lft-out 80.9%

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

      6. associate-*l/ 81.0%

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

      7. *-lft-identity 81.0%

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

      8. fma-def 81.1%

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

      9. *-commutative 81.1%

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

      10. distribute-rgt-out-- 81.2%

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

      11. *-lft-identity 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in normAngle around 0 96.7%

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

   5. Taylor expanded in n0_i around 0 64.9%

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

   if -4.99999998e-13 < n1_i < 7.19999987e-22

   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. *-commutative 97.7%

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

      2. associate-*l* 86.1%

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

      3. *-commutative 86.1%

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

      4. associate-*l* 66.1%

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

      5. distribute-lft-out 66.1%

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

      6. associate-*l/ 66.3%

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

      7. *-lft-identity 66.3%

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

      8. fma-def 66.3%

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

      9. *-commutative 66.3%

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

      10. distribute-rgt-out-- 66.2%

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

      11. *-lft-identity 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in u around 0 86.1%

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

      1. fma-def 86.1%

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

      2. associate-/l* 90.2%

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

      3. associate-/l* 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in u around 0 62.5%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 7.199999872485958 \cdot 10^{-22}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 8: 98.0% 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 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*l* 85.3%

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

   3. *-commutative 85.3%

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

   4. associate-*l* 71.9%

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

   5. distribute-lft-out 72.0%

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

   6. associate-*l/ 72.2%

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

   7. *-lft-identity 72.2%

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

   8. fma-def 72.2%

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

   9. *-commutative 72.2%

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

   10. distribute-rgt-out-- 72.2%

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

   11. *-lft-identity 72.2%

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

3. Simplified 72.2%

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

4. Taylor expanded in normAngle around 0 97.1%

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

5. Taylor expanded in u around -inf 97.3%

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

   1. mul-1-neg 97.3%

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

   2. unsub-neg 97.3%

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

   3. mul-1-neg 97.3%

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

   4. unsub-neg 97.3%

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

7. Simplified 97.3%

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

8. Final simplification 97.3%

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

Alternative 9: 47.5% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*l* 85.3%

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

   3. *-commutative 85.3%

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

   4. associate-*l* 71.9%

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

   5. distribute-lft-out 72.0%

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

   6. associate-*l/ 72.2%

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

   7. *-lft-identity 72.2%

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

   8. fma-def 72.2%

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

   9. *-commutative 72.2%

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

   10. distribute-rgt-out-- 72.2%

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

   11. *-lft-identity 72.2%

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

3. Simplified 72.2%

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

4. Taylor expanded in u around 0 87.4%

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

   1. fma-def 87.4%

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

   2. associate-/l* 90.0%

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

   3. associate-/l* 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in u around 0 47.0%

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

8. Final simplification 47.0%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023297 
(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)))