Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.0% → 98.1%

Time: 17.2s

Alternatives: 8

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: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.1% 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 97.7%

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

   1. *-commutative 97.7%

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

   2. associate-*l* 82.4%

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

   3. *-commutative 82.4%

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

   4. associate-*l* 75.9%

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

   5. distribute-lft-out 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in normAngle around 0 98.3%

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

   1. +-commutative 98.3%

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

   2. fma-def 98.5%

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

   3. *-commutative 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in n1_i around 0 98.3%

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

   1. sub-neg 98.3%

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

   2. distribute-lft-out 98.5%

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

   3. *-rgt-identity 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-+l+ 98.7%

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

   6. *-commutative 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. distribute-rgt-neg-in 98.7%

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

   9. distribute-lft-in 98.7%

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

   10. +-commutative 98.7%

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

   11. neg-mul-1 98.7%

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

   12. +-commutative 98.7%

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

   13. fma-def 98.8%

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

   14. neg-mul-1 98.8%

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

   15. unsub-neg 98.8%

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

10. Simplified 98.8%

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

11. Final simplification 98.8%

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

Alternative 2: 69.5% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -2.100000078344988 \cdot 10^{-12} \lor \neg \left(n1_i \leq 1.000000013351432 \cdot 10^{-10}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -2.100000078344988e-12)
         (not (<= n1_i 1.000000013351432e-10)))
   (* u n1_i)
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -2.100000078344988e-12f) || !(n1_i <= 1.000000013351432e-10f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-2.100000078344988e-12)) .or. (.not. (n1_i <= 1.000000013351432e-10))) then
        tmp = u * n1_i
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-2.100000078344988e-12)) || !(n1_i <= Float32(1.000000013351432e-10)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-2.100000078344988e-12)) || ~((n1_i <= single(1.000000013351432e-10))))
		tmp = u * n1_i;
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -2.10000008e-12 or 1.00000001e-10 < n1_i

   1. Initial program 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*l* 95.6%

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

      3. *-commutative 95.6%

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

      4. associate-*l* 87.1%

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

      5. distribute-lft-out 87.0%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in normAngle around 0 97.1%

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

      1. +-commutative 97.1%

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

      2. fma-def 97.3%

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

      3. *-commutative 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in n1_i around inf 67.6%

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

      1. *-commutative 67.6%

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

   10. Simplified 67.6%

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

   if -2.10000008e-12 < n1_i < 1.00000001e-10

   1. Initial program 97.6%

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

      1. *-commutative 97.6%

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

      2. associate-*l* 77.5%

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

      3. *-commutative 77.5%

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

      4. associate-*l* 71.7%

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

      5. distribute-lft-out 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in normAngle around 0 98.8%

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

      1. +-commutative 98.8%

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

      2. fma-def 98.9%

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

      3. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in n1_i around 0 98.8%

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

      1. sub-neg 98.8%

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

      2. distribute-lft-out 99.1%

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

      3. *-rgt-identity 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-+l+ 99.2%

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

      6. *-commutative 99.2%

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

      7. distribute-lft-neg-out 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-lft-in 99.2%

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

      10. +-commutative 99.2%

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

      11. neg-mul-1 99.2%

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

      12. +-commutative 99.2%

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

      13. fma-def 99.3%

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

      14. neg-mul-1 99.3%

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

      15. unsub-neg 99.3%

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

   10. Simplified 99.3%

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

   11. Taylor expanded in n1_i around 0 77.4%

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

       1. *-rgt-identity 77.4%

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

       2. mul-1-neg 77.4%

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

       3. distribute-rgt-neg-in 77.4%

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

       4. mul-1-neg 77.4%

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

       5. distribute-lft-in 77.2%

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

       6. mul-1-neg 77.2%

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

       7. sub-neg 77.2%

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

   13. Simplified 77.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -2.100000078344988 \cdot 10^{-12} \lor \neg \left(n1_i \leq 1.000000013351432 \cdot 10^{-10}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -4.0000001089808046 \cdot 10^{-27} \lor \neg \left(n1_i \leq 2.5000000488537034 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -4.0000001089808046e-27)
         (not (<= n1_i 2.5000000488537034e-26)))
   (+ n0_i (* u n1_i))
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -4.0000001089808046e-27f) || !(n1_i <= 2.5000000488537034e-26f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-4.0000001089808046e-27)) .or. (.not. (n1_i <= 2.5000000488537034e-26))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-4.0000001089808046e-27)) || !(n1_i <= Float32(2.5000000488537034e-26)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-4.0000001089808046e-27)) || ~((n1_i <= single(2.5000000488537034e-26))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.00000011e-27 or 2.50000005e-26 < n1_i

   1. Initial program 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-*l* 88.4%

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

      3. *-commutative 88.4%

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

      4. associate-*l* 78.6%

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

      5. distribute-lft-out 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in normAngle around 0 98.0%

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

      1. +-commutative 98.0%

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

      2. fma-def 98.2%

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

      3. *-commutative 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in n1_i around 0 98.0%

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

      1. sub-neg 98.0%

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

      2. distribute-lft-out 98.2%

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

      3. *-rgt-identity 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-+l+ 98.3%

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

      6. *-commutative 98.3%

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

      7. distribute-lft-neg-out 98.3%

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

      8. distribute-rgt-neg-in 98.3%

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

      9. distribute-lft-in 98.2%

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

      10. +-commutative 98.2%

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

      11. neg-mul-1 98.2%

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

      12. +-commutative 98.2%

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

      13. fma-def 98.4%

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

      14. neg-mul-1 98.4%

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

      15. unsub-neg 98.4%

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

   10. Simplified 98.4%

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

   11. Taylor expanded in n1_i around 0 98.3%

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

   12. Taylor expanded in n0_i around 0 86.0%

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

   if -4.00000011e-27 < n1_i < 2.50000005e-26

   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%

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

      2. associate-*l* 71.1%

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

      3. *-commutative 71.1%

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

      4. associate-*l* 70.8%

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

      5. distribute-lft-out 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in normAngle around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. fma-def 99.0%

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

      3. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in n1_i around 0 99.0%

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

      1. sub-neg 99.0%

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

      2. distribute-lft-out 99.3%

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

      3. *-rgt-identity 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-+l+ 99.4%

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

      6. *-commutative 99.4%

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

      7. distribute-lft-neg-out 99.4%

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

      8. distribute-rgt-neg-in 99.4%

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

      9. distribute-lft-in 99.4%

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

      10. +-commutative 99.4%

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

      11. neg-mul-1 99.4%

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

      12. +-commutative 99.4%

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

      13. fma-def 99.4%

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

      14. neg-mul-1 99.4%

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

      15. unsub-neg 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in n1_i around 0 94.5%

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

       1. *-rgt-identity 94.5%

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

       2. mul-1-neg 94.5%

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

       3. distribute-rgt-neg-in 94.5%

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

       4. mul-1-neg 94.5%

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

       5. distribute-lft-in 94.3%

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

       6. mul-1-neg 94.3%

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

       7. sub-neg 94.3%

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

   13. Simplified 94.3%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -4.0000001089808046 \cdot 10^{-27} \lor \neg \left(n1_i \leq 2.5000000488537034 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -4.0000001089808046 \cdot 10^{-27} \lor \neg \left(n1_i \leq 2.5000000488537034 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -4.0000001089808046e-27)
         (not (<= n1_i 2.5000000488537034e-26)))
   (+ n0_i (* u n1_i))
   (- n0_i (* u n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -4.0000001089808046e-27f) || !(n1_i <= 2.5000000488537034e-26f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i - (u * n0_i);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-4.0000001089808046e-27)) .or. (.not. (n1_i <= 2.5000000488537034e-26))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i - (u * n0_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-4.0000001089808046e-27)) || !(n1_i <= Float32(2.5000000488537034e-26)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i - Float32(u * n0_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-4.0000001089808046e-27)) || ~((n1_i <= single(2.5000000488537034e-26))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i - (u * n0_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.00000011e-27 or 2.50000005e-26 < n1_i

   1. Initial program 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-*l* 88.4%

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

      3. *-commutative 88.4%

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

      4. associate-*l* 78.6%

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

      5. distribute-lft-out 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in normAngle around 0 98.0%

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

      1. +-commutative 98.0%

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

      2. fma-def 98.2%

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

      3. *-commutative 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in n1_i around 0 98.0%

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

      1. sub-neg 98.0%

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

      2. distribute-lft-out 98.2%

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

      3. *-rgt-identity 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-+l+ 98.3%

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

      6. *-commutative 98.3%

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

      7. distribute-lft-neg-out 98.3%

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

      8. distribute-rgt-neg-in 98.3%

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

      9. distribute-lft-in 98.2%

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

      10. +-commutative 98.2%

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

      11. neg-mul-1 98.2%

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

      12. +-commutative 98.2%

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

      13. fma-def 98.4%

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

      14. neg-mul-1 98.4%

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

      15. unsub-neg 98.4%

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

   10. Simplified 98.4%

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

   11. Taylor expanded in n1_i around 0 98.3%

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

   12. Taylor expanded in n0_i around 0 86.0%

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

   if -4.00000011e-27 < n1_i < 2.50000005e-26

   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%

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

      2. associate-*l* 71.1%

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

      3. *-commutative 71.1%

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

      4. associate-*l* 70.8%

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

      5. distribute-lft-out 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in normAngle around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. fma-def 99.0%

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

      3. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in n1_i around 0 99.0%

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

      1. sub-neg 99.0%

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

      2. distribute-lft-out 99.3%

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

      3. *-rgt-identity 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-+l+ 99.4%

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

      6. *-commutative 99.4%

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

      7. distribute-lft-neg-out 99.4%

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

      8. distribute-rgt-neg-in 99.4%

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

      9. distribute-lft-in 99.4%

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

      10. +-commutative 99.4%

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

      11. neg-mul-1 99.4%

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

      12. +-commutative 99.4%

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

      13. fma-def 99.4%

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

      14. neg-mul-1 99.4%

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

      15. unsub-neg 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in n1_i around 0 99.4%

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

   12. Taylor expanded in n0_i around inf 94.3%

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

       1. mul-1-neg 94.3%

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

       2. unsub-neg 94.3%

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

       3. distribute-lft-out-- 94.5%

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

       4. *-rgt-identity 94.5%

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

       5. *-commutative 94.5%

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

   14. Simplified 94.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -4.0000001089808046 \cdot 10^{-27} \lor \neg \left(n1_i \leq 2.5000000488537034 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.0% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 2.0000000063421537 \cdot 10^{-28}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -2.00000006274879e-22)
   n0_i
   (if (<= n0_i 2.0000000063421537e-28) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -2.00000006274879e-22f) {
		tmp = n0_i;
	} else if (n0_i <= 2.0000000063421537e-28f) {
		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 <= (-2.00000006274879e-22)) then
        tmp = n0_i
    else if (n0_i <= 2.0000000063421537e-28) 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(-2.00000006274879e-22))
		tmp = n0_i;
	elseif (n0_i <= Float32(2.0000000063421537e-28))
		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(-2.00000006274879e-22))
		tmp = n0_i;
	elseif (n0_i <= single(2.0000000063421537e-28))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -2.00000006e-22 or 2.00000001e-28 < n0_i

   1. Initial program 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*l* 85.0%

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

      3. *-commutative 85.0%

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

      4. associate-*l* 83.7%

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

      5. distribute-lft-out 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in u around 0 63.4%

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

   if -2.00000006e-22 < n0_i < 2.00000001e-28

   1. Initial program 97.2%

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

      1. *-commutative 97.2%

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

      2. associate-*l* 77.6%

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

      3. *-commutative 77.6%

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

      4. associate-*l* 61.2%

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

      5. distribute-lft-out 61.2%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in normAngle around 0 97.7%

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

      1. +-commutative 97.7%

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

      2. fma-def 97.9%

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

      3. *-commutative 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in n1_i around inf 64.3%

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

      1. *-commutative 64.3%

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

   10. Simplified 64.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 2.0000000063421537 \cdot 10^{-28}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 46.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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%

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

   2. associate-*l* 82.4%

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

   3. *-commutative 82.4%

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

   4. associate-*l* 75.9%

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

   5. distribute-lft-out 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in normAngle around 0 98.3%

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

   1. +-commutative 98.3%

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

   2. fma-def 98.5%

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

   3. *-commutative 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in n1_i around 0 98.3%

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

   1. sub-neg 98.3%

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

   2. distribute-lft-out 98.5%

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

   3. *-rgt-identity 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-+l+ 98.7%

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

   6. *-commutative 98.7%

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

   7. distribute-lft-neg-out 98.7%

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

   8. distribute-rgt-neg-in 98.7%

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

   9. distribute-lft-in 98.7%

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

   10. +-commutative 98.7%

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

   11. neg-mul-1 98.7%

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

   12. +-commutative 98.7%

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

   13. fma-def 98.8%

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

   14. neg-mul-1 98.8%

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

   15. unsub-neg 98.8%

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

10. Simplified 98.8%

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

11. Taylor expanded in n1_i around 0 98.7%

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

    1. +-commutative 98.7%

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

    2. mul-1-neg 98.7%

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

    3. unsub-neg 98.7%

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

    4. *-commutative 98.7%

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

13. Applied egg-rr 98.7%

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

14. Final simplification 98.7%

    \[\leadsto n0_i + \left(u \cdot n1_i - u \cdot n0_i\right) \]
15. Add Preprocessing

Alternative 7: 97.9% accurate, 60.1× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i - (u * (n0_i - 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 - n1_i))
end function

Julia

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

MATLAB

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

Derivation

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%

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

   2. associate-*l* 82.4%

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

   3. *-commutative 82.4%

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

   4. associate-*l* 75.9%

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

   5. distribute-lft-out 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in normAngle around 0 98.3%

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

   1. +-commutative 98.3%

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

   2. fma-def 98.5%

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

   3. *-commutative 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in u around -inf 98.7%

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

   1. mul-1-neg 98.7%

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

   2. unsub-neg 98.7%

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

   3. mul-1-neg 98.7%

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

   4. unsub-neg 98.7%

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

10. Simplified 98.7%

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

11. Final simplification 98.7%

    \[\leadsto n0_i - u \cdot \left(n0_i - n1_i\right) \]
12. Add Preprocessing

Alternative 8: 46.6% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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%

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

   2. associate-*l* 82.4%

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

   3. *-commutative 82.4%

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

   4. associate-*l* 75.9%

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

   5. distribute-lft-out 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in u around 0 51.8%

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

6. Final simplification 51.8%

   \[\leadsto n0_i \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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)))