Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 98.1%

Time: 13.1s

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: 97.3% 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: 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 97.6%

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

   1. fma-def 97.6%

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

   2. associate-*r/ 98.0%

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

   3. *-rgt-identity 98.0%

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

   4. associate-*r/ 98.1%

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

   5. *-rgt-identity 98.1%

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

3. Simplified 98.1%

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

4. Taylor expanded in normAngle around 0 98.3%

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

5. Taylor expanded in u around 0 98.6%

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

   1. *-commutative 98.6%

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

   2. fma-def 98.8%

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

   3. mul-1-neg 98.8%

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

   4. unsub-neg 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 2: 70.6% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -4.0000001089808046 \cdot 10^{-27} \lor \neg \left(n0_i \leq 5.000000229068525 \cdot 10^{-19}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -4.0000001089808046e-27)
         (not (<= n0_i 5.000000229068525e-19)))
   (* n0_i (- 1.0 u))
   (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -4.0000001089808046e-27f) || !(n0_i <= 5.000000229068525e-19f)) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = u * n1_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n0_i <= (-4.0000001089808046e-27)) .or. (.not. (n0_i <= 5.000000229068525e-19))) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-4.0000001089808046e-27)) || !(n0_i <= Float32(5.000000229068525e-19)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-4.0000001089808046e-27)) || ~((n0_i <= single(5.000000229068525e-19))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.00000011e-27 or 5.00000023e-19 < n0_i

   1. Initial program 97.2%

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

      1. fma-def 97.2%

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

      2. associate-*r/ 97.7%

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

      3. *-rgt-identity 97.7%

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

      4. associate-*r/ 97.8%

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

      5. *-rgt-identity 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in normAngle around 0 98.9%

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

   5. Taylor expanded in n1_i around 0 75.4%

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

   if -4.00000011e-27 < n0_i < 5.00000023e-19

   1. Initial program 98.1%

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

      1. fma-def 98.2%

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

      2. associate-*r/ 98.4%

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

      3. *-rgt-identity 98.4%

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

      4. associate-*r/ 98.5%

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

      5. *-rgt-identity 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in normAngle around 0 97.8%

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

   5. Taylor expanded in u around inf 68.4%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -4.0000001089808046 \cdot 10^{-27} \lor \neg \left(n0_i \leq 5.000000229068525 \cdot 10^{-19}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 3: 70.7% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -4.0000001089808046 \cdot 10^{-27} \lor \neg \left(n0_i \leq 5.000000229068525 \cdot 10^{-19}\right):\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -4.0000001089808046e-27)
         (not (<= n0_i 5.000000229068525e-19)))
   (- n0_i (* u n0_i))
   (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -4.0000001089808046e-27f) || !(n0_i <= 5.000000229068525e-19f)) {
		tmp = n0_i - (u * 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 ((n0_i <= (-4.0000001089808046e-27)) .or. (.not. (n0_i <= 5.000000229068525e-19))) then
        tmp = n0_i - (u * 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 ((n0_i <= Float32(-4.0000001089808046e-27)) || !(n0_i <= Float32(5.000000229068525e-19)))
		tmp = Float32(n0_i - Float32(u * 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 ((n0_i <= single(-4.0000001089808046e-27)) || ~((n0_i <= single(5.000000229068525e-19))))
		tmp = n0_i - (u * n0_i);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.00000011e-27 or 5.00000023e-19 < n0_i

   1. Initial program 97.2%

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

      1. +-commutative 97.2%

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

      2. *-commutative 97.2%

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

      3. associate-*r* 92.7%

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

      4. *-commutative 92.7%

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

      5. associate-*r* 81.7%

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

      6. distribute-rgt-out 81.7%

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

      7. *-commutative 81.7%

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

      8. associate-*r/ 81.8%

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

      9. associate-/l* 81.8%

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

      10. *-commutative 81.8%

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

      11. fma-def 81.9%

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

      12. *-commutative 81.9%

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

      13. /-rgt-identity 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in normAngle around 0 80.6%

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

      1. *-commutative 80.6%

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

      2. *-commutative 80.6%

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

      3. fma-def 80.7%

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

   6. Simplified 80.7%

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

   7. Taylor expanded in u around -inf 80.4%

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

      1. mul-1-neg 80.4%

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

      2. unsub-neg 80.4%

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

      3. associate-*r* 80.5%

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

      4. *-commutative 80.5%

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

      5. +-commutative 80.5%

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

      6. mul-1-neg 80.5%

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

      7. unsub-neg 80.5%

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

   9. Simplified 80.5%

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

   10. Taylor expanded in normAngle around 0 99.2%

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

   11. Taylor expanded in n0_i around inf 75.4%

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

       1. *-commutative 75.4%

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

       2. distribute-rgt-out-- 75.7%

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

       3. *-lft-identity 75.7%

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

   13. Simplified 75.7%

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

   if -4.00000011e-27 < n0_i < 5.00000023e-19

   1. Initial program 98.1%

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

      1. fma-def 98.2%

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

      2. associate-*r/ 98.4%

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

      3. *-rgt-identity 98.4%

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

      4. associate-*r/ 98.5%

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

      5. *-rgt-identity 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in normAngle around 0 97.8%

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

   5. Taylor expanded in u around inf 68.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -4.0000001089808046 \cdot 10^{-27} \lor \neg \left(n0_i \leq 5.000000229068525 \cdot 10^{-19}\right):\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 4: 98.0% accurate, 46.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. +-commutative 97.6%

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

   2. *-commutative 97.6%

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

   3. associate-*r* 87.6%

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

   4. *-commutative 87.6%

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

   5. associate-*r* 76.9%

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

   6. distribute-rgt-out 77.0%

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

   7. *-commutative 77.0%

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

   8. associate-*r/ 77.1%

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

   9. associate-/l* 77.1%

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

   10. *-commutative 77.1%

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

   11. fma-def 77.2%

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

   12. *-commutative 77.2%

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

   13. /-rgt-identity 77.2%

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

3. Simplified 77.2%

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

4. Taylor expanded in normAngle around 0 75.7%

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

   1. *-commutative 75.7%

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

   2. *-commutative 75.7%

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

   3. fma-def 75.8%

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

6. Simplified 75.8%

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

7. Taylor expanded in u around -inf 75.5%

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

   1. mul-1-neg 75.5%

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

   2. unsub-neg 75.5%

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

   3. associate-*r* 75.6%

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

   4. *-commutative 75.6%

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

   5. +-commutative 75.6%

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

   6. mul-1-neg 75.6%

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

   7. unsub-neg 75.6%

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

9. Simplified 75.6%

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

10. Taylor expanded in normAngle around 0 98.6%

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

    1. *-commutative 98.6%

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

    2. sub-neg 98.6%

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

    3. distribute-lft-in 98.6%

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

12. Applied egg-rr 98.6%

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

13. Final simplification 98.6%

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

Alternative 5: 60.2% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 3.550000049935294 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -9.99999983775159e-18)
   (* u n1_i)
   (if (<= n1_i 3.550000049935294e-16) n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -9.99999983775159e-18f) {
		tmp = u * n1_i;
	} else if (n1_i <= 3.550000049935294e-16f) {
		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 <= (-9.99999983775159e-18)) then
        tmp = u * n1_i
    else if (n1_i <= 3.550000049935294e-16) 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(-9.99999983775159e-18))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(3.550000049935294e-16))
		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(-9.99999983775159e-18))
		tmp = u * n1_i;
	elseif (n1_i <= single(3.550000049935294e-16))
		tmp = n0_i;
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -9.99999984e-18 or 3.55000005e-16 < n1_i

   1. Initial program 96.5%

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

      1. fma-def 96.5%

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

      2. associate-*r/ 96.7%

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

      3. *-rgt-identity 96.7%

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

      4. associate-*r/ 97.0%

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

      5. *-rgt-identity 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in normAngle around 0 98.5%

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

   5. Taylor expanded in u around inf 65.2%

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

   if -9.99999984e-18 < n1_i < 3.55000005e-16

   1. Initial program 98.6%

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

      1. fma-def 98.6%

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

      2. associate-*r/ 99.1%

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

      3. *-rgt-identity 99.1%

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

      4. associate-*r/ 99.0%

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

      5. *-rgt-identity 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in u around 0 62.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 3.550000049935294 \cdot 10^{-16}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 6: 83.6% accurate, 59.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -4.999999858590343e-10) (- n0_i (* u n0_i)) (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -4.999999858590343e-10f) {
		tmp = n0_i - (u * n0_i);
	} else {
		tmp = n0_i + (u * n1_i);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-4.999999858590343e-10)) then
        tmp = n0_i - (u * n0_i)
    else
        tmp = n0_i + (u * n1_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-4.999999858590343e-10))
		tmp = Float32(n0_i - Float32(u * n0_i));
	else
		tmp = Float32(n0_i + Float32(u * n1_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-4.999999858590343e-10))
		tmp = n0_i - (u * n0_i);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.99999986e-10

   1. Initial program 98.6%

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

      1. +-commutative 98.6%

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

      2. *-commutative 98.6%

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

      3. associate-*r* 96.9%

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

      4. *-commutative 96.9%

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

      5. associate-*r* 89.9%

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

      6. distribute-rgt-out 89.8%

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

      7. *-commutative 89.8%

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

      8. associate-*r/ 89.8%

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

      9. associate-/l* 89.8%

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

      10. *-commutative 89.8%

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

      11. fma-def 89.9%

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

      12. *-commutative 89.9%

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

      13. /-rgt-identity 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in normAngle around 0 90.1%

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

      1. *-commutative 90.1%

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

      2. *-commutative 90.1%

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

      3. fma-def 90.1%

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

   6. Simplified 90.1%

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

   7. Taylor expanded in u around -inf 90.0%

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

      1. mul-1-neg 90.0%

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

      2. unsub-neg 90.0%

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

      3. associate-*r* 89.9%

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

      4. *-commutative 89.9%

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

      5. +-commutative 89.9%

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

      6. mul-1-neg 89.9%

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

      7. unsub-neg 89.9%

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

   9. Simplified 89.9%

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

   10. Taylor expanded in normAngle around 0 99.7%

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

   11. Taylor expanded in n0_i around inf 89.5%

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

       1. *-commutative 89.5%

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

       2. distribute-rgt-out-- 89.7%

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

       3. *-lft-identity 89.7%

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

   13. Simplified 89.7%

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

   if -4.99999986e-10 < n0_i

   1. Initial program 97.4%

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

      1. +-commutative 97.4%

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

      2. *-commutative 97.4%

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

      3. associate-*r* 85.8%

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

      4. *-commutative 85.8%

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

      5. associate-*r* 74.5%

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

      6. distribute-rgt-out 74.6%

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

      7. *-commutative 74.6%

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

      8. associate-*r/ 74.7%

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

      9. associate-/l* 74.7%

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

      10. *-commutative 74.7%

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

      11. fma-def 74.7%

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

      12. *-commutative 74.7%

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

      13. /-rgt-identity 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in normAngle around 0 72.9%

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

      1. *-commutative 72.9%

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

      2. *-commutative 72.9%

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

      3. fma-def 73.0%

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

   6. Simplified 73.0%

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

   7. Taylor expanded in u around -inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. unsub-neg 72.7%

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

      3. associate-*r* 72.8%

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

      4. *-commutative 72.8%

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

      5. +-commutative 72.8%

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

      6. mul-1-neg 72.8%

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

      7. unsub-neg 72.8%

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

   9. Simplified 72.8%

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

   10. Taylor expanded in normAngle around 0 98.4%

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

   11. Taylor expanded in n0_i around 0 86.7%

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

       1. mul-1-neg 86.7%

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

       2. distribute-lft-neg-out 86.7%

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

       3. *-commutative 86.7%

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

   13. Simplified 86.7%

       \[\leadsto n0_i - \color{blue}{u \cdot \left(-n1_i\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \]

Alternative 7: 98.0% 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 97.6%

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

   1. fma-def 97.6%

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

   2. associate-*r/ 98.0%

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

   3. *-rgt-identity 98.0%

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

   4. associate-*r/ 98.1%

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

   5. *-rgt-identity 98.1%

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

3. Simplified 98.1%

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

4. Taylor expanded in normAngle around 0 98.3%

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

5. Taylor expanded in u around -inf 98.6%

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

   1. +-commutative 98.6%

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

   2. mul-1-neg 98.6%

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

   3. unsub-neg 98.6%

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

   4. +-commutative 98.6%

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

   5. mul-1-neg 98.6%

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

   6. unsub-neg 98.6%

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

7. Simplified 98.6%

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

8. Final simplification 98.6%

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

Alternative 8: 47.7% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. fma-def 97.6%

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

   2. associate-*r/ 98.0%

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

   3. *-rgt-identity 98.0%

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

   4. associate-*r/ 98.1%

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

   5. *-rgt-identity 98.1%

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

3. Simplified 98.1%

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

4. Taylor expanded in u around 0 45.8%

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

5. Final simplification 45.8%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023255 
(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)))