Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.1%

Time: 12.7s

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: 99.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Taylor expanded in normAngle around 0 97.6%

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

3. Taylor expanded in u around 0 90.8%

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

   1. +-commutative 90.8%

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

   2. mul-1-neg 90.8%

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

   3. unsub-neg 90.8%

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

   4. associate-/l* 99.5%

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

5. Simplified 99.5%

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

   1. sub-neg 99.5%

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

   2. div-inv 99.4%

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

   3. clear-num 99.5%

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

7. Applied egg-rr 99.5%

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

   1. unsub-neg 99.5%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

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

Alternative 2: 98.2% 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 97.4%

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

2. Taylor expanded in normAngle around 0 97.6%

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

3. Taylor expanded in u around 0 90.8%

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

   1. +-commutative 90.8%

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

   2. mul-1-neg 90.8%

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

   3. unsub-neg 90.8%

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

   4. associate-/l* 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in normAngle around 0 98.4%

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

   1. +-commutative 98.4%

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

   2. fma-def 98.6%

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

8. Simplified 98.6%

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

9. Final simplification 98.6%

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

Alternative 3: 67.9% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -2.900000062311392 \cdot 10^{-8} \lor \neg \left(n1_i \leq 1.999999987845058 \cdot 10^{-8}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -2.900000062311392e-8) (not (<= n1_i 1.999999987845058e-8)))
   (* 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 <= -2.900000062311392e-8f) || !(n1_i <= 1.999999987845058e-8f)) {
		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 <= (-2.900000062311392e-8)) .or. (.not. (n1_i <= 1.999999987845058e-8))) 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(-2.900000062311392e-8)) || !(n1_i <= Float32(1.999999987845058e-8)))
		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(-2.900000062311392e-8)) || ~((n1_i <= single(1.999999987845058e-8))))
		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 < -2.90000006e-8 or 1.99999999e-8 < n1_i

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-*l* 95.4%

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

      3. *-commutative 95.4%

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

      4. associate-*l* 92.6%

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

      5. distribute-lft-out 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in normAngle around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. *-commutative 98.6%

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

      3. fma-def 98.9%

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

      4. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in n1_i around inf 74.6%

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

      1. *-commutative 74.6%

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

   9. Simplified 74.6%

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

   if -2.90000006e-8 < n1_i < 1.99999999e-8

   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. Taylor expanded in normAngle around 0 97.8%

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

   3. Taylor expanded in u around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

      4. associate-/l* 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in n0_i around inf 70.6%

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

      1. mul-1-neg 70.6%

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

      2. sub-neg 70.6%

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

   8. Simplified 70.6%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -2.900000062311392 \cdot 10^{-8} \lor \neg \left(n1_i \leq 1.999999987845058 \cdot 10^{-8}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 4: 68.1% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -2.900000062311392 \cdot 10^{-8} \lor \neg \left(n1_i \leq 1.999999987845058 \cdot 10^{-8}\right):\\ \;\;\;\;u \cdot \left(n0_i + n1_i\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -2.900000062311392e-8) (not (<= n1_i 1.999999987845058e-8)))
   (* u (+ n0_i 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 <= -2.900000062311392e-8f) || !(n1_i <= 1.999999987845058e-8f)) {
		tmp = u * (n0_i + 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 <= (-2.900000062311392e-8)) .or. (.not. (n1_i <= 1.999999987845058e-8))) then
        tmp = u * (n0_i + 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(-2.900000062311392e-8)) || !(n1_i <= Float32(1.999999987845058e-8)))
		tmp = Float32(u * Float32(n0_i + 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(-2.900000062311392e-8)) || ~((n1_i <= single(1.999999987845058e-8))))
		tmp = u * (n0_i + 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 < -2.90000006e-8 or 1.99999999e-8 < n1_i

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-*l* 95.4%

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

      3. *-commutative 95.4%

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

      4. associate-*l* 92.6%

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

      5. distribute-lft-out 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in normAngle around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. *-commutative 98.6%

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

      3. fma-def 98.9%

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

      4. *-commutative 98.9%

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

   6. Simplified 98.9%

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

      1. fma-udef 98.6%

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

      2. *-commutative 98.6%

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

      3. *-commutative 98.6%

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

      4. +-commutative 98.6%

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

      5. *-commutative 98.6%

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

      6. sub-neg 98.6%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod 93.4%

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

      9. sqr-neg 93.4%

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

      10. sqrt-unprod 93.4%

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

      11. add-sqr-sqrt 93.4%

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

      12. *-commutative 93.4%

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

   8. Applied egg-rr 93.4%

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

   9. Taylor expanded in u around inf 75.3%

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

       1. +-commutative 75.3%

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

   11. Simplified 75.3%

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

   if -2.90000006e-8 < n1_i < 1.99999999e-8

   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. Taylor expanded in normAngle around 0 97.8%

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

   3. Taylor expanded in u around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

      4. associate-/l* 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in n0_i around inf 70.6%

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

      1. mul-1-neg 70.6%

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

      2. sub-neg 70.6%

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

   8. Simplified 70.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -2.900000062311392 \cdot 10^{-8} \lor \neg \left(n1_i \leq 1.999999987845058 \cdot 10^{-8}\right):\\ \;\;\;\;u \cdot \left(n0_i + n1_i\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 5: 61.0% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 7.99999974612418 \cdot 10^{-19}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -9.999999998199587e-24)
   n0_i
   (if (<= n0_i 7.99999974612418e-19) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -9.999999998199587e-24f) {
		tmp = n0_i;
	} else if (n0_i <= 7.99999974612418e-19f) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-9.999999998199587e-24)) then
        tmp = n0_i
    else if (n0_i <= 7.99999974612418e-19) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-9.999999998199587e-24))
		tmp = n0_i;
	elseif (n0_i <= Float32(7.99999974612418e-19))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-9.999999998199587e-24))
		tmp = n0_i;
	elseif (n0_i <= single(7.99999974612418e-19))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1e-23 or 7.99999975e-19 < n0_i

   1. Initial program 97.3%

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

   2. Taylor expanded in normAngle around 0 97.5%

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

   3. Taylor expanded in u around 0 97.7%

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

      1. +-commutative 97.7%

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

      2. mul-1-neg 97.7%

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

      3. unsub-neg 97.7%

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

      4. associate-/l* 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in u around 0 60.2%

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

   if -1e-23 < n0_i < 7.99999975e-19

   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* 75.8%

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

      3. *-commutative 75.8%

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

      4. associate-*l* 60.5%

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

      5. distribute-lft-out 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in normAngle around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. *-commutative 98.3%

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

      3. fma-def 98.3%

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

      4. *-commutative 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in n1_i around inf 60.9%

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

      1. *-commutative 60.9%

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

   9. Simplified 60.9%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 7.99999974612418 \cdot 10^{-19}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 6: 98.1% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 81.4%

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

   3. *-commutative 81.4%

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

   4. associate-*l* 74.7%

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

   5. distribute-lft-out 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in normAngle around 0 98.4%

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

   1. +-commutative 98.4%

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

   2. *-commutative 98.4%

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

   3. fma-def 98.5%

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

   4. *-commutative 98.5%

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

6. Simplified 98.5%

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

7. Taylor expanded in u around -inf 98.4%

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

   1. mul-1-neg 98.4%

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

   2. unsub-neg 98.4%

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

   3. neg-mul-1 98.4%

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

   4. unsub-neg 98.4%

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

9. Simplified 98.4%

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

10. Final simplification 98.4%

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

Alternative 7: 82.2% accurate, 84.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 81.4%

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

   3. *-commutative 81.4%

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

   4. associate-*l* 74.7%

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

   5. distribute-lft-out 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in normAngle around 0 98.4%

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

   1. +-commutative 98.4%

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

   2. *-commutative 98.4%

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

   3. fma-def 98.5%

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

   4. *-commutative 98.5%

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

6. Simplified 98.5%

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

   1. fma-udef 98.4%

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

   2. *-commutative 98.4%

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

   3. *-commutative 98.4%

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

   4. +-commutative 98.4%

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

   5. *-commutative 98.4%

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

   6. sub-neg 98.4%

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

   7. add-sqr-sqrt -0.0%

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

   8. sqrt-unprod 79.9%

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

   9. sqr-neg 79.9%

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

   10. sqrt-unprod 79.9%

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

   11. add-sqr-sqrt 79.9%

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

   12. *-commutative 79.9%

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

8. Applied egg-rr 79.9%

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

9. Taylor expanded in u around 0 79.9%

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

    1. +-commutative 79.9%

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

11. Simplified 79.9%

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

12. Taylor expanded in n1_i around inf 81.3%

    \[\leadsto n0_i + \color{blue}{n1_i \cdot u} \]
13. Step-by-step derivation

    1. *-commutative 81.3%

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

14. Simplified 81.3%

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

15. Final simplification 81.3%

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

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 97.4%

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

2. Taylor expanded in normAngle around 0 97.6%

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

3. Taylor expanded in u around 0 90.8%

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

   1. +-commutative 90.8%

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

   2. mul-1-neg 90.8%

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

   3. unsub-neg 90.8%

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

   4. associate-/l* 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in u around 0 47.6%

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

7. Final simplification 47.6%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023313 
(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)))