Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 98.1%

Time: 14.7s

Alternatives: 6

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, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(normAngle \cdot \left(1 - u\right)\right)}{\sin normAngle} \cdot n0\_i + u \cdot n1\_i \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((sinf((normAngle * (1.0f - u))) / sinf(normAngle)) * 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 = ((sin((normangle * (1.0e0 - u))) / sin(normangle)) * n0_i) + (u * n1_i)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.4%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Add Preprocessing

3. Taylor expanded in normAngle around 0 97.9%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{u} \cdot n1\_i \]

4. Taylor expanded in u around inf 98.3%

   \[\leadsto \color{blue}{\frac{\sin \left(normAngle \cdot \left(1 - u\right)\right)}{\sin normAngle}} \cdot n0\_i + u \cdot n1\_i \]
5. Add Preprocessing

Alternative 2: 69.3% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -7.000000383309403 \cdot 10^{-27} \lor \neg \left(n0\_i \leq 3.999999935100636 \cdot 10^{-17}\right):\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -7.000000383309403e-27)
         (not (<= n0_i 3.999999935100636e-17)))
   (* (- 1.0 u) n0_i)
   (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -7.000000383309403e-27f) || !(n0_i <= 3.999999935100636e-17f)) {
		tmp = (1.0f - 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 <= (-7.000000383309403e-27)) .or. (.not. (n0_i <= 3.999999935100636e-17))) then
        tmp = (1.0e0 - 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(-7.000000383309403e-27)) || !(n0_i <= Float32(3.999999935100636e-17)))
		tmp = Float32(Float32(Float32(1.0) - 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(-7.000000383309403e-27)) || ~((n0_i <= single(3.999999935100636e-17))))
		tmp = (single(1.0) - u) * n0_i;
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -7.00000038e-27 or 3.99999994e-17 < 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%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

      2. associate-*l* 84.5%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

      3. *-commutative 84.5%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1\_i \]

      4. associate-*l* 83.3%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

      5. distribute-lft-out 83.3%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in normAngle around 0 98.8%

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]

   6. Taylor expanded in u around 0 99.0%

      \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + -1 \cdot n0\_i\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.0%

         \[\leadsto n0\_i + u \cdot \left(n1\_i + \color{blue}{\left(-n0\_i\right)}\right) \]

      2. unsub-neg 99.0%

         \[\leadsto n0\_i + u \cdot \color{blue}{\left(n1\_i - n0\_i\right)} \]

   8. Simplified 99.0%

      \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i - n0\_i\right)} \]

   9. Taylor expanded in n0_i around inf 78.9%

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 + -1 \cdot u\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 78.9%

          \[\leadsto n0\_i \cdot \left(1 + \color{blue}{\left(-u\right)}\right) \]

       2. sub-neg 78.9%

          \[\leadsto n0\_i \cdot \color{blue}{\left(1 - u\right)} \]

   11. Simplified 78.9%

       \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]

   if -7.00000038e-27 < n0_i < 3.99999994e-17

   1. Initial program 95.6%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   2. Step-by-step derivation

      1. *-commutative 95.6%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

      2. associate-*l* 75.4%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

      3. *-commutative 75.4%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1\_i \]

      4. associate-*l* 59.8%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

      5. distribute-lft-out 59.7%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

   3. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in normAngle around 0 97.4%

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]

   6. Taylor expanded in n0_i around 0 65.0%

      \[\leadsto \color{blue}{n1\_i \cdot u} \]
   7. Step-by-step derivation

      1. *-commutative 65.0%

         \[\leadsto \color{blue}{u \cdot n1\_i} \]

   8. Simplified 65.0%

      \[\leadsto \color{blue}{u \cdot n1\_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -7.000000383309403 \cdot 10^{-27} \lor \neg \left(n0\_i \leq 3.999999935100636 \cdot 10^{-17}\right):\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.5% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -7.000000383309403 \cdot 10^{-27}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -7.000000383309403e-27)
   n0_i
   (if (<= n0_i 3.999999935100636e-17) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -7.000000383309403e-27f) {
		tmp = n0_i;
	} else if (n0_i <= 3.999999935100636e-17f) {
		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 <= (-7.000000383309403e-27)) then
        tmp = n0_i
    else if (n0_i <= 3.999999935100636e-17) 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(-7.000000383309403e-27))
		tmp = n0_i;
	elseif (n0_i <= Float32(3.999999935100636e-17))
		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(-7.000000383309403e-27))
		tmp = n0_i;
	elseif (n0_i <= single(3.999999935100636e-17))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -7.00000038e-27 or 3.99999994e-17 < 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%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

      2. associate-*l* 84.5%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

      3. *-commutative 84.5%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1\_i \]

      4. associate-*l* 83.3%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

      5. distribute-lft-out 83.3%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in normAngle around 0 98.8%

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]

   6. Taylor expanded in u around 0 99.0%

      \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + -1 \cdot n0\_i\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.0%

         \[\leadsto n0\_i + u \cdot \left(n1\_i + \color{blue}{\left(-n0\_i\right)}\right) \]

      2. unsub-neg 99.0%

         \[\leadsto n0\_i + u \cdot \color{blue}{\left(n1\_i - n0\_i\right)} \]

   8. Simplified 99.0%

      \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i - n0\_i\right)} \]

   9. Taylor expanded in u around 0 65.9%

      \[\leadsto \color{blue}{n0\_i} \]

   if -7.00000038e-27 < n0_i < 3.99999994e-17

   1. Initial program 95.6%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   2. Step-by-step derivation

      1. *-commutative 95.6%

         \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

      2. associate-*l* 75.4%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

      3. *-commutative 75.4%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1\_i \]

      4. associate-*l* 59.8%

         \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

      5. distribute-lft-out 59.7%

         \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

   3. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in normAngle around 0 97.4%

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]

   6. Taylor expanded in n0_i around 0 65.0%

      \[\leadsto \color{blue}{n1\_i \cdot u} \]
   7. Step-by-step derivation

      1. *-commutative 65.0%

         \[\leadsto \color{blue}{u \cdot n1\_i} \]

   8. Simplified 65.0%

      \[\leadsto \color{blue}{u \cdot n1\_i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.2% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(n1\_i - n0\_i\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.4%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Step-by-step derivation

   1. *-commutative 96.4%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

   2. associate-*l* 80.0%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

   3. *-commutative 80.0%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1\_i \]

   4. associate-*l* 71.7%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

   5. distribute-lft-out 71.7%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

3. Simplified 71.7%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]
4. Add Preprocessing

5. Taylor expanded in normAngle around 0 98.1%

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]

6. Taylor expanded in u around 0 98.3%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + -1 \cdot n0\_i\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 98.3%

      \[\leadsto n0\_i + u \cdot \left(n1\_i + \color{blue}{\left(-n0\_i\right)}\right) \]

   2. unsub-neg 98.3%

      \[\leadsto n0\_i + u \cdot \color{blue}{\left(n1\_i - n0\_i\right)} \]

8. Simplified 98.3%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i - n0\_i\right)} \]
9. Add Preprocessing

Alternative 5: 81.5% 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 96.4%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Step-by-step derivation

   1. fma-define 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-*l* 96.3%

      \[\leadsto \mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0\_i, \color{blue}{\sin \left(u \cdot normAngle\right) \cdot \left(\frac{1}{\sin normAngle} \cdot n1\_i\right)}\right) \]

3. Simplified 96.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0\_i, \sin \left(u \cdot normAngle\right) \cdot \left(\frac{1}{\sin normAngle} \cdot n1\_i\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in u around 0 83.4%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, n0\_i, \sin \left(u \cdot normAngle\right) \cdot \left(\frac{1}{\sin normAngle} \cdot n1\_i\right)\right) \]

6. Taylor expanded in normAngle around 0 85.6%

   \[\leadsto \color{blue}{n0\_i + n1\_i \cdot u} \]
7. Step-by-step derivation

   1. *-commutative 85.6%

      \[\leadsto n0\_i + \color{blue}{u \cdot n1\_i} \]

8. Simplified 85.6%

   \[\leadsto \color{blue}{n0\_i + u \cdot n1\_i} \]
9. Add Preprocessing

Alternative 6: 47.5% accurate, 421.0× speedup

Math

\[\begin{array}{l} \\ n0\_i \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.4%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Step-by-step derivation

   1. *-commutative 96.4%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

   2. associate-*l* 80.0%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

   3. *-commutative 80.0%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1\_i \]

   4. associate-*l* 71.7%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

   5. distribute-lft-out 71.7%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

3. Simplified 71.7%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]
4. Add Preprocessing

5. Taylor expanded in normAngle around 0 98.1%

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]

6. Taylor expanded in u around 0 98.3%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + -1 \cdot n0\_i\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 98.3%

      \[\leadsto n0\_i + u \cdot \left(n1\_i + \color{blue}{\left(-n0\_i\right)}\right) \]

   2. unsub-neg 98.3%

      \[\leadsto n0\_i + u \cdot \color{blue}{\left(n1\_i - n0\_i\right)} \]

8. Simplified 98.3%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i - n0\_i\right)} \]

9. Taylor expanded in u around 0 47.4%

   \[\leadsto \color{blue}{n0\_i} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(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)))