Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.1%

Time: 15.6s

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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.1%

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

3. Taylor expanded in u around 0 96.6%

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

   1. associate-/l* 98.0%

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

5. Simplified 98.0%

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

6. Taylor expanded in u around 0 92.7%

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

   1. mul-1-neg 92.7%

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

   2. unsub-neg 92.7%

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

   3. associate-/l* 98.7%

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

   4. associate-*r* 98.7%

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

   5. *-commutative 98.7%

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

8. Simplified 98.7%

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

Alternative 2: 98.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

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 = ((normangle * (u / sin(normangle))) * n1_i) + (n0_i - (n0_i * u))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.1%

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

3. Taylor expanded in u around 0 96.6%

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

   1. associate-/l* 98.0%

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

5. Simplified 98.0%

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

6. Taylor expanded in u around 0 92.7%

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

   1. mul-1-neg 92.7%

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

   2. unsub-neg 92.7%

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

   3. associate-/l* 98.7%

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

   4. associate-*r* 98.7%

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

   5. *-commutative 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in normAngle around 0 98.6%

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

    1. *-commutative 98.6%

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

11. Simplified 98.6%

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

12. Final simplification 98.6%

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

Alternative 3: 98.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

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 = ((normangle * (u / sin(normangle))) * n1_i) + (n0_i * (1.0e0 - u))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.1%

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

3. Taylor expanded in u around 0 96.6%

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

   1. associate-/l* 98.0%

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

5. Simplified 98.0%

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

6. Taylor expanded in normAngle around 0 98.4%

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

7. Final simplification 98.4%

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

Alternative 4: 71.7% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.499999973677505 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 7.499999762508407 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.499999973677505e-14)
         (not (<= n1_i 7.499999762508407e-16)))
   (* u n1_i)
   (- n0_i (* n0_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.499999973677505e-14f) || !(n1_i <= 7.499999762508407e-16f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i - (n0_i * u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-1.499999973677505e-14)) .or. (.not. (n1_i <= 7.499999762508407e-16))) then
        tmp = u * n1_i
    else
        tmp = n0_i - (n0_i * u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-1.499999973677505e-14)) || !(n1_i <= Float32(7.499999762508407e-16)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(n0_i - Float32(n0_i * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-1.499999973677505e-14)) || ~((n1_i <= single(7.499999762508407e-16))))
		tmp = u * n1_i;
	else
		tmp = n0_i - (n0_i * u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.49999997e-14 or 7.49999976e-16 < n1_i

   1. Initial program 96.6%

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

      1. associate-*l* 96.4%

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

      2. cancel-sign-sub 96.4%

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

      3. *-commutative 96.4%

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

      4. associate-*r* 91.4%

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

      5. associate-*r/ 91.4%

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

      6. *-rgt-identity 91.4%

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

      7. sin-neg 91.4%

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

      8. distribute-lft-neg-out 91.4%

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

      9. associate-*l* 91.6%

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

      10. *-commutative 91.6%

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

      11. distribute-lft-neg-out 91.6%

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

      12. distribute-rgt-neg-out 91.6%

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

      13. associate-*r/ 91.6%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in u around 0 81.5%

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

      1. *-commutative 81.5%

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

   7. Simplified 81.5%

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

   8. Taylor expanded in u around inf 53.1%

      \[\leadsto \frac{\color{blue}{n1\_i \cdot \left(normAngle \cdot u\right)}}{\sin normAngle} \]
   9. Step-by-step derivation

      1. *-commutative 53.1%

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

      2. *-commutative 53.1%

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

      3. associate-*r* 53.3%

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

   10. Simplified 53.3%

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

   11. Taylor expanded in normAngle around 0 61.5%

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

   if -1.49999997e-14 < n1_i < 7.49999976e-16

   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. fma-define 97.5%

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

      2. associate-*l* 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in n0_i around inf 63.0%

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

   6. Taylor expanded in u around 0 73.7%

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

      1. mul-1-neg 73.7%

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

      2. unsub-neg 73.7%

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

      3. associate-/l* 79.4%

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

      4. associate-*r* 79.4%

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

      5. *-commutative 79.4%

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

   8. Simplified 79.4%

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

   9. Taylor expanded in normAngle around 0 79.3%

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

       1. *-commutative 98.8%

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

   11. Simplified 79.3%

       \[\leadsto n0\_i - \color{blue}{u \cdot n0\_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.499999973677505 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 7.499999762508407 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.6% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.499999973677505 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 7.499999762508407 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.499999973677505e-14)
         (not (<= n1_i 7.499999762508407e-16)))
   (* 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 <= -1.499999973677505e-14f) || !(n1_i <= 7.499999762508407e-16f)) {
		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 <= (-1.499999973677505e-14)) .or. (.not. (n1_i <= 7.499999762508407e-16))) 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(-1.499999973677505e-14)) || !(n1_i <= Float32(7.499999762508407e-16)))
		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(-1.499999973677505e-14)) || ~((n1_i <= single(7.499999762508407e-16))))
		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 < -1.49999997e-14 or 7.49999976e-16 < n1_i

   1. Initial program 96.6%

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

      1. associate-*l* 96.4%

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

      2. cancel-sign-sub 96.4%

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

      3. *-commutative 96.4%

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

      4. associate-*r* 91.4%

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

      5. associate-*r/ 91.4%

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

      6. *-rgt-identity 91.4%

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

      7. sin-neg 91.4%

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

      8. distribute-lft-neg-out 91.4%

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

      9. associate-*l* 91.6%

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

      10. *-commutative 91.6%

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

      11. distribute-lft-neg-out 91.6%

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

      12. distribute-rgt-neg-out 91.6%

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

      13. associate-*r/ 91.6%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in u around 0 81.5%

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

      1. *-commutative 81.5%

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

   7. Simplified 81.5%

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

   8. Taylor expanded in u around inf 53.1%

      \[\leadsto \frac{\color{blue}{n1\_i \cdot \left(normAngle \cdot u\right)}}{\sin normAngle} \]
   9. Step-by-step derivation

      1. *-commutative 53.1%

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

      2. *-commutative 53.1%

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

      3. associate-*r* 53.3%

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

   10. Simplified 53.3%

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

   11. Taylor expanded in normAngle around 0 61.5%

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

   if -1.49999997e-14 < n1_i < 7.49999976e-16

   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. fma-define 97.5%

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

      2. associate-*l* 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in n0_i around inf 63.0%

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

   6. Taylor expanded in normAngle around 0 79.1%

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.499999973677505 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 7.499999762508407 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.7% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.499999973677505 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 7.499999762508407 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.499999973677505e-14)
         (not (<= n1_i 7.499999762508407e-16)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.499999973677505e-14f) || !(n1_i <= 7.499999762508407e-16f)) {
		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 ((n1_i <= (-1.499999973677505e-14)) .or. (.not. (n1_i <= 7.499999762508407e-16))) 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 ((n1_i <= Float32(-1.499999973677505e-14)) || !(n1_i <= Float32(7.499999762508407e-16)))
		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 ((n1_i <= single(-1.499999973677505e-14)) || ~((n1_i <= single(7.499999762508407e-16))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.49999997e-14 or 7.49999976e-16 < n1_i

   1. Initial program 96.6%

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

      1. associate-*l* 96.4%

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

      2. cancel-sign-sub 96.4%

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

      3. *-commutative 96.4%

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

      4. associate-*r* 91.4%

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

      5. associate-*r/ 91.4%

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

      6. *-rgt-identity 91.4%

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

      7. sin-neg 91.4%

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

      8. distribute-lft-neg-out 91.4%

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

      9. associate-*l* 91.6%

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

      10. *-commutative 91.6%

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

      11. distribute-lft-neg-out 91.6%

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

      12. distribute-rgt-neg-out 91.6%

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

      13. associate-*r/ 91.6%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in u around 0 81.5%

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

      1. *-commutative 81.5%

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

   7. Simplified 81.5%

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

   8. Taylor expanded in u around inf 53.1%

      \[\leadsto \frac{\color{blue}{n1\_i \cdot \left(normAngle \cdot u\right)}}{\sin normAngle} \]
   9. Step-by-step derivation

      1. *-commutative 53.1%

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

      2. *-commutative 53.1%

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

      3. associate-*r* 53.3%

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

   10. Simplified 53.3%

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

   11. Taylor expanded in normAngle around 0 61.5%

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

   if -1.49999997e-14 < n1_i < 7.49999976e-16

   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. fma-define 97.5%

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

      2. associate-*l* 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in u around 0 60.1%

      \[\leadsto \color{blue}{n0\_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.499999973677505 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 7.499999762508407 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.1%

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

3. Taylor expanded in u around 0 96.6%

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

   1. associate-/l* 98.0%

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

5. Simplified 98.0%

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

6. Taylor expanded in u around 0 92.7%

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

   1. mul-1-neg 92.7%

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

   2. unsub-neg 92.7%

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

   3. associate-/l* 98.7%

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

   4. associate-*r* 98.7%

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

   5. *-commutative 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in normAngle around 0 97.3%

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

    1. associate--l+ 97.4%

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

    2. distribute-rgt-out-- 97.4%

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

11. Simplified 97.4%

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

Alternative 8: 47.2% 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 97.1%

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

   1. fma-define 97.1%

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

   2. associate-*l* 97.1%

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

3. Simplified 97.1%

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

5. Taylor expanded in u around 0 47.3%

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

Reproduce

herbie shell --seed 2024185 
(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)))