Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 97.8%

Time: 15.2s

Alternatives: 12

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: 97.2% 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: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. +-commutative 97.8%

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

   2. fma-define 97.8%

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

   3. associate-*r/ 98.0%

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

   4. *-rgt-identity 98.0%

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

   5. *-commutative 98.0%

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

   6. associate-*r* 82.5%

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

   7. associate-*r/ 82.6%

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

   8. *-rgt-identity 82.6%

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

3. Simplified 82.6%

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

   1. *-commutative 82.6%

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

   2. associate-*l/ 98.3%

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

   3. *-commutative 98.3%

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

6. Applied egg-rr 98.3%

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

Alternative 2: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (normangle * ((u * cos(normangle)) / sin(normangle))))) + (n1_i * (sin((u * normangle)) * (1.0e0 / sin(normangle))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in u around 0 92.6%

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

   1. mul-1-neg 92.6%

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

   2. unsub-neg 92.6%

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

   3. associate-/l* 98.1%

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

   4. associate-/l* 98.1%

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

5. Simplified 98.1%

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

6. Final simplification 98.1%

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

Alternative 3: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ n1\_i \cdot \left(\sin \left(u \cdot normAngle\right) \cdot t\_0\right) + n0\_i \cdot \left(t\_0 \cdot \sin \left(normAngle \cdot \left(1 - u\right)\right)\right) \end{array} \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = 1.0f / sinf(normAngle);
	return (n1_i * (sinf((u * normAngle)) * t_0)) + (n0_i * (t_0 * sinf((normAngle * (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
    real(4) :: t_0
    t_0 = 1.0e0 / sin(normangle)
    code = (n1_i * (sin((u * normangle)) * t_0)) + (n0_i * (t_0 * sin((normangle * (1.0e0 - u)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

3. Final simplification 97.8%

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

Alternative 4: 97.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in normAngle around 0 97.6%

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

4. Final simplification 97.6%

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

Alternative 5: 70.9% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -8.00000025099516 \cdot 10^{-22} \lor \neg \left(n0\_i \leq 3.999999954906409 \cdot 10^{-26}\right):\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(n1\_i + n0\_i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -8.00000025099516e-22) (not (<= n0_i 3.999999954906409e-26)))
   (* (- 1.0 u) n0_i)
   (* u (+ n1_i n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -8.00000025099516e-22f) || !(n0_i <= 3.999999954906409e-26f)) {
		tmp = (1.0f - u) * n0_i;
	} else {
		tmp = u * (n1_i + 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 <= (-8.00000025099516e-22)) .or. (.not. (n0_i <= 3.999999954906409e-26))) then
        tmp = (1.0e0 - u) * n0_i
    else
        tmp = u * (n1_i + 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(-8.00000025099516e-22)) || !(n0_i <= Float32(3.999999954906409e-26)))
		tmp = Float32(Float32(Float32(1.0) - u) * n0_i);
	else
		tmp = Float32(u * Float32(n1_i + 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(-8.00000025099516e-22)) || ~((n0_i <= single(3.999999954906409e-26))))
		tmp = (single(1.0) - u) * n0_i;
	else
		tmp = u * (n1_i + n0_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -8.00000025e-22 or 3.99999995e-26 < n0_i

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

      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* 82.7%

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

      5. distribute-lft-out 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in normAngle around 0 98.7%

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

   6. Taylor expanded in u around 0 98.8%

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

      1. mul-1-neg 98.8%

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

      2. unsub-neg 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in n0_i around inf 71.9%

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

       1. neg-mul-1 71.9%

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

       2. sub-neg 71.9%

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

   11. Simplified 71.9%

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

   if -8.00000025e-22 < n0_i < 3.99999995e-26

   1. Initial program 96.7%

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

      1. *-commutative 96.7%

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

      2. associate-*l* 79.2%

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

      3. *-commutative 79.2%

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

      4. associate-*l* 59.4%

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

      5. distribute-lft-out 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in normAngle around 0 95.3%

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

      1. pow1 95.3%

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

      2. *-commutative 95.3%

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

      3. sub-neg 95.3%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 88.9%

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

      6. sqr-neg 88.9%

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

      7. sqrt-unprod 88.9%

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

      8. add-sqr-sqrt 88.9%

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

   7. Applied egg-rr 88.9%

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

      1. unpow1 88.9%

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

      2. *-commutative 88.9%

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

   9. Simplified 88.9%

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

   10. Taylor expanded in u around inf 71.2%

       \[\leadsto \color{blue}{u \cdot \left(n0\_i + n1\_i\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -8.00000025099516 \cdot 10^{-22} \lor \neg \left(n0\_i \leq 3.999999954906409 \cdot 10^{-26}\right):\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(n1\_i + n0\_i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.6% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -8.00000025099516 \cdot 10^{-22} \lor \neg \left(n0\_i \leq 3.999999954906409 \cdot 10^{-26}\right):\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -8.00000025099516e-22) (not (<= n0_i 3.999999954906409e-26)))
   (* (- 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 <= -8.00000025099516e-22f) || !(n0_i <= 3.999999954906409e-26f)) {
		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 <= (-8.00000025099516e-22)) .or. (.not. (n0_i <= 3.999999954906409e-26))) 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(-8.00000025099516e-22)) || !(n0_i <= Float32(3.999999954906409e-26)))
		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(-8.00000025099516e-22)) || ~((n0_i <= single(3.999999954906409e-26))))
		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 < -8.00000025e-22 or 3.99999995e-26 < n0_i

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

      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* 82.7%

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

      5. distribute-lft-out 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in normAngle around 0 98.7%

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

   6. Taylor expanded in u around 0 98.8%

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

      1. mul-1-neg 98.8%

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

      2. unsub-neg 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in n0_i around inf 71.9%

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

       1. neg-mul-1 71.9%

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

       2. sub-neg 71.9%

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

   11. Simplified 71.9%

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

   if -8.00000025e-22 < n0_i < 3.99999995e-26

   1. Initial program 96.7%

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

      1. *-commutative 96.7%

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

      2. associate-*l* 79.2%

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

      3. *-commutative 79.2%

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

      4. associate-*l* 59.4%

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

      5. distribute-lft-out 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in normAngle around 0 95.3%

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

   6. Taylor expanded in n0_i around 0 70.8%

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

      1. *-commutative 70.8%

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

   8. Simplified 70.8%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -8.00000025099516 \cdot 10^{-22} \lor \neg \left(n0\_i \leq 3.999999954906409 \cdot 10^{-26}\right):\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.3% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 3.999999954906409 \cdot 10^{-26}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.999999936531045e-20)
   n0_i
   (if (<= n0_i 3.999999954906409e-26) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.999999936531045e-20f) {
		tmp = n0_i;
	} else if (n0_i <= 3.999999954906409e-26f) {
		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 <= (-1.999999936531045e-20)) then
        tmp = n0_i
    else if (n0_i <= 3.999999954906409e-26) 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(-1.999999936531045e-20))
		tmp = n0_i;
	elseif (n0_i <= Float32(3.999999954906409e-26))
		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(-1.999999936531045e-20))
		tmp = n0_i;
	elseif (n0_i <= single(3.999999954906409e-26))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.99999994e-20 or 3.99999995e-26 < n0_i

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*l* 84.6%

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

      3. *-commutative 84.6%

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

      4. associate-*l* 82.7%

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

      5. distribute-lft-out 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in normAngle around 0 98.7%

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

   6. Taylor expanded in u around 0 98.8%

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

      1. mul-1-neg 98.8%

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

      2. unsub-neg 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in u around 0 56.9%

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

   if -1.99999994e-20 < n0_i < 3.99999995e-26

   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* 79.4%

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

      3. *-commutative 79.4%

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

      4. associate-*l* 60.7%

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

      5. distribute-lft-out 60.7%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in normAngle around 0 95.5%

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

   6. Taylor expanded in n0_i around 0 69.3%

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

      1. *-commutative 69.3%

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

   8. Simplified 69.3%

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

Alternative 8: 97.9% accurate, 46.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. associate-*l* 82.3%

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

   3. *-commutative 82.3%

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

   4. associate-*l* 73.2%

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

   5. distribute-lft-out 73.1%

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

3. Simplified 73.1%

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

5. Taylor expanded in normAngle around 0 97.3%

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

6. Taylor expanded in u around 0 97.4%

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

   1. mul-1-neg 97.4%

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

   2. unsub-neg 97.4%

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

8. Simplified 97.4%

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

   1. sub-neg 97.4%

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

   2. distribute-rgt-in 97.5%

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

   3. *-commutative 97.5%

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

10. Applied egg-rr 97.5%

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

11. Final simplification 97.5%

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

Alternative 9: 97.9% 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 97.8%

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

   1. *-commutative 97.8%

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

   2. associate-*l* 82.3%

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

   3. *-commutative 82.3%

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

   4. associate-*l* 73.2%

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

   5. distribute-lft-out 73.1%

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

3. Simplified 73.1%

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

5. Taylor expanded in normAngle around 0 97.3%

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

6. Taylor expanded in u around 0 97.4%

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

   1. mul-1-neg 97.4%

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

   2. unsub-neg 97.4%

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

8. Simplified 97.4%

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

Alternative 10: 81.4% accurate, 60.1× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return u * (n1_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 = u * (n1_i + (n0_i / u))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. associate-*l* 82.3%

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

   3. *-commutative 82.3%

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

   4. associate-*l* 73.2%

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

   5. distribute-lft-out 73.1%

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

3. Simplified 73.1%

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

5. Taylor expanded in normAngle around 0 97.3%

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

6. Taylor expanded in u around inf 97.1%

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

   1. +-commutative 97.1%

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

   2. +-commutative 97.1%

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

   3. associate-+l+ 97.1%

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

   4. +-commutative 97.1%

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

   5. mul-1-neg 97.1%

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

   6. unsub-neg 97.1%

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

8. Simplified 97.1%

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

9. Taylor expanded in n1_i around inf 81.0%

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

10. Final simplification 81.0%

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

Alternative 11: 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 97.8%

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

3. Taylor expanded in u around 0 81.2%

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

4. Taylor expanded in normAngle around 0 81.0%

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

   1. *-commutative 81.0%

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

6. Simplified 81.0%

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

Alternative 12: 46.5% 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 97.8%

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

   1. *-commutative 97.8%

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

   2. associate-*l* 82.3%

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

   3. *-commutative 82.3%

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

   4. associate-*l* 73.2%

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

   5. distribute-lft-out 73.1%

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

3. Simplified 73.1%

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

5. Taylor expanded in normAngle around 0 97.3%

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

6. Taylor expanded in u around 0 97.4%

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

   1. mul-1-neg 97.4%

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

   2. unsub-neg 97.4%

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

8. Simplified 97.4%

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

9. Taylor expanded in u around 0 42.7%

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

Reproduce

herbie shell --seed 2024139 
(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)))