Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 97.4%

Time: 14.1s

Alternatives: 6

Speedup: 60.1×

Specification

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0\_i \land n0\_i \leq 1\right)\right) \land \left(-1 \leq n1\_i \land n1\_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = 1.0f / sinf(normAngle);
	return ((sinf(((1.0f - u) * normAngle)) * t_0) * n0_i) + ((sinf((u * normAngle)) * t_0) * n1_i);
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: t_0
    t_0 = 1.0e0 / sin(normangle)
    code = ((sin(((1.0e0 - u) * normangle)) * t_0) * n0_i) + ((sin((u * normangle)) * t_0) * n1_i)
end function

Julia

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

MATLAB

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

Initial Program: 97.1% 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: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.3%

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

   1. +-commutative 96.3%

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

   2. fma-define 96.5%

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

   3. associate-*r/ 96.7%

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

   4. *-rgt-identity 96.7%

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

   5. *-commutative 96.7%

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

   6. associate-*r* 83.6%

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

   7. associate-*r/ 83.8%

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

   8. *-rgt-identity 83.8%

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

3. Simplified 83.8%

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

   1. associate-/l* 97.1%

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

   2. un-div-inv 96.7%

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

   3. *-commutative 96.7%

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

   4. fma-define 96.5%

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

   5. div-inv 96.3%

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

   6. +-commutative 96.3%

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

   7. associate-*l* 96.3%

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

   8. associate-*l/ 96.4%

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

   9. *-un-lft-identity 96.4%

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

   10. associate-*r* 96.3%

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

   11. associate-*l/ 96.8%

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

6. Applied egg-rr 96.8%

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

Alternative 2: 70.9% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.99999983775159 \cdot 10^{-18} \lor \neg \left(n1\_i \leq 9.999999960041972 \cdot 10^{-12}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -9.99999983775159e-18) (not (<= n1_i 9.999999960041972e-12)))
   (* u n1_i)
   (* (- 1.0 u) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -9.99999983775159e-18f) || !(n1_i <= 9.999999960041972e-12f)) {
		tmp = u * n1_i;
	} else {
		tmp = (1.0f - u) * 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 <= (-9.99999983775159e-18)) .or. (.not. (n1_i <= 9.999999960041972e-12))) then
        tmp = u * n1_i
    else
        tmp = (1.0e0 - u) * 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(-9.99999983775159e-18)) || !(n1_i <= Float32(9.999999960041972e-12)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(Float32(Float32(1.0) - u) * 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(-9.99999983775159e-18)) || ~((n1_i <= single(9.999999960041972e-12))))
		tmp = u * n1_i;
	else
		tmp = (single(1.0) - u) * n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -9.99999984e-18 or 9.99999996e-12 < n1_i

   1. Initial program 94.1%

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

      1. +-commutative 94.1%

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

      2. fma-define 94.5%

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

      3. associate-*r/ 94.7%

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

      4. *-rgt-identity 94.7%

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

      5. *-commutative 94.7%

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

      6. associate-*r* 92.1%

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

      7. associate-*r/ 92.1%

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

      8. *-rgt-identity 92.1%

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

   3. Simplified 92.1%

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

      1. associate-/l* 94.6%

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

      2. un-div-inv 94.7%

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

      3. *-commutative 94.7%

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

      4. fma-define 94.4%

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

      5. div-inv 94.1%

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

      6. +-commutative 94.1%

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

      7. associate-*l* 94.0%

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

      8. associate-*l/ 94.0%

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

      9. *-un-lft-identity 94.0%

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

      10. associate-*r* 93.9%

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

      11. associate-*l/ 94.0%

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

   6. Applied egg-rr 94.0%

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

   7. Taylor expanded in normAngle around 0 95.2%

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

      1. +-commutative 95.2%

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

      2. *-commutative 95.2%

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

      3. fma-define 95.6%

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

   9. Simplified 95.6%

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

   10. Taylor expanded in n1_i around inf 66.7%

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

       1. *-commutative 66.7%

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

   12. Simplified 66.7%

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

   if -9.99999984e-18 < n1_i < 9.99999996e-12

   1. Initial program 97.9%

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

   3. Taylor expanded in n0_i around inf 58.3%

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

      1. associate-/l* 74.5%

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

   5. Simplified 74.5%

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

   6. Taylor expanded in normAngle around 0 74.5%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.99999983775159 \cdot 10^{-18} \lor \neg \left(n1\_i \leq 9.999999960041972 \cdot 10^{-12}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.9% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.99999983775159 \cdot 10^{-18} \lor \neg \left(n1\_i \leq 1.5000000583807998 \cdot 10^{-17}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -9.99999983775159e-18)
         (not (<= n1_i 1.5000000583807998e-17)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -9.99999983775159e-18f) || !(n1_i <= 1.5000000583807998e-17f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-9.99999983775159e-18)) .or. (.not. (n1_i <= 1.5000000583807998e-17))) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-9.99999983775159e-18)) || !(n1_i <= Float32(1.5000000583807998e-17)))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-9.99999983775159e-18)) || ~((n1_i <= single(1.5000000583807998e-17))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -9.99999984e-18 or 1.5000001e-17 < n1_i

   1. Initial program 94.9%

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

      1. +-commutative 94.9%

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

      2. fma-define 95.2%

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

      3. associate-*r/ 95.5%

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

      4. *-rgt-identity 95.5%

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

      5. *-commutative 95.5%

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

      6. associate-*r* 92.6%

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

      7. associate-*r/ 92.6%

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

      8. *-rgt-identity 92.6%

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

   3. Simplified 92.6%

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

      1. associate-/l* 95.3%

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

      2. un-div-inv 95.5%

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

      3. *-commutative 95.5%

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

      4. fma-define 95.2%

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

      5. div-inv 94.9%

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

      6. +-commutative 94.9%

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

      7. associate-*l* 94.7%

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

      8. associate-*l/ 94.7%

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

      9. *-un-lft-identity 94.7%

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

      10. associate-*r* 94.6%

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

      11. associate-*l/ 94.7%

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

   6. Applied egg-rr 94.7%

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

   7. Taylor expanded in normAngle around 0 95.3%

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

      1. +-commutative 95.3%

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

      2. *-commutative 95.3%

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

      3. fma-define 95.6%

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

   9. Simplified 95.6%

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

   10. Taylor expanded in n1_i around inf 62.5%

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

       1. *-commutative 62.5%

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

   12. Simplified 62.5%

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

   if -9.99999984e-18 < n1_i < 1.5000001e-17

   1. Initial program 97.7%

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

   3. Taylor expanded in u around 0 61.2%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.99999983775159 \cdot 10^{-18} \lor \neg \left(n1\_i \leq 1.5000000583807998 \cdot 10^{-17}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.3%

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

   1. +-commutative 96.3%

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

   2. fma-define 96.5%

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

   3. associate-*r/ 96.7%

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

   4. *-rgt-identity 96.7%

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

   5. *-commutative 96.7%

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

   6. associate-*r* 83.6%

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

   7. associate-*r/ 83.8%

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

   8. *-rgt-identity 83.8%

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

3. Simplified 83.8%

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

   1. associate-/l* 97.1%

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

   2. un-div-inv 96.7%

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

   3. *-commutative 96.7%

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

   4. fma-define 96.5%

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

   5. div-inv 96.3%

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

   6. +-commutative 96.3%

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

   7. associate-*l* 96.3%

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

   8. associate-*l/ 96.4%

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

   9. *-un-lft-identity 96.4%

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

   10. associate-*r* 96.3%

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

   11. associate-*l/ 96.8%

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

6. Applied egg-rr 96.8%

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

7. Taylor expanded in normAngle around 0 96.3%

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

   1. +-commutative 96.3%

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

   2. *-commutative 96.3%

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

   3. fma-define 96.5%

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

9. Simplified 96.5%

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

10. Taylor expanded in u around 0 96.5%

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

    1. neg-mul-1 96.5%

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

12. Simplified 96.5%

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

13. Taylor expanded in u around 0 96.5%

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

Alternative 5: 81.7% accurate, 84.2× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot n1\_i \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.3%

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

   1. fma-define 96.4%

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

   2. associate-*l* 96.3%

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

3. Simplified 96.3%

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

5. Taylor expanded in u around 0 81.4%

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

6. Taylor expanded in normAngle around 0 82.0%

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

   1. *-commutative 82.0%

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

8. Simplified 82.0%

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

Alternative 6: 47.8% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in u around 0 43.3%

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

Reproduce

herbie shell --seed 2024165 
(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)))