Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 98.9%

Time: 10.2s

Alternatives: 8

Speedup: 27.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n1_i * ((normAngle / sinf(normAngle)) * u)) + (sinf(((1.0f - u) * normAngle)) * (n0_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 = (n1_i * ((normangle / sin(normangle)) * u)) + (sin(((1.0e0 - u) * normangle)) * (n0_i / sin(normangle)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in u around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.3

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

5. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. associate-*l/ N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 98.6

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.6

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

7. Applied rewrites 98.6%

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

8. Final simplification 98.6%

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

Alternative 2: 98.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in u around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in normAngle around 0

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

   1. lower--.f32 98.5

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

8. Applied rewrites 98.5%

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

9. Final simplification 98.5%

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

Alternative 3: 98.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in u around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in normAngle around 0

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

   1. lower--.f32 98.5

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in u around 0

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

    1. associate-/l* N/A

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

    2. lower-*.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-sin.f32 98.4

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

11. Applied rewrites 98.4%

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

12. Final simplification 98.4%

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

Alternative 4: 67.9% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{elif}\;n1\_i \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(u, n1\_i, \left(1 - u\right) \cdot n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;n1\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -1.99999996490334e-14)
   (* n1_i u)
   (if (<= n1_i 4.99999991225835e-14)
     (fma u n1_i (* (- 1.0 u) n0_i))
     (* n1_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -1.99999996490334e-14f) {
		tmp = n1_i * u;
	} else if (n1_i <= 4.99999991225835e-14f) {
		tmp = fmaf(u, n1_i, ((1.0f - u) * n0_i));
	} else {
		tmp = n1_i * u;
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-1.99999996490334e-14))
		tmp = Float32(n1_i * u);
	elseif (n1_i <= Float32(4.99999991225835e-14))
		tmp = fma(u, n1_i, Float32(Float32(Float32(1.0) - u) * n0_i));
	else
		tmp = Float32(n1_i * u);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.99999996e-14 or 4.99999991e-14 < n1_i

   1. Initial program 95.4%

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

   3. Taylor expanded in normAngle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 73.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 73.1%

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

   9. Taylor expanded in n0_i around inf

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

   11. Applied rewrites 8.1%

       \[\leadsto \left(-n0\_i\right) \cdot u \]

   12. Taylor expanded in n0_i around 0

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

   14. Applied rewrites 73.6%

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

   if -1.99999996e-14 < n1_i < 4.99999991e-14

   1. Initial program 98.5%

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

   3. Taylor expanded in normAngle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 17.7

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

   5. Applied rewrites 17.7%

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

   7. Applied rewrites 17.7%

      \[\leadsto \mathsf{fma}\left(1 - u, n0\_i, u \cdot n1\_i\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 81.0%

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

   10. Taylor expanded in u around 0

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

   12. Applied rewrites 81.0%

       \[\leadsto \mathsf{fma}\left(u, n1\_i, \left(1 - u\right) \cdot n0\_i\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.3% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{elif}\;n1\_i \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(n1\_i, u, \left(1 - u\right) \cdot n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;n1\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -1.99999996490334e-14)
   (* n1_i u)
   (if (<= n1_i 4.99999991225835e-14)
     (fma n1_i u (* (- 1.0 u) n0_i))
     (* n1_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -1.99999996490334e-14f) {
		tmp = n1_i * u;
	} else if (n1_i <= 4.99999991225835e-14f) {
		tmp = fmaf(n1_i, u, ((1.0f - u) * n0_i));
	} else {
		tmp = n1_i * u;
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-1.99999996490334e-14))
		tmp = Float32(n1_i * u);
	elseif (n1_i <= Float32(4.99999991225835e-14))
		tmp = fma(n1_i, u, Float32(Float32(Float32(1.0) - u) * n0_i));
	else
		tmp = Float32(n1_i * u);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.99999996e-14 or 4.99999991e-14 < n1_i

   1. Initial program 95.4%

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

   3. Taylor expanded in normAngle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 73.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 73.1%

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

   9. Taylor expanded in n0_i around inf

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

   11. Applied rewrites 8.1%

       \[\leadsto \left(-n0\_i\right) \cdot u \]

   12. Taylor expanded in n0_i around 0

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

   14. Applied rewrites 73.6%

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

   if -1.99999996e-14 < n1_i < 4.99999991e-14

   1. Initial program 98.5%

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

   3. Taylor expanded in normAngle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 17.7

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

   5. Applied rewrites 17.7%

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

   7. Applied rewrites 17.7%

      \[\leadsto \mathsf{fma}\left(1 - u, n0\_i, u \cdot n1\_i\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 81.0%

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

   11. Applied rewrites 80.9%

       \[\leadsto \mathsf{fma}\left(n1\_i, \color{blue}{u}, n0\_i \cdot \left(1 - u\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{elif}\;n1\_i \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(n1\_i, u, \left(1 - u\right) \cdot n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;n1\_i \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.5% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{elif}\;n1\_i \leq 6.000000233523199 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;n1\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -1.99999996490334e-14)
   (* n1_i u)
   (if (<= n1_i 6.000000233523199e-17) (fma (- n1_i n0_i) u n0_i) (* n1_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -1.99999996490334e-14f) {
		tmp = n1_i * u;
	} else if (n1_i <= 6.000000233523199e-17f) {
		tmp = fmaf((n1_i - n0_i), u, n0_i);
	} else {
		tmp = n1_i * u;
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-1.99999996490334e-14))
		tmp = Float32(n1_i * u);
	elseif (n1_i <= Float32(6.000000233523199e-17))
		tmp = fma(Float32(n1_i - n0_i), u, n0_i);
	else
		tmp = Float32(n1_i * u);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.99999996e-14 or 6.0000002e-17 < n1_i

   1. Initial program 95.6%

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

   3. Taylor expanded in normAngle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 72.4

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 71.7%

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

   9. Taylor expanded in n0_i around inf

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

   11. Applied rewrites 8.1%

       \[\leadsto \left(-n0\_i\right) \cdot u \]

   12. Taylor expanded in n0_i around 0

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

   14. Applied rewrites 72.4%

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

   if -1.99999996e-14 < n1_i < 6.0000002e-17

   1. Initial program 98.5%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift--.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in normAngle around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate-+l+ N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. distribute-lft-neg-out N/A

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f32 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(n1\_i + \color{blue}{\left(\mathsf{neg}\left(n0\_i\right)\right)}, u, n0\_i\right) \]

      14. unsub-neg N/A

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

      15. lower--.f32 64.0

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

   7. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.8% accurate, 27.0× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((1.0f - u) * n0_i) + (n1_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 = ((1.0e0 - u) * n0_i) + (n1_i * u)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in normAngle around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 38.4

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

5. Applied rewrites 38.4%

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

7. Applied rewrites 97.6%

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

8. Final simplification 97.6%

   \[\leadsto \left(1 - u\right) \cdot n0\_i + n1\_i \cdot u \]
9. Add Preprocessing

Alternative 8: 38.0% accurate, 76.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in normAngle around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 38.4

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

5. Applied rewrites 38.1%

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

6. Taylor expanded in u around inf

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

8. Applied rewrites 37.0%

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

9. Taylor expanded in n0_i around inf

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

11. Applied rewrites 8.2%

    \[\leadsto \left(-n0\_i\right) \cdot u \]

12. Taylor expanded in n0_i around 0

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

14. Applied rewrites 38.4%

    \[\leadsto n1\_i \cdot \color{blue}{u} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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)))