Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 98.9%

Time: 11.4s

Alternatives: 9

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. 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 98.4

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

5. Applied rewrites 98.4%

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

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f32 N/A

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

   11. lower-/.f32 99.2

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 99.2

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

7. Applied rewrites 99.2%

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

8. Final simplification 99.2%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. 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 98.4

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

5. Applied rewrites 98.4%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\left(\frac{normAngle}{\sin normAngle} \cdot u\right)} \cdot n1\_i \]
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. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f32 N/A

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

   8. un-div-inv 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. lower-/.f32 99.0

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

   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 99.0

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

7. Applied rewrites 99.0%

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

8. Final simplification 99.0%

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

Alternative 3: 98.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. 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 98.4

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

5. Applied rewrites 98.4%

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

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f32 N/A

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

   11. lower-/.f32 99.2

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in normAngle around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 98.9

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

10. Applied rewrites 98.9%

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

11. Final simplification 98.9%

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

Alternative 4: 66.8% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot n0\_i\\ \mathbf{if}\;n0\_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) n0_i)))
   (if (<= n0_i -4.999999980020986e-13)
     t_0
     (if (<= n0_i 4.999999999099794e-24) (* n1_i u) t_0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = (1.0f - u) * n0_i;
	float tmp;
	if (n0_i <= -4.999999980020986e-13f) {
		tmp = t_0;
	} else if (n0_i <= 4.999999999099794e-24f) {
		tmp = n1_i * u;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = (1.0e0 - u) * n0_i
    if (n0_i <= (-4.999999980020986e-13)) then
        tmp = t_0
    else if (n0_i <= 4.999999999099794e-24) then
        tmp = n1_i * u
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(Float32(1.0) - u) * n0_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-4.999999980020986e-13))
		tmp = t_0;
	elseif (n0_i <= Float32(4.999999999099794e-24))
		tmp = Float32(n1_i * u);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	t_0 = (single(1.0) - u) * n0_i;
	tmp = single(0.0);
	if (n0_i <= single(-4.999999980020986e-13))
		tmp = t_0;
	elseif (n0_i <= single(4.999999999099794e-24))
		tmp = n1_i * u;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.99999998e-13 or 5e-24 < n0_i

   1. Initial program 98.2%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   2. 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. *-commutative N/A

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

      5. lower-*.f32 44.1

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

   5. Applied rewrites 13.9%

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

   6. Taylor expanded in n0_i around inf

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

   8. Applied rewrites 86.0%

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

   if -4.99999998e-13 < n0_i < 5e-24

   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 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. *-commutative N/A

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

      5. lower-*.f32 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in n0_i around 0

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

   8. Applied rewrites 66.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{elif}\;n0\_i \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.2% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq 4.00000018325482 \cdot 10^{-18}:\\ \;\;\;\;n1\_i \cdot u + 1 \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i 4.00000018325482e-18)
   (+ (* n1_i u) (* 1.0 n0_i))
   (* (- 1.0 u) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= 4.00000018325482e-18f) {
		tmp = (n1_i * u) + (1.0f * n0_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 (n0_i <= 4.00000018325482e-18) then
        tmp = (n1_i * u) + (1.0e0 * n0_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 (n0_i <= Float32(4.00000018325482e-18))
		tmp = Float32(Float32(n1_i * u) + Float32(Float32(1.0) * n0_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 (n0_i <= single(4.00000018325482e-18))
		tmp = (n1_i * u) + (single(1.0) * n0_i);
	else
		tmp = (single(1.0) - u) * n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < 4.00000018e-18

   1. Initial program 97.3%

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

   3. Taylor expanded in u around 0

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\frac{normAngle \cdot u}{\sin normAngle}} \cdot n1\_i \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 86.9%

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

   9. Taylor expanded in normAngle around 0

      \[\leadsto 1 \cdot n0\_i + \color{blue}{n1\_i \cdot u} \]
   10. Step-by-step derivation

       1. lower-*.f32 86.4

          \[\leadsto 1 \cdot n0\_i + \color{blue}{n1\_i \cdot u} \]

   11. Applied rewrites 86.4%

       \[\leadsto 1 \cdot n0\_i + \color{blue}{n1\_i \cdot u} \]

   if 4.00000018e-18 < n0_i

   1. Initial program 97.0%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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. *-commutative N/A

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

      5. lower-*.f32 9.7

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

   5. Applied rewrites 9.8%

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

   6. Taylor expanded in n0_i around inf

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

   8. Applied rewrites 91.9%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq 4.00000018325482 \cdot 10^{-18}:\\ \;\;\;\;n1\_i \cdot u + 1 \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.8% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;1 \cdot n0\_i\\ \mathbf{elif}\;n0\_i \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;1 \cdot n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -4.999999980020986e-13)
   (* 1.0 n0_i)
   (if (<= n0_i 1.5000000786160286e-23) (* n1_i u) (* 1.0 n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -4.999999980020986e-13f) {
		tmp = 1.0f * n0_i;
	} else if (n0_i <= 1.5000000786160286e-23f) {
		tmp = n1_i * u;
	} else {
		tmp = 1.0f * 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 <= (-4.999999980020986e-13)) then
        tmp = 1.0e0 * n0_i
    else if (n0_i <= 1.5000000786160286e-23) then
        tmp = n1_i * u
    else
        tmp = 1.0e0 * 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(-4.999999980020986e-13))
		tmp = Float32(Float32(1.0) * n0_i);
	elseif (n0_i <= Float32(1.5000000786160286e-23))
		tmp = Float32(n1_i * u);
	else
		tmp = Float32(Float32(1.0) * 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(-4.999999980020986e-13))
		tmp = single(1.0) * n0_i;
	elseif (n0_i <= single(1.5000000786160286e-23))
		tmp = n1_i * u;
	else
		tmp = single(1.0) * n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.99999998e-13 or 1.50000008e-23 < n0_i

   1. Initial program 98.1%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      5. lower-*.f32 13.3

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

   5. Applied rewrites 13.3%

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

   6. Taylor expanded in n0_i around inf

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

   8. Applied rewrites 86.6%

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

   9. Taylor expanded in u around 0

      \[\leadsto 1 \cdot n0\_i \]
   10. Step-by-step derivation

   11. Applied rewrites 68.6%

       \[\leadsto 1 \cdot n0\_i \]

   if -4.99999998e-13 < n0_i < 1.50000008e-23

   1. Initial program 96.4%

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

   3. Taylor expanded in normAngle around 0

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
   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. *-commutative N/A

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

      5. lower-*.f32 66.0

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in n0_i around 0

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

   8. Applied rewrites 66.0%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;1 \cdot n0\_i\\ \mathbf{elif}\;n0\_i \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;1 \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 27.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. 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. *-commutative N/A

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

   5. lower-*.f32 44.2

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

5. Applied rewrites 41.7%

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

7. Applied rewrites 98.7%

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 8: 97.9% 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 97.2%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. 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. *-commutative N/A

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

   5. lower-*.f32 41.7

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

5. Applied rewrites 41.7%

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 9: 38.6% 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 97.2%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. 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. *-commutative N/A

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

   5. lower-*.f32 41.7

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

5. Applied rewrites 41.7%

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

6. Taylor expanded in n0_i around 0

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

8. Applied rewrites 41.7%

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

9. Final simplification 41.7%

   \[\leadsto n1\_i \cdot u \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(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)))