Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.3%

Time: 11.0s

Alternatives: 8

Speedup: 38.3×

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: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n1_i * ((normAngle / sinf(normAngle)) * u)) + (n0_i * (((cosf((u * normAngle)) * sinf(normAngle)) / sinf(normAngle)) - ((sinf((u * normAngle)) * cosf(normAngle)) / 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)) + (n0_i * (((cos((u * normangle)) * sin(normangle)) / sin(normangle)) - ((sin((u * normangle)) * cos(normangle)) / 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(n0_i * Float32(Float32(Float32(cos(Float32(u * normAngle)) * sin(normAngle)) / sin(normAngle)) - Float32(Float32(sin(Float32(u * normAngle)) * cos(normAngle)) / sin(normAngle)))))
end

MATLAB

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

Derivation

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

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\frac{normAngle \cdot u}{\sin normAngle}} \cdot n1\_i \]
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 98.8

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

5. Applied rewrites 98.8%

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

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. sub-neg N/A

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

   7. lift-neg.f32 N/A

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

   8. +-commutative N/A

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

   9. distribute-lft1-in N/A

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

   10. lift-*.f32 N/A

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

   11. sin-sum N/A

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

   12. distribute-rgt-in N/A

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

   13. lower-+.f32 N/A

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

7. Applied rewrites 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\frac{normAngle \cdot u}{\sin normAngle}} \cdot n1\_i \]
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 98.8

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

5. Applied rewrites 98.8%

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

6. Final simplification 98.8%

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

Alternative 3: 98.9% 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 97.7%

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

3. Taylor expanded in u around 0

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \color{blue}{\frac{normAngle \cdot u}{\sin normAngle}} \cdot n1\_i \]
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 98.8

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

5. Applied rewrites 98.8%

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

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

   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 98.7

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 4: 98.9% 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 97.7%

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

3. Taylor expanded in normAngle around 0

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

   1. lower--.f32 97.2

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in u around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.3

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

8. Applied rewrites 98.3%

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

9. Final simplification 98.3%

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

Alternative 5: 70.3% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -2.799999967805335 \cdot 10^{-15} \lor \neg \left(n1\_i \leq 4.99999991225835 \cdot 10^{-15}\right):\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -2.799999967805335e-15) (not (<= n1_i 4.99999991225835e-15)))
   (* n1_i u)
   (* (- 1.0 u) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -2.799999967805335e-15f) || !(n1_i <= 4.99999991225835e-15f)) {
		tmp = n1_i * u;
	} 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 <= (-2.799999967805335e-15)) .or. (.not. (n1_i <= 4.99999991225835e-15))) then
        tmp = n1_i * u
    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(-2.799999967805335e-15)) || !(n1_i <= Float32(4.99999991225835e-15)))
		tmp = Float32(n1_i * u);
	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(-2.799999967805335e-15)) || ~((n1_i <= single(4.99999991225835e-15))))
		tmp = n1_i * u;
	else
		tmp = (single(1.0) - u) * n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -2.79999997e-15 or 4.99999991e-15 < n1_i

   1. Initial program 96.9%

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

   3. Taylor expanded in 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 67.4

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in n0_i around 0

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

   8. Applied rewrites 67.4%

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

   if -2.79999997e-15 < n1_i < 4.99999991e-15

   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 19.3

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

   5. Applied rewrites 19.4%

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

   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 80.1%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -2.799999967805335 \cdot 10^{-15} \lor \neg \left(n1\_i \leq 4.99999991225835 \cdot 10^{-15}\right):\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.0% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -5.000000229068525 \cdot 10^{-19} \lor \neg \left(n1\_i \leq 4.99999991225835 \cdot 10^{-15}\right):\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;1 \cdot n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -5.000000229068525e-19) (not (<= n1_i 4.99999991225835e-15)))
   (* n1_i u)
   (* 1.0 n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -5.000000229068525e-19f) || !(n1_i <= 4.99999991225835e-15f)) {
		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 ((n1_i <= (-5.000000229068525e-19)) .or. (.not. (n1_i <= 4.99999991225835e-15))) 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 ((n1_i <= Float32(-5.000000229068525e-19)) || !(n1_i <= Float32(4.99999991225835e-15)))
		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 ((n1_i <= single(-5.000000229068525e-19)) || ~((n1_i <= single(4.99999991225835e-15))))
		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 n1_i < -5.00000023e-19 or 4.99999991e-15 < n1_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 64.6

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in n0_i around 0

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

   8. Applied rewrites 64.6%

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

   if -5.00000023e-19 < n1_i < 4.99999991e-15

   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 17.7

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

   5. Applied rewrites 17.8%

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

   6. Taylor expanded in n0_i around inf

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 63.3%

       \[\leadsto 1 \cdot n0\_i \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -5.000000229068525 \cdot 10^{-19} \lor \neg \left(n1\_i \leq 4.99999991225835 \cdot 10^{-15}\right):\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;1 \cdot n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 38.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 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 39.1

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

5. Applied rewrites 46.8%

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

6. Taylor expanded in n0_i around 0

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

8. Applied rewrites 37.0%

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

9. Taylor expanded in u around -inf

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

11. Applied rewrites 96.9%

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

12. Taylor expanded in u around 0

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

14. Applied rewrites 97.3%

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

15. Final simplification 97.3%

    \[\leadsto n0\_i - \left(n0\_i - n1\_i\right) \cdot u \]
16. 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 97.7%

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

3. Taylor expanded in 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 37.0

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

5. Applied rewrites 37.0%

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

6. Taylor expanded in n0_i around 0

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

8. Applied rewrites 37.0%

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

9. Final simplification 37.0%

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

Reproduce

herbie shell --seed 2024271 
(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)))