Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 98.7%

Time: 9.5s

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: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = ((sin(((single(1.0) - u) * normAngle)) * (single(1.0) / sin(normAngle))) * n0_i) + (((normAngle / sin(normAngle)) * u) * n1_i);
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 99.0

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

5. Applied rewrites 99.0%

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

Alternative 2: 98.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = ((single(1.0) - u) * n0_i) + (((normAngle / sin(normAngle)) * u) * n1_i);
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 99.0

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

5. Applied rewrites 99.0%

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

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. Add Preprocessing

Alternative 3: 67.4% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -7.99999999855967 \cdot 10^{-23} \lor \neg \left(n0\_i \leq 3.99999987306209 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{fma}\left(n1\_i, u, n0\_i \cdot \left(1 - u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n1\_i - n0\_i\right) \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -7.99999999855967e-23) (not (<= n0_i 3.99999987306209e-20)))
   (fma n1_i u (* n0_i (- 1.0 u)))
   (* (- n1_i n0_i) u)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -7.99999999855967e-23f) || !(n0_i <= 3.99999987306209e-20f)) {
		tmp = fmaf(n1_i, u, (n0_i * (1.0f - u)));
	} else {
		tmp = (n1_i - n0_i) * u;
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-7.99999999855967e-23)) || !(n0_i <= Float32(3.99999987306209e-20)))
		tmp = fma(n1_i, u, Float32(n0_i * Float32(Float32(1.0) - u)));
	else
		tmp = Float32(Float32(n1_i - n0_i) * u);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -8e-23 or 3.99999987e-20 < n0_i

   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. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lift-*.f32 N/A

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

      7. lower--.f32 N/A

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 75.0

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

   7. Applied rewrites 75.0%

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

   8. Taylor expanded in normAngle around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 77.3

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

   10. Applied rewrites 76.6%

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

   if -8e-23 < n0_i < 3.99999987e-20

   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. lower-*.f32 64.5

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

   5. Applied rewrites 63.9%

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

   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 64.1%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -7.99999999855967 \cdot 10^{-23} \lor \neg \left(n0\_i \leq 3.99999987306209 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{fma}\left(n1\_i, u, n0\_i \cdot \left(1 - u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n1\_i - n0\_i\right) \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.2% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -7.99999999855967 \cdot 10^{-23} \lor \neg \left(n0\_i \leq 3.99999987306209 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{fma}\left(u, n1\_i, n0\_i \cdot \left(1 - u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n1\_i - n0\_i\right) \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -7.99999999855967e-23) (not (<= n0_i 3.99999987306209e-20)))
   (fma u n1_i (* n0_i (- 1.0 u)))
   (* (- n1_i n0_i) u)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -7.99999999855967e-23f) || !(n0_i <= 3.99999987306209e-20f)) {
		tmp = fmaf(u, n1_i, (n0_i * (1.0f - u)));
	} else {
		tmp = (n1_i - n0_i) * u;
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-7.99999999855967e-23)) || !(n0_i <= Float32(3.99999987306209e-20)))
		tmp = fma(u, n1_i, Float32(n0_i * Float32(Float32(1.0) - u)));
	else
		tmp = Float32(Float32(n1_i - n0_i) * u);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -8e-23 or 3.99999987e-20 < n0_i

   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 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 19.2

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

   5. Applied rewrites 19.1%

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

   7. Applied rewrites 76.6%

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

   if -8e-23 < n0_i < 3.99999987e-20

   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. lower-*.f32 64.5

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

   5. Applied rewrites 63.9%

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

   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 64.1%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -7.99999999855967 \cdot 10^{-23} \lor \neg \left(n0\_i \leq 3.99999987306209 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{fma}\left(u, n1\_i, n0\_i \cdot \left(1 - u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n1\_i - n0\_i\right) \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.5% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -7.99999999855967 \cdot 10^{-23} \lor \neg \left(n0\_i \leq 3.99999987306209 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n1\_i - n0\_i\right) \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -7.99999999855967e-23) (not (<= n0_i 3.99999987306209e-20)))
   (fma (- n1_i n0_i) u n0_i)
   (* (- n1_i n0_i) u)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -7.99999999855967e-23f) || !(n0_i <= 3.99999987306209e-20f)) {
		tmp = fmaf((n1_i - n0_i), u, n0_i);
	} else {
		tmp = (n1_i - n0_i) * u;
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-7.99999999855967e-23)) || !(n0_i <= Float32(3.99999987306209e-20)))
		tmp = fma(Float32(n1_i - n0_i), u, n0_i);
	else
		tmp = Float32(Float32(n1_i - n0_i) * u);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -8e-23 or 3.99999987e-20 < n0_i

   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 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 19.2

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

   5. Applied rewrites 19.2%

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

   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 16.5%

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

   9. Taylor expanded in n0_i around inf

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

   11. Applied rewrites 5.7%

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

   12. Taylor expanded in u around 0

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

   14. Applied rewrites 60.2%

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

   if -8e-23 < n0_i < 3.99999987e-20

   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. lower-*.f32 64.5

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

   5. Applied rewrites 64.5%

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

   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 64.1%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -7.99999999855967 \cdot 10^{-23} \lor \neg \left(n0\_i \leq 3.99999987306209 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n1\_i - n0\_i\right) \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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

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

   1. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in normAngle around 0

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

   1. lower--.f32 97.5

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

8. Applied rewrites 97.5%

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

Alternative 7: 36.6% accurate, 51.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n1_i - n0_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. lower-*.f32 42.2

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

5. Applied rewrites 41.9%

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

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 40.7%

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

9. Final simplification 40.7%

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

Alternative 8: 7.8% accurate, 57.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = -n0_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. lower-*.f32 42.2

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

5. Applied rewrites 41.9%

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

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 40.7%

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

9. Taylor expanded in n0_i around inf

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

11. Applied rewrites 8.2%

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

12. Final simplification 8.2%

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

Reproduce

herbie shell --seed 2024339 
(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)))