Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 98.0%

Time: 11.2s

Alternatives: 4

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: 97.2% 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: 98.0% accurate, 38.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 48.4

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

5. Applied rewrites 48.3%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lower-+.f32 98.2

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

7. Applied rewrites 98.2%

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

   1. sub-neg N/A

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

   2. lift-neg.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt1-in N/A

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

   5. lift-neg.f32 N/A

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

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

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

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 98.4

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

9. Applied rewrites 98.4%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-+l- N/A

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

    4. lower--.f32 N/A

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

    5. lift-*.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. distribute-lft-out-- N/A

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

    8. lower-*.f32 N/A

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

    9. lower--.f32 98.5

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

11. Applied rewrites 98.5%

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

12. Final simplification 98.5%

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

Alternative 2: 70.5% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n0\_i \cdot \left(1 - u\right)\\ \mathbf{if}\;n0\_i \leq -1.4999999523982838 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* n0_i (- 1.0 u))))
   (if (<= n0_i -1.4999999523982838e-21)
     t_0
     (if (<= n0_i 4.999999841327613e-22) (* u n1_i) t_0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = n0_i * (1.0f - u);
	float tmp;
	if (n0_i <= -1.4999999523982838e-21f) {
		tmp = t_0;
	} else if (n0_i <= 4.999999841327613e-22f) {
		tmp = u * n1_i;
	} 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 = n0_i * (1.0e0 - u)
    if (n0_i <= (-1.4999999523982838e-21)) then
        tmp = t_0
    else if (n0_i <= 4.999999841327613e-22) then
        tmp = u * n1_i
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(n0_i * Float32(Float32(1.0) - u))
	tmp = Float32(0.0)
	if (n0_i <= Float32(-1.4999999523982838e-21))
		tmp = t_0;
	elseif (n0_i <= Float32(4.999999841327613e-22))
		tmp = Float32(u * n1_i);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	t_0 = n0_i * (single(1.0) - u);
	tmp = single(0.0);
	if (n0_i <= single(-1.4999999523982838e-21))
		tmp = t_0;
	elseif (n0_i <= single(4.999999841327613e-22))
		tmp = u * n1_i;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.5e-21 or 4.9999998e-22 < n0_i

   1. Initial program 98.4%

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

   3. Taylor expanded in n0_i around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

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

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f32 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f32 N/A

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

      15. lower-sin.f32 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in normAngle around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f32 86.1

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

   8. Applied rewrites 86.1%

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

   if -1.5e-21 < n0_i < 4.9999998e-22

   1. Initial program 96.8%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 29.1

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

   5. Applied rewrites 28.4%

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

   6. Taylor expanded in n0_i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 64.0

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

   8. Applied rewrites 64.0%

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

Alternative 3: 60.6% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.4999999523982838 \cdot 10^{-21}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.4999999523982838e-21)
   n0_i
   (if (<= n0_i 4.999999841327613e-22) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.4999999523982838e-21f) {
		tmp = n0_i;
	} else if (n0_i <= 4.999999841327613e-22f) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-1.4999999523982838e-21)) then
        tmp = n0_i
    else if (n0_i <= 4.999999841327613e-22) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-1.4999999523982838e-21))
		tmp = n0_i;
	elseif (n0_i <= Float32(4.999999841327613e-22))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-1.4999999523982838e-21))
		tmp = n0_i;
	elseif (n0_i <= single(4.999999841327613e-22))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.5e-21 or 4.9999998e-22 < n0_i

   1. Initial program 98.4%

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

   3. Taylor expanded in n0_i around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

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

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

      10. distribute-rgt-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f32 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f32 N/A

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

      15. lower-sin.f32 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in u around 0

      \[\leadsto n0\_i \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.7%

      \[\leadsto n0\_i \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 67.7

         \[\leadsto \color{blue}{n0\_i} \]

   10. Applied rewrites 67.7%

       \[\leadsto \color{blue}{n0\_i} \]

   if -1.5e-21 < n0_i < 4.9999998e-22

   1. Initial program 96.8%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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. lower-fma.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 29.1

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

   5. Applied rewrites 28.4%

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

   6. Taylor expanded in n0_i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 64.0

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

   8. Applied rewrites 64.0%

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

Alternative 4: 47.5% accurate, 459.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in n0_i around inf

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. *-commutative N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-lft1-in N/A

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

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

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

   10. distribute-rgt-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. lower-fma.f32 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f32 N/A

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

   15. lower-sin.f32 45.6

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

5. Applied rewrites 45.6%

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

6. Taylor expanded in u around 0

   \[\leadsto n0\_i \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 48.4%

   \[\leadsto n0\_i \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 48.4

      \[\leadsto \color{blue}{n0\_i} \]

10. Applied rewrites 48.4%

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

Reproduce

herbie shell --seed 2024216 
(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)))