Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 98.8%

Time: 10.5s

Alternatives: 5

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: 98.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n1_i * ((normAngle / sin(normAngle)) * u)) + (n0_i * (single(1.0) - 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 98.9

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

5. Applied rewrites 98.9%

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

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 99.2

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

8. Applied rewrites 99.2%

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

9. Final simplification 99.2%

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

Alternative 2: 98.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (((u / sin(normAngle)) * normAngle) * n1_i) + (n0_i * (single(1.0) - 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 98.9

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

5. Applied rewrites 98.9%

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

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 99.2

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

8. Applied rewrites 99.2%

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

10. Applied rewrites 99.2%

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

11. Final simplification 99.2%

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

Alternative 3: 70.7% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n0\_i \cdot \left(1 - u\right)\\ \mathbf{if}\;n0\_i \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;\left(n1\_i - n0\_i\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* n0_i (- 1.0 u))))
   (if (<= n0_i -1.999999936531045e-21)
     t_0
     (if (<= n0_i 1.9999999774532045e-26) (* (- n1_i n0_i) u) 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 <= -1.999999936531045e-21f) {
		tmp = t_0;
	} else if (n0_i <= 1.9999999774532045e-26f) {
		tmp = (n1_i - n0_i) * u;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: t_0
    real(4) :: tmp
    t_0 = n0_i * (1.0e0 - u)
    if (n0_i <= (-1.999999936531045e-21)) then
        tmp = t_0
    else if (n0_i <= 1.9999999774532045e-26) then
        tmp = (n1_i - n0_i) * u
    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(-1.999999936531045e-21))
		tmp = t_0;
	elseif (n0_i <= Float32(1.9999999774532045e-26))
		tmp = Float32(Float32(n1_i - n0_i) * u);
	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(-1.999999936531045e-21))
		tmp = t_0;
	elseif (n0_i <= single(1.9999999774532045e-26))
		tmp = (n1_i - n0_i) * u;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.9999999e-21 or 1.99999998e-26 < n0_i

   1. Initial program 98.5%

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

   3. Taylor expanded in 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 98.5

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

   5. Applied rewrites 98.5%

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

   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 99.2

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in normAngle around 0

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

       1. sub-neg N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

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

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

       5. mul-1-neg N/A

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

       6. associate-+r+ N/A

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

       7. +-commutative N/A

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

       8. associate-*r* N/A

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

       9. distribute-rgt-in N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f32 N/A

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

       13. mul-1-neg N/A

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

       14. sub-neg N/A

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

       15. lower--.f32 58.0

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

   11. Applied rewrites 58.0%

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

   12. Taylor expanded in n0_i around inf

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

   14. Applied rewrites 76.2%

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

   if -1.9999999e-21 < n0_i < 1.99999998e-26

   1. Initial program 96.7%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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. lower-*.f32 65.8

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

   5. Applied rewrites 65.7%

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

   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 65.3%

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

Alternative 4: 98.0% accurate, 38.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = ((n1_i - n0_i) * u) + n0_i;
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 98.9

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

5. Applied rewrites 98.9%

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

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 99.2

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in normAngle around 0

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

    1. sub-neg N/A

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

    2. distribute-lft-in N/A

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

    3. *-rgt-identity N/A

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

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

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

    5. mul-1-neg N/A

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

    6. associate-+r+ N/A

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

    7. +-commutative N/A

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

    8. associate-*r* N/A

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

    9. distribute-rgt-in N/A

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

    10. +-commutative N/A

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

    11. *-commutative N/A

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

    12. lower-fma.f32 N/A

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

    13. mul-1-neg N/A

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

    14. sub-neg N/A

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

    15. lower--.f32 43.8

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

11. Applied rewrites 43.8%

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

13. Applied rewrites 97.8%

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

Alternative 5: 37.0% 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 97.7%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
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. lower-*.f32 40.8

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

5. Applied rewrites 40.8%

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

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 39.1%

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

Reproduce

herbie shell --seed 2024296 
(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)))