Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.0%

Time: 5.6s

Alternatives: 5

Speedup: 19.5×

Specification

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0\_i \land n0\_i \leq 1\right)\right) \land \left(-1 \leq n1\_i \land n1\_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (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)
use fmin_fmax_functions
    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

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0\_i \land n0\_i \leq 1\right)\right) \land \left(-1 \leq n1\_i \land n1\_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (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)
use fmin_fmax_functions
    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: 99.0% accurate, 6.7× speedup

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0\_i \land n0\_i \leq 1\right)\right) \land \left(-1 \leq n1\_i \land n1\_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

\[\mathsf{fma}\left(\left(n1\_i - n0\_i\right) - \mathsf{fma}\left(-0.3333333333333333, n0\_i, -0.16666666666666666 \cdot n1\_i\right) \cdot \left(normAngle \cdot normAngle\right), u, n0\_i\right) \]

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (fma
 (-
  (- n1_i n0_i)
  (*
   (fma -0.3333333333333333 n0_i (* -0.16666666666666666 n1_i))
   (* normAngle normAngle)))
 u
 n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(((n1_i - n0_i) - (fmaf(-0.3333333333333333f, n0_i, (-0.16666666666666666f * n1_i)) * (normAngle * normAngle))), u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(Float32(n1_i - n0_i) - Float32(fma(Float32(-0.3333333333333333), n0_i, Float32(Float32(-0.16666666666666666) * n1_i)) * Float32(normAngle * normAngle))), u, n0_i)
end

Derivation

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. Taylor expanded in normAngle around 0

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - \left(\frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right) \]
3. Step-by-step derivation

4. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \mathsf{fma}\left(n1\_i, u, {normAngle}^{2} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, n0\_i \cdot {\left(1 - u\right)}^{3}, -0.16666666666666666 \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - \mathsf{fma}\left(-0.16666666666666666, n0\_i \cdot \left(1 - u\right), -0.16666666666666666 \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right) \]

5. Taylor expanded in u around 0

   \[\leadsto n0\_i + u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{1}{2} \cdot n0\_i - \left(\frac{-1}{6} \cdot n1\_i + \frac{1}{6} \cdot n0\_i\right)\right)\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 98.9%

   \[\leadsto n0\_i + u \cdot \left(n1\_i + \mathsf{fma}\left(-1, n0\_i, {normAngle}^{2} \cdot \left(0.5 \cdot n0\_i - \mathsf{fma}\left(-0.16666666666666666, n1\_i, 0.16666666666666666 \cdot n0\_i\right)\right)\right)\right) \]

9. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(\left(n1\_i - n0\_i\right) - \mathsf{fma}\left(-0.16666666666666666, n1\_i - n0\_i, -0.5 \cdot n0\_i\right) \cdot \left(normAngle \cdot normAngle\right), u, n0\_i\right) \]

10. Taylor expanded in n0_i around 0

    \[\leadsto \mathsf{fma}\left(\left(n1\_i - n0\_i\right) - \left(\frac{-1}{3} \cdot n0\_i + \frac{-1}{6} \cdot n1\_i\right) \cdot \left(normAngle \cdot normAngle\right), u, n0\_i\right) \]
11. Step-by-step derivation

12. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(\left(n1\_i - n0\_i\right) - \mathsf{fma}\left(-0.3333333333333333, n0\_i, -0.16666666666666666 \cdot n1\_i\right) \cdot \left(normAngle \cdot normAngle\right), u, n0\_i\right) \]
13. Add Preprocessing

Alternative 2: 98.1% accurate, 19.5× speedup

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0\_i \land n0\_i \leq 1\right)\right) \land \left(-1 \leq n1\_i \land n1\_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (fma (- n1_i n0_i) u n0_i))

C

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

Julia

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

Derivation

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. Taylor expanded in u around 0

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

4. Applied rewrites 88.8%

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

5. Taylor expanded in normAngle around 0

   \[\leadsto n0\_i + u \cdot \left(n1\_i + -1 \cdot n0\_i\right) \]
6. Step-by-step derivation

7. Applied rewrites 98.0%

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

9. Applied rewrites 98.1%

   \[\leadsto \mathsf{fma}\left(n1\_i - n0\_i, u, n0\_i\right) \]
10. Add Preprocessing

Alternative 3: 81.8% accurate, 28.0× speedup

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0\_i \land n0\_i \leq 1\right)\right) \land \left(-1 \leq n1\_i \land n1\_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (fma u n1_i n0_i))

C

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

Julia

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

Derivation

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. Taylor expanded in u around 0

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

4. Applied rewrites 81.3%

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

5. Taylor expanded in normAngle around 0

   \[\leadsto n0\_i + u \cdot n1\_i \]
6. Step-by-step derivation

7. Applied rewrites 81.7%

   \[\leadsto n0\_i + u \cdot n1\_i \]
8. Step-by-step derivation

9. Applied rewrites 81.8%

   \[\leadsto \mathsf{fma}\left(u, n1\_i, n0\_i\right) \]
10. Add Preprocessing

Alternative 4: 60.1% accurate, 12.3× speedup

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0\_i \land n0\_i \leq 1\right)\right) \land \left(-1 \leq n1\_i \land n1\_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := -\left(-u\right) \cdot n1\_i\\ \mathbf{if}\;n1\_i \leq -1.4564692236958598 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n1\_i \leq 1.1249517491201857 \cdot 10^{-22}:\\ \;\;\;\;\left(-n0\_i\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (let* ((t_0 (- (* (- u) n1_i))))
  (if (<= n1_i -1.4564692236958598e-18)
    t_0
    (if (<= n1_i 1.1249517491201857e-22) (* (- n0_i) -1.0) t_0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = -(-u * n1_i);
	float tmp;
	if (n1_i <= -1.4564692236958598e-18f) {
		tmp = t_0;
	} else if (n1_i <= 1.1249517491201857e-22f) {
		tmp = -n0_i * -1.0f;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
use fmin_fmax_functions
    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 = -(-u * n1_i)
    if (n1_i <= (-1.4564692236958598e-18)) then
        tmp = t_0
    else if (n1_i <= 1.1249517491201857e-22) then
        tmp = -n0_i * (-1.0e0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(-Float32(Float32(-u) * n1_i))
	tmp = Float32(0.0)
	if (n1_i <= Float32(-1.4564692236958598e-18))
		tmp = t_0;
	elseif (n1_i <= Float32(1.1249517491201857e-22))
		tmp = Float32(Float32(-n0_i) * Float32(-1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	t_0 = -(-u * n1_i);
	tmp = single(0.0);
	if (n1_i <= single(-1.4564692236958598e-18))
		tmp = t_0;
	elseif (n1_i <= single(1.1249517491201857e-22))
		tmp = -n0_i * single(-1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.45646922e-18 or 1.12495175e-22 < n1_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. Taylor expanded in normAngle around 0

      \[\leadsto n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u \]
   3. Step-by-step derivation

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in n0_i around -inf

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in n0_i around 0

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

   10. Applied rewrites 37.9%

       \[\leadsto -1 \cdot \left(-1 \cdot \left(n1\_i \cdot u\right)\right) \]

   12. Applied rewrites 37.9%

       \[\leadsto -\left(-u\right) \cdot n1\_i \]

   if -1.45646922e-18 < n1_i < 1.12495175e-22

   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. Taylor expanded in normAngle around 0

      \[\leadsto n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u \]
   3. Step-by-step derivation

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in n0_i around -inf

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in u around 0

      \[\leadsto -1 \cdot \left(n0\_i \cdot -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 47.5%

       \[\leadsto -1 \cdot \left(n0\_i \cdot -1\right) \]

   12. Applied rewrites 47.5%

       \[\leadsto \left(-n0\_i\right) \cdot -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 47.5% accurate, 33.6× speedup

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0\_i \land n0\_i \leq 1\right)\right) \land \left(-1 \leq n1\_i \land n1\_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

\[\left(-n0\_i\right) \cdot -1 \]

FPCore

(FPCore (normAngle u n0_i n1_i)
  :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)))
  (* (- n0_i) -1.0))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. Taylor expanded in normAngle around 0

   \[\leadsto n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u \]
3. Step-by-step derivation

4. Applied rewrites 97.8%

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

5. Taylor expanded in n0_i around -inf

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

7. Applied rewrites 97.6%

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

8. Taylor expanded in u around 0

   \[\leadsto -1 \cdot \left(n0\_i \cdot -1\right) \]
9. Step-by-step derivation

10. Applied rewrites 47.5%

    \[\leadsto -1 \cdot \left(n0\_i \cdot -1\right) \]

12. Applied rewrites 47.5%

    \[\leadsto \left(-n0\_i\right) \cdot -1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2026070 
(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)))