Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 98.9%

Time: 37.3s

Alternatives: 7

Speedup: 18.2×

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: 97.2% 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: 98.9% accurate, 1.6× 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

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

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
 n0_i
 (- 1.0 u)
 (fma
  n1_i
  u
  (*
   (pow normAngle 2.0)
   (-
    (fma
     -0.16666666666666666
     (* n0_i (pow (- 1.0 u) 3.0))
     (* -0.16666666666666666 (* n1_i (pow u 3.0))))
    (fma
     -0.16666666666666666
     (* n0_i (- 1.0 u))
     (* -0.16666666666666666 (* n1_i u))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(n0_i, (1.0f - u), fmaf(n1_i, u, (powf(normAngle, 2.0f) * (fmaf(-0.16666666666666666f, (n0_i * powf((1.0f - u), 3.0f)), (-0.16666666666666666f * (n1_i * powf(u, 3.0f)))) - fmaf(-0.16666666666666666f, (n0_i * (1.0f - u)), (-0.16666666666666666f * (n1_i * u)))))));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(n0_i, Float32(Float32(1.0) - u), fma(n1_i, u, Float32((normAngle ^ Float32(2.0)) * Float32(fma(Float32(-0.16666666666666666), Float32(n0_i * (Float32(Float32(1.0) - u) ^ Float32(3.0))), Float32(Float32(-0.16666666666666666) * Float32(n1_i * (u ^ Float32(3.0))))) - fma(Float32(-0.16666666666666666), Float32(n0_i * Float32(Float32(1.0) - u)), Float32(Float32(-0.16666666666666666) * Float32(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. Taylor expanded in normAngle around 0

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

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. Add Preprocessing

Alternative 2: 98.9% accurate, 5.1× 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

\[n0\_i + u \cdot \mathsf{fma}\left(\mathsf{fma}\left(u, -0.5 \cdot n0\_i, \mathsf{fma}\left(n0\_i, 0.3333333333333333, n1\_i \cdot 0.16666666666666666\right)\right), normAngle \cdot normAngle, n1\_i - n0\_i\right) \]

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)))
  (+
 n0_i
 (*
  u
  (fma
   (fma
    u
    (* -0.5 n0_i)
    (fma n0_i 0.3333333333333333 (* n1_i 0.16666666666666666)))
   (* normAngle normAngle)
   (- n1_i n0_i)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * fmaf(fmaf(u, (-0.5f * n0_i), fmaf(n0_i, 0.3333333333333333f, (n1_i * 0.16666666666666666f))), (normAngle * normAngle), (n1_i - n0_i)));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * fma(fma(u, Float32(Float32(-0.5) * n0_i), fma(n0_i, Float32(0.3333333333333333), Float32(n1_i * Float32(0.16666666666666666)))), Float32(normAngle * normAngle), Float32(n1_i - n0_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. Taylor expanded in u around 0

   \[\leadsto n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) \]

4. Applied rewrites 88.4%

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

5. Taylor expanded in normAngle around 0

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

7. Applied rewrites 98.9%

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

9. Applied rewrites 98.9%

   \[\leadsto n0\_i + u \cdot \mathsf{fma}\left(\mathsf{fma}\left(u, -0.5 \cdot n0\_i, \mathsf{fma}\left(n0\_i, 0.3333333333333333, n1\_i \cdot 0.16666666666666666\right)\right), normAngle \cdot normAngle, n1\_i - n0\_i\right) \]
10. Add Preprocessing

Alternative 3: 98.9% 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

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

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)))
  (+
 n0_i
 (*
  u
  (fma
   (fma 0.16666666666666666 n1_i (* 0.3333333333333333 n0_i))
   (* normAngle normAngle)
   (- n1_i n0_i)))))

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * fma(fma(Float32(0.16666666666666666), n1_i, Float32(Float32(0.3333333333333333) * n0_i)), Float32(normAngle * normAngle), Float32(n1_i - n0_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. Taylor expanded in u around 0

   \[\leadsto n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \left(\frac{-1}{2} \cdot \left(n0\_i \cdot \left({normAngle}^{2} \cdot u\right)\right) + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)\right) \]

4. Applied rewrites 88.4%

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

5. Taylor expanded in normAngle around 0

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

7. Applied rewrites 98.9%

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

9. Applied rewrites 98.9%

   \[\leadsto n0\_i + u \cdot \mathsf{fma}\left(\mathsf{fma}\left(u, -0.5 \cdot n0\_i, \mathsf{fma}\left(n0\_i, 0.3333333333333333, n1\_i \cdot 0.16666666666666666\right)\right), normAngle \cdot normAngle, n1\_i - n0\_i\right) \]

10. Taylor expanded in u around 0

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

12. Applied rewrites 98.9%

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

Alternative 4: 98.5% accurate, 7.2× 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

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

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
 n0_i
 (- 1.0 u)
 (*
  u
  (- n1_i (* (* -0.16666666666666666 n1_i) (* normAngle normAngle))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(n0_i, (1.0f - u), (u * (n1_i - ((-0.16666666666666666f * n1_i) * (normAngle * normAngle)))));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(n0_i, Float32(Float32(1.0) - u), Float32(u * Float32(n1_i - Float32(Float32(Float32(-0.16666666666666666) * n1_i) * Float32(normAngle * normAngle)))))
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. Taylor expanded in normAngle around 0

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

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

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, u \cdot \left(n1\_i + {normAngle}^{2} \cdot \left(\frac{1}{2} \cdot n0\_i - \left(\frac{-1}{6} \cdot n1\_i + \frac{1}{6} \cdot n0\_i\right)\right)\right)\right) \]

7. Applied rewrites 98.7%

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

9. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, u \cdot \left(n1\_i - \left(-0.16666666666666666 \cdot n1\_i - n0\_i \cdot 0.3333333333333333\right) \cdot \left(normAngle \cdot normAngle\right)\right)\right) \]

10. Taylor expanded in n0_i around 0

    \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, u \cdot \left(n1\_i - \left(\frac{-1}{6} \cdot n1\_i\right) \cdot \left(normAngle \cdot normAngle\right)\right)\right) \]

12. Applied rewrites 98.5%

    \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, u \cdot \left(n1\_i - \left(-0.16666666666666666 \cdot n1\_i\right) \cdot \left(normAngle \cdot normAngle\right)\right)\right) \]
13. Add Preprocessing

Alternative 5: 98.0% accurate, 18.2× 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

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

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)))
  (+ 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)
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 + (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 97.2%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 88.3%

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

5. Taylor expanded in normAngle around 0

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

7. Applied rewrites 98.0%

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

9. Applied rewrites 98.0%

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

Alternative 6: 83.9% accurate, 16.1× 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

\[\begin{array}{l} \mathbf{if}\;n0\_i \leq 3.5105799560744555 \cdot 10^{-13}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]

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)))
  (if (<= n0_i 3.5105799560744555e-13)
  (+ n0_i (* u n1_i))
  (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= 3.5105799560744555e-13f) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i * (1.0f - u);
	}
	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) :: tmp
    if (n0_i <= 3.5105799560744555e-13) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(3.5105799560744555e-13))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(3.5105799560744555e-13))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < 3.51057996e-13

   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. Taylor expanded in u around 0

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

   4. Applied rewrites 81.2%

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

   5. Taylor expanded in normAngle around 0

      \[\leadsto n0\_i + u \cdot n1\_i \]

   7. Applied rewrites 81.7%

      \[\leadsto n0\_i + u \cdot n1\_i \]

   if 3.51057996e-13 < n0_i

   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. Taylor expanded in normAngle around 0

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

   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 n0_i around inf

      \[\leadsto n0\_i \cdot \left(\left(1 + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot {\left(1 - u\right)}^{3} - \frac{-1}{6} \cdot \left(1 - u\right)\right)\right) - u\right) \]

   7. Applied rewrites 59.0%

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

   8. Taylor expanded in normAngle around 0

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

   10. Applied rewrites 58.8%

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

Alternative 7: 58.8% accurate, 25.4× 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

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

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)))
  (* n0_i (- 1.0 u)))

C

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

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 - u)
end function

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i * (single(1.0) - 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. Taylor expanded in normAngle around 0

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

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 n0_i around inf

   \[\leadsto n0\_i \cdot \left(\left(1 + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot {\left(1 - u\right)}^{3} - \frac{-1}{6} \cdot \left(1 - u\right)\right)\right) - u\right) \]

7. Applied rewrites 59.0%

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

8. Taylor expanded in normAngle around 0

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

10. Applied rewrites 58.8%

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

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(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)))