Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.4% → 99.0%

Time: 5.7s

Alternatives: 13

Speedup: 14.5×

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
  (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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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: 97.4% accurate, 1.0× speedup

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
  (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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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, 2.8× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - u\right) \cdot n0\_i\\ t_1 := \mathsf{fma}\left(n1\_i, u, t\_0\right)\\ \mathsf{fma}\left(\left(-0.16666666666666666 \cdot \left(\mathsf{fma}\left(1 + -2 \cdot u, t\_0, \left(n1\_i \cdot \left(u \cdot u\right)\right) \cdot u\right) - t\_1\right)\right) \cdot normAngle, normAngle, t\_1\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
  :precision binary32
  (let* ((t_0 (* (- 1.0 u) n0_i)) (t_1 (fma n1_i u t_0)))
  (fma
   (*
    (*
     -0.16666666666666666
     (- (fma (+ 1.0 (* -2.0 u)) t_0 (* (* n1_i (* u u)) u)) t_1))
    normAngle)
   normAngle
   t_1)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = (1.0f - u) * n0_i;
	float t_1 = fmaf(n1_i, u, t_0);
	return fmaf(((-0.16666666666666666f * (fmaf((1.0f + (-2.0f * u)), t_0, ((n1_i * (u * u)) * u)) - t_1)) * normAngle), normAngle, t_1);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(Float32(1.0) - u) * n0_i)
	t_1 = fma(n1_i, u, t_0)
	return fma(Float32(Float32(Float32(-0.16666666666666666) * Float32(fma(Float32(Float32(1.0) + Float32(Float32(-2.0) * u)), t_0, Float32(Float32(n1_i * Float32(u * u)) * u)) - t_1)) * normAngle), normAngle, t_1)
end

Derivation

1. Initial program 97.4%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in u around 0

   \[\leadsto \mathsf{fma}\left(\left(-0.16666666666666666 \cdot \left(\mathsf{fma}\left(1 + -2 \cdot u, \left(1 - u\right) \cdot n0\_i, \left(n1\_i \cdot \left(u \cdot u\right)\right) \cdot u\right) - \mathsf{fma}\left(n1\_i, u, \left(1 - u\right) \cdot n0\_i\right)\right)\right) \cdot normAngle, normAngle, \mathsf{fma}\left(n1\_i, u, \left(1 - u\right) \cdot n0\_i\right)\right) \]
8. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. lower-*.f32 98.9%

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

9. Applied rewrites 98.9%

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

Alternative 2: 99.0% accurate, 3.4× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :precision binary32
  (fma
 (*
  (*
   (*
    (fma
     (fma 3.0 n0_i (* (- n1_i n0_i) u))
     u
     (fma -3.0 n0_i (- n0_i n1_i)))
    u)
   normAngle)
  -0.16666666666666666)
 normAngle
 (fma n1_i u (* (- 1.0 u) n0_i))))

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(Float32(Float32(fma(fma(Float32(3.0), n0_i, Float32(Float32(n1_i - n0_i) * u)), u, fma(Float32(-3.0), n0_i, Float32(n0_i - n1_i))) * u) * normAngle) * Float32(-0.16666666666666666)), normAngle, fma(n1_i, u, Float32(Float32(Float32(1.0) - u) * n0_i)))
end

Derivation

1. Initial program 97.4%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in u around 0

   \[\leadsto \mathsf{fma}\left(\left(-0.16666666666666666 \cdot \left(u \cdot \left(\left(-3 \cdot n0\_i + u \cdot \left(3 \cdot n0\_i + u \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right) - \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \cdot normAngle, normAngle, \mathsf{fma}\left(n1\_i, u, \left(1 - u\right) \cdot n0\_i\right)\right) \]
8. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 99.0%

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

9. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(\left(-0.16666666666666666 \cdot \left(u \cdot \left(\mathsf{fma}\left(-3, n0\_i, u \cdot \mathsf{fma}\left(3, n0\_i, u \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right) - \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \cdot normAngle, normAngle, \mathsf{fma}\left(n1\_i, u, \left(1 - u\right) \cdot n0\_i\right)\right) \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

11. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(3, n0\_i, \left(n1\_i - n0\_i\right) \cdot u\right), u, \mathsf{fma}\left(-3, n0\_i, n0\_i - n1\_i\right)\right) \cdot u\right) \cdot normAngle\right) \cdot -0.16666666666666666, normAngle, \mathsf{fma}\left(n1\_i, u, \left(1 - u\right) \cdot n0\_i\right)\right) \]
12. Add Preprocessing

Alternative 3: 98.9% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :precision binary32
  (+
 n0_i
 (*
  u
  (+
   n1_i
   (fma
    -1.0
    n0_i
    (*
     (pow normAngle 2.0)
     (-
      (* 0.5 n0_i)
      (fma
       -0.16666666666666666
       n1_i
       (* 0.16666666666666666 n0_i)))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * (n1_i + fmaf(-1.0f, n0_i, (powf(normAngle, 2.0f) * ((0.5f * n0_i) - fmaf(-0.16666666666666666f, n1_i, (0.16666666666666666f * n0_i)))))));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(n1_i + fma(Float32(-1.0), n0_i, Float32((normAngle ^ Float32(2.0)) * Float32(Float32(Float32(0.5) * n0_i) - fma(Float32(-0.16666666666666666), n1_i, Float32(Float32(0.16666666666666666) * n0_i))))))))
end

Derivation

1. Initial program 97.4%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in n1_i around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-*.f32 98.6%

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in u around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 99.0%

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

10. Applied rewrites 99.0%

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

Alternative 4: 98.9% accurate, 3.7× speedup

Math

\[\mathsf{fma}\left(\left(u \cdot \mathsf{fma}\left(-0.5, n0\_i \cdot u, -0.16666666666666666 \cdot \left(-3 \cdot n0\_i - \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \cdot normAngle, normAngle, \mathsf{fma}\left(n1\_i, u, \left(1 - u\right) \cdot n0\_i\right)\right) \]

FPCore

(FPCore (normAngle u n0_i n1_i)
  :precision binary32
  (fma
 (*
  (*
   u
   (fma
    -0.5
    (* n0_i u)
    (*
     -0.16666666666666666
     (- (* -3.0 n0_i) (+ n1_i (* -1.0 n0_i))))))
  normAngle)
 normAngle
 (fma n1_i u (* (- 1.0 u) n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(((u * fmaf(-0.5f, (n0_i * u), (-0.16666666666666666f * ((-3.0f * n0_i) - (n1_i + (-1.0f * n0_i)))))) * normAngle), normAngle, fmaf(n1_i, u, ((1.0f - u) * n0_i)));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(Float32(u * fma(Float32(-0.5), Float32(n0_i * u), Float32(Float32(-0.16666666666666666) * Float32(Float32(Float32(-3.0) * n0_i) - Float32(n1_i + Float32(Float32(-1.0) * n0_i)))))) * normAngle), normAngle, fma(n1_i, u, Float32(Float32(Float32(1.0) - u) * n0_i)))
end

Derivation

1. Initial program 97.4%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower-*.f32 98.9%

      \[\leadsto \mathsf{fma}\left(\left(u \cdot \mathsf{fma}\left(-0.5, n0\_i \cdot u, -0.16666666666666666 \cdot \left(-3 \cdot n0\_i - \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \cdot normAngle, normAngle, \mathsf{fma}\left(n1\_i, u, \left(1 - u\right) \cdot n0\_i\right)\right) \]

9. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(\left(u \cdot \mathsf{fma}\left(-0.5, n0\_i \cdot u, -0.16666666666666666 \cdot \left(-3 \cdot n0\_i - \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \cdot normAngle, normAngle, \mathsf{fma}\left(n1\_i, u, \left(1 - u\right) \cdot n0\_i\right)\right) \]
10. Add Preprocessing

Alternative 5: 98.8% accurate, 4.6× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :precision binary32
  (fma
 (*
  (*
   -0.16666666666666666
   (* u (- (* -3.0 n0_i) (+ n1_i (* -1.0 n0_i)))))
  normAngle)
 normAngle
 (fma n1_i u (* (- 1.0 u) n0_i))))

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(Float32(Float32(-0.16666666666666666) * Float32(u * Float32(Float32(Float32(-3.0) * n0_i) - Float32(n1_i + Float32(Float32(-1.0) * n0_i))))) * normAngle), normAngle, fma(n1_i, u, Float32(Float32(Float32(1.0) - u) * n0_i)))
end

Derivation

1. Initial program 97.4%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in u around 0

   \[\leadsto \mathsf{fma}\left(\left(-0.16666666666666666 \cdot \left(u \cdot \left(-3 \cdot n0\_i - \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \cdot normAngle, normAngle, \mathsf{fma}\left(n1\_i, u, \left(1 - u\right) \cdot n0\_i\right)\right) \]
8. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

   5. lower-*.f32 98.8%

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

9. Applied rewrites 98.8%

   \[\leadsto \mathsf{fma}\left(\left(-0.16666666666666666 \cdot \left(u \cdot \left(-3 \cdot n0\_i - \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \cdot normAngle, normAngle, \mathsf{fma}\left(n1\_i, u, \left(1 - u\right) \cdot n0\_i\right)\right) \]
10. Add Preprocessing

Alternative 6: 98.6% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :precision binary32
  (fma
 n0_i
 (- 1.0 u)
 (fma
  (* (- (* (* u u) u) u) -0.16666666666666666)
  (* (* normAngle normAngle) n1_i)
  (* n1_i u))))

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(n0_i, Float32(Float32(1.0) - u), fma(Float32(Float32(Float32(Float32(u * u) * u) - u) * Float32(-0.16666666666666666)), Float32(Float32(normAngle * normAngle) * n1_i), Float32(n1_i * u)))
end

Derivation

1. Initial program 97.4%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in n1_i around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-*.f32 98.6%

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

7. Applied rewrites 98.6%

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, n1\_i \cdot \left(u + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {u}^{3} - -0.16666666666666666 \cdot u\right)\right)\right) \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lower-fma.f32 N/A

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

9. Applied rewrites 98.6%

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

Alternative 7: 98.6% accurate, 5.4× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :precision binary32
  (fma
 (- 1.0 u)
 n0_i
 (*
  (fma
   (* (* normAngle normAngle) -0.16666666666666666)
   (- (* (* u u) u) u)
   u)
  n1_i)))

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(Float32(1.0) - u), n0_i, Float32(fma(Float32(Float32(normAngle * normAngle) * Float32(-0.16666666666666666)), Float32(Float32(Float32(u * u) * u) - u), u) * n1_i))
end

Derivation

1. Initial program 97.4%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in n1_i around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-*.f32 98.6%

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

7. Applied rewrites 98.6%

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, n1\_i \cdot \left(u + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {u}^{3} - -0.16666666666666666 \cdot u\right)\right)\right) \]
8. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 98.6%

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

   4. lift-*.f32 N/A

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

9. Applied rewrites 98.6%

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

Alternative 8: 98.6% accurate, 5.6× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :precision binary32
  (fma
 n0_i
 (- 1.0 u)
 (*
  n1_i
  (*
   (-
    1.0
    (/ (* (* -0.16666666666666666 u) (* normAngle normAngle)) u))
   u))))

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(n0_i, Float32(Float32(1.0) - u), Float32(n1_i * Float32(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(-0.16666666666666666) * u) * Float32(normAngle * normAngle)) / u)) * u)))
end

Derivation

1. Initial program 97.4%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in n1_i around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-*.f32 98.6%

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

7. Applied rewrites 98.6%

   \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, n1\_i \cdot \left(u + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {u}^{3} - -0.16666666666666666 \cdot u\right)\right)\right) \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. add-flip N/A

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

   3. sub-to-mult N/A

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

   4. lower-unsound-*.f32 N/A

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

9. Applied rewrites 98.6%

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

10. Taylor expanded in u around 0

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

    1. lower-*.f32 98.6%

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

12. Applied rewrites 98.6%

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

Alternative 9: 98.2% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 + ((-1.0e0) * n0_i)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 98.0%

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in u around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 98.2%

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

7. Applied rewrites 98.2%

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

Alternative 10: 98.0% accurate, 14.5× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :precision binary32
  (fma u n1_i (* (- 1.0 u) n0_i)))

C

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

Julia

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

Derivation

1. Initial program 97.4%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 98.0%

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

4. Applied rewrites 98.0%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 98.0%

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 98.0%

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

6. Applied rewrites 98.0%

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

Alternative 11: 98.0% accurate, 14.5× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :precision binary32
  (fma n0_i (- 1.0 u) (* n1_i u)))

C

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

Julia

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

Derivation

1. Initial program 97.4%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 98.0%

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

4. Applied rewrites 98.0%

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

Alternative 12: 82.1% accurate, 18.6× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :precision binary32
  (fma u n1_i (* 1.0 n0_i)))

C

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

Julia

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

Derivation

1. Initial program 97.4%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 98.0%

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 82.0%

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

   1. lift-fma.f32 N/A

      \[\leadsto n0\_i \cdot 1 + \color{blue}{n1\_i \cdot u} \]

   2. +-commutative N/A

      \[\leadsto n1\_i \cdot u + \color{blue}{n0\_i \cdot 1} \]

   3. lift-*.f32 N/A

      \[\leadsto n1\_i \cdot u + \color{blue}{n0\_i} \cdot 1 \]

   4. *-commutative N/A

      \[\leadsto u \cdot n1\_i + \color{blue}{n0\_i} \cdot 1 \]

   5. lower-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 82.1%

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

9. Applied rewrites 82.1%

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

Alternative 13: 82.0% accurate, 18.6× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
  :precision binary32
  (fma n0_i 1.0 (* n1_i u)))

C

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

Julia

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

Derivation

1. Initial program 97.4%

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

2. Taylor expanded in normAngle around 0

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

   1. lower-fma.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 98.0%

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 82.0%

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

Reproduce

herbie shell --seed 2025212 
(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)))