Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.4%

Time: 11.4s

Alternatives: 7

Speedup: 45.9×

Specification

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

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sin normAngle))))
   (+
    (* (* (sin (* (- 1.0 u) normAngle)) t_0) n0_i)
    (* (* (sin (* u normAngle)) t_0) n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = 1.0f / sinf(normAngle);
	return ((sinf(((1.0f - u) * normAngle)) * t_0) * n0_i) + ((sinf((u * normAngle)) * t_0) * n1_i);
}

Fortran

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: 97.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sin normAngle))))
   (+
    (* (* (sin (* (- 1.0 u) normAngle)) t_0) n0_i)
    (* (* (sin (* u normAngle)) t_0) n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = 1.0f / sinf(normAngle);
	return ((sinf(((1.0f - u) * normAngle)) * t_0) * n0_i) + ((sinf((u * normAngle)) * t_0) * n1_i);
}

Fortran

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: 99.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\\ t_1 := \mathsf{fma}\left(0.5, n0\_i, t\_0\right)\\ t_2 := \mathsf{fma}\left(t\_1, -0.16666666666666666, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot 0.008333333333333333\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot n0\_i - \mathsf{fma}\left(-0.041666666666666664 \cdot n0\_i - t\_2, -0.16666666666666666, \mathsf{fma}\left(t\_1, 0.008333333333333333, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot -0.0001984126984126984\right)\right), normAngle \cdot normAngle, -0.041666666666666664 \cdot n0\_i\right) - t\_2, normAngle \cdot normAngle, t\_0\right)\right), normAngle \cdot normAngle, -n0\_i\right) + n1\_i, u, n0\_i\right) \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* 0.16666666666666666 (fma -1.0 n0_i n1_i)))
        (t_1 (fma 0.5 n0_i t_0))
        (t_2
         (fma
          t_1
          -0.16666666666666666
          (* (fma -1.0 n0_i n1_i) 0.008333333333333333))))
   (fma
    (+
     (fma
      (fma
       0.5
       n0_i
       (fma
        (-
         (fma
          (-
           (* 0.001388888888888889 n0_i)
           (fma
            (- (* -0.041666666666666664 n0_i) t_2)
            -0.16666666666666666
            (fma
             t_1
             0.008333333333333333
             (* (fma -1.0 n0_i n1_i) -0.0001984126984126984))))
          (* normAngle normAngle)
          (* -0.041666666666666664 n0_i))
         t_2)
        (* normAngle normAngle)
        t_0))
      (* normAngle normAngle)
      (- n0_i))
     n1_i)
    u
    n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = 0.16666666666666666f * fmaf(-1.0f, n0_i, n1_i);
	float t_1 = fmaf(0.5f, n0_i, t_0);
	float t_2 = fmaf(t_1, -0.16666666666666666f, (fmaf(-1.0f, n0_i, n1_i) * 0.008333333333333333f));
	return fmaf((fmaf(fmaf(0.5f, n0_i, fmaf((fmaf(((0.001388888888888889f * n0_i) - fmaf(((-0.041666666666666664f * n0_i) - t_2), -0.16666666666666666f, fmaf(t_1, 0.008333333333333333f, (fmaf(-1.0f, n0_i, n1_i) * -0.0001984126984126984f)))), (normAngle * normAngle), (-0.041666666666666664f * n0_i)) - t_2), (normAngle * normAngle), t_0)), (normAngle * normAngle), -n0_i) + n1_i), u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(0.16666666666666666) * fma(Float32(-1.0), n0_i, n1_i))
	t_1 = fma(Float32(0.5), n0_i, t_0)
	t_2 = fma(t_1, Float32(-0.16666666666666666), Float32(fma(Float32(-1.0), n0_i, n1_i) * Float32(0.008333333333333333)))
	return fma(Float32(fma(fma(Float32(0.5), n0_i, fma(Float32(fma(Float32(Float32(Float32(0.001388888888888889) * n0_i) - fma(Float32(Float32(Float32(-0.041666666666666664) * n0_i) - t_2), Float32(-0.16666666666666666), fma(t_1, Float32(0.008333333333333333), Float32(fma(Float32(-1.0), n0_i, n1_i) * Float32(-0.0001984126984126984))))), Float32(normAngle * normAngle), Float32(Float32(-0.041666666666666664) * n0_i)) - t_2), Float32(normAngle * normAngle), t_0)), Float32(normAngle * normAngle), Float32(-n0_i)) + n1_i), 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. Add Preprocessing

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 88.8%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{2} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{24} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{1}{720} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{1}{2} \cdot n0\_i - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right) + \frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right) + \left(\frac{-1}{5040} \cdot \left(n1\_i + -1 \cdot n0\_i\right) + \frac{1}{120} \cdot \left(\frac{1}{2} \cdot n0\_i - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{1}{2} \cdot n0\_i - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right) + \frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right), u, n0\_i\right) \]

8. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot n0\_i - \mathsf{fma}\left(-0.041666666666666664 \cdot n0\_i - \mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right), -0.16666666666666666, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot 0.008333333333333333\right), -0.16666666666666666, \mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right), 0.008333333333333333, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot -0.0001984126984126984\right)\right), normAngle \cdot normAngle, -0.041666666666666664 \cdot n0\_i\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right), -0.16666666666666666, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot 0.008333333333333333\right), normAngle \cdot normAngle, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right)\right), normAngle \cdot normAngle, -n0\_i\right) + n1\_i, u, n0\_i\right) \]
9. Add Preprocessing

Alternative 2: 99.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, \mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, normAngle \cdot normAngle, 0.019444444444444445\right) \cdot n1\_i, normAngle \cdot normAngle, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right)\right), normAngle \cdot normAngle, -n0\_i\right) + n1\_i, u, n0\_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (fma
  (+
   (fma
    (fma
     0.5
     n0_i
     (fma
      (*
       (fma 0.00205026455026455 (* normAngle normAngle) 0.019444444444444445)
       n1_i)
      (* normAngle normAngle)
      (* 0.16666666666666666 (fma -1.0 n0_i n1_i))))
    (* normAngle normAngle)
    (- n0_i))
   n1_i)
  u
  n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf((fmaf(fmaf(0.5f, n0_i, fmaf((fmaf(0.00205026455026455f, (normAngle * normAngle), 0.019444444444444445f) * n1_i), (normAngle * normAngle), (0.16666666666666666f * fmaf(-1.0f, n0_i, n1_i)))), (normAngle * normAngle), -n0_i) + n1_i), u, n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(fma(fma(Float32(0.5), n0_i, fma(Float32(fma(Float32(0.00205026455026455), Float32(normAngle * normAngle), Float32(0.019444444444444445)) * n1_i), Float32(normAngle * normAngle), Float32(Float32(0.16666666666666666) * fma(Float32(-1.0), n0_i, n1_i)))), Float32(normAngle * normAngle), Float32(-n0_i)) + n1_i), 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. Add Preprocessing

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 88.8%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{2} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{24} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{1}{720} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{1}{2} \cdot n0\_i - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right) + \frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right) + \left(\frac{-1}{5040} \cdot \left(n1\_i + -1 \cdot n0\_i\right) + \frac{1}{120} \cdot \left(\frac{1}{2} \cdot n0\_i - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{1}{2} \cdot n0\_i - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right) + \frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right), u, n0\_i\right) \]

8. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot n0\_i - \mathsf{fma}\left(-0.041666666666666664 \cdot n0\_i - \mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right), -0.16666666666666666, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot 0.008333333333333333\right), -0.16666666666666666, \mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right), 0.008333333333333333, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot -0.0001984126984126984\right)\right), normAngle \cdot normAngle, -0.041666666666666664 \cdot n0\_i\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right), -0.16666666666666666, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot 0.008333333333333333\right), normAngle \cdot normAngle, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right)\right), normAngle \cdot normAngle, -n0\_i\right) + n1\_i, u, n0\_i\right) \]

9. Taylor expanded in n1_i around inf

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, n0\_i, \mathsf{fma}\left(n1\_i \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {normAngle}^{2}\right), normAngle \cdot normAngle, \frac{1}{6} \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right)\right), normAngle \cdot normAngle, -n0\_i\right) + n1\_i, u, n0\_i\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, \mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, normAngle \cdot normAngle, 0.019444444444444445\right) \cdot n1\_i, normAngle \cdot normAngle, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right)\right), normAngle \cdot normAngle, -n0\_i\right) + n1\_i, u, n0\_i\right) \]
12. Add Preprocessing

Alternative 3: 99.1% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in normAngle around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 98.5%

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

9. Final simplification 98.5%

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

Alternative 4: 99.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(normAngle \cdot normAngle, \mathsf{fma}\left(\mathsf{fma}\left(-1, n0\_i, n1\_i\right), 0.16666666666666666, 0.5 \cdot n0\_i\right), n1\_i\right) - n0\_i, u, n0\_i\right) \end{array} \]

FPCore

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

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(fma(Float32(normAngle * normAngle), fma(fma(Float32(-1.0), n0_i, n1_i), Float32(0.16666666666666666), Float32(Float32(0.5) * n0_i)), n1_i) - n0_i), 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. Add Preprocessing

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 88.8%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{2} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{24} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{1}{720} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{1}{2} \cdot n0\_i - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right) + \frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right) + \left(\frac{-1}{5040} \cdot \left(n1\_i + -1 \cdot n0\_i\right) + \frac{1}{120} \cdot \left(\frac{1}{2} \cdot n0\_i - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{1}{2} \cdot n0\_i - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right) + \frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right), u, n0\_i\right) \]

8. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot n0\_i - \mathsf{fma}\left(-0.041666666666666664 \cdot n0\_i - \mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right), -0.16666666666666666, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot 0.008333333333333333\right), -0.16666666666666666, \mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right), 0.008333333333333333, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot -0.0001984126984126984\right)\right), normAngle \cdot normAngle, -0.041666666666666664 \cdot n0\_i\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right), -0.16666666666666666, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot 0.008333333333333333\right), normAngle \cdot normAngle, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right)\right), normAngle \cdot normAngle, -n0\_i\right) + n1\_i, u, n0\_i\right) \]

9. Taylor expanded in n1_i around inf

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, n0\_i, \mathsf{fma}\left(n1\_i \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {normAngle}^{2}\right), normAngle \cdot normAngle, \frac{1}{6} \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right)\right), normAngle \cdot normAngle, -n0\_i\right) + n1\_i, u, n0\_i\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, \mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, normAngle \cdot normAngle, 0.019444444444444445\right) \cdot n1\_i, normAngle \cdot normAngle, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right)\right), normAngle \cdot normAngle, -n0\_i\right) + n1\_i, u, n0\_i\right) \]

12. Taylor expanded in normAngle around 0

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

14. Applied rewrites 98.5%

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

Alternative 5: 70.8% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.4999999523982838 \cdot 10^{-21} \lor \neg \left(n0\_i \leq 5.000000097707407 \cdot 10^{-26}\right):\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -1.4999999523982838e-21)
         (not (<= n0_i 5.000000097707407e-26)))
   (* (- 1.0 u) n0_i)
   (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -1.4999999523982838e-21f) || !(n0_i <= 5.000000097707407e-26f)) {
		tmp = (1.0f - u) * n0_i;
	} else {
		tmp = u * n1_i;
	}
	return tmp;
}

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) :: tmp
    if ((n0_i <= (-1.4999999523982838e-21)) .or. (.not. (n0_i <= 5.000000097707407e-26))) then
        tmp = (1.0e0 - u) * n0_i
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-1.4999999523982838e-21)) || !(n0_i <= Float32(5.000000097707407e-26)))
		tmp = Float32(Float32(Float32(1.0) - u) * n0_i);
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-1.4999999523982838e-21)) || ~((n0_i <= single(5.000000097707407e-26))))
		tmp = (single(1.0) - u) * n0_i;
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.5e-21 or 5.0000001e-26 < n0_i

   1. Initial program 98.6%

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

   3. Taylor expanded in normAngle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 15.0%

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

   9. Taylor expanded in n0_i around inf

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

   11. Applied rewrites 79.2%

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

   if -1.5e-21 < n0_i < 5.0000001e-26

   1. Initial program 95.4%

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

   3. Taylor expanded in n0_i around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-sin.f32 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in normAngle around 0

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

   8. Applied rewrites 63.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -1.4999999523982838 \cdot 10^{-21} \lor \neg \left(n0\_i \leq 5.000000097707407 \cdot 10^{-26}\right):\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 45.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 88.8%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto \mathsf{fma}\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{2} \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{-1}{24} \cdot n0\_i + {normAngle}^{2} \cdot \left(\frac{1}{720} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{-1}{24} \cdot n0\_i - \left(\frac{-1}{6} \cdot \left(\frac{1}{2} \cdot n0\_i - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right) + \frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right) + \left(\frac{-1}{5040} \cdot \left(n1\_i + -1 \cdot n0\_i\right) + \frac{1}{120} \cdot \left(\frac{1}{2} \cdot n0\_i - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{1}{2} \cdot n0\_i - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right) + \frac{1}{120} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right), u, n0\_i\right) \]

8. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot n0\_i - \mathsf{fma}\left(-0.041666666666666664 \cdot n0\_i - \mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right), -0.16666666666666666, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot 0.008333333333333333\right), -0.16666666666666666, \mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right), 0.008333333333333333, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot -0.0001984126984126984\right)\right), normAngle \cdot normAngle, -0.041666666666666664 \cdot n0\_i\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right), -0.16666666666666666, \mathsf{fma}\left(-1, n0\_i, n1\_i\right) \cdot 0.008333333333333333\right), normAngle \cdot normAngle, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right)\right), normAngle \cdot normAngle, -n0\_i\right) + n1\_i, u, n0\_i\right) \]

9. Taylor expanded in n1_i around inf

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, n0\_i, \mathsf{fma}\left(n1\_i \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {normAngle}^{2}\right), normAngle \cdot normAngle, \frac{1}{6} \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right)\right), normAngle \cdot normAngle, -n0\_i\right) + n1\_i, u, n0\_i\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, n0\_i, \mathsf{fma}\left(\mathsf{fma}\left(0.00205026455026455, normAngle \cdot normAngle, 0.019444444444444445\right) \cdot n1\_i, normAngle \cdot normAngle, 0.16666666666666666 \cdot \mathsf{fma}\left(-1, n0\_i, n1\_i\right)\right)\right), normAngle \cdot normAngle, -n0\_i\right) + n1\_i, u, n0\_i\right) \]

12. Taylor expanded in normAngle around 0

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

14. Applied rewrites 97.3%

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

Alternative 7: 37.9% accurate, 76.5× speedup

Math

\[\begin{array}{l} \\ u \cdot n1\_i \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return u * 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
    code = u * n1_i
end function

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = 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. Add Preprocessing

3. Taylor expanded in n0_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-sin.f32 35.2

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

5. Applied rewrites 35.2%

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

6. Taylor expanded in normAngle around 0

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

8. Applied rewrites 35.4%

   \[\leadsto u \cdot \color{blue}{n1\_i} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025010 
(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)))