Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 98.9%

Time: 14.0s

Alternatives: 6

Speedup: 60.1×

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

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

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: t_0
    t_0 = 1.0e0 / sin(normangle)
    code = ((sin(((1.0e0 - u) * normangle)) * t_0) * n0_i) + ((sin((u * normangle)) * t_0) * n1_i)
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(1.0) / sin(normAngle))
	return Float32(Float32(Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * t_0) * n0_i) + Float32(Float32(sin(Float32(u * normAngle)) * t_0) * n1_i))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	t_0 = single(1.0) / sin(normAngle);
	tmp = ((sin(((single(1.0) - u) * normAngle)) * t_0) * n0_i) + ((sin((u * normAngle)) * t_0) * n1_i);
end

Initial Program: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: t_0
    t_0 = 1.0e0 / sin(normangle)
    code = ((sin(((1.0e0 - u) * normangle)) * t_0) * n0_i) + ((sin((u * normangle)) * t_0) * n1_i)
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(1.0) / sin(normAngle))
	return Float32(Float32(Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * t_0) * n0_i) + Float32(Float32(sin(Float32(u * normAngle)) * t_0) * n1_i))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	t_0 = single(1.0) / sin(normAngle);
	tmp = ((sin(((single(1.0) - u) * normAngle)) * t_0) * n0_i) + ((sin((u * normAngle)) * t_0) * n1_i);
end

Alternative 1: 98.9% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i + (u * ((n1_i - n0_i) + ((normangle * normangle) * ((n1_i * 0.16666666666666666e0) + ((n0_i * 0.5e0) + (n0_i * (-0.16666666666666666e0)))))))
end function

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * ((n1_i - n0_i) + ((normAngle * normAngle) * ((n1_i * single(0.16666666666666666)) + ((n0_i * single(0.5)) + (n0_i * single(-0.16666666666666666)))))));
end

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.8%

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

6. Taylor expanded in u around 0

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

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \color{blue}{\left(u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)}\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)}\right)\right) \]

   3. associate-+r+ N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\left(n1\_i + -1 \cdot n0\_i\right) + \color{blue}{{normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i + -1 \cdot n0\_i\right), \color{blue}{\left({normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)}\right)\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i + \left(\mathsf{neg}\left(n0\_i\right)\right)\right), \left({normAngle}^{\color{blue}{2}} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i - n0\_i\right), \left(\color{blue}{{normAngle}^{2}} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   7. --lowering--.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \left(\color{blue}{{normAngle}^{2}} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\left({normAngle}^{2}\right), \color{blue}{\left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\left(normAngle \cdot normAngle\right), \left(\color{blue}{\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right)} - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\color{blue}{\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right)} - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   11. associate--l+ N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\frac{1}{6} \cdot n1\_i + \color{blue}{\left(\frac{1}{2} \cdot n0\_i - \frac{1}{6} \cdot n0\_i\right)}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\left(\frac{1}{6} \cdot n1\_i\right), \color{blue}{\left(\frac{1}{2} \cdot n0\_i - \frac{1}{6} \cdot n0\_i\right)}\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\left(n1\_i \cdot \frac{1}{6}\right), \left(\color{blue}{\frac{1}{2} \cdot n0\_i} - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \left(\color{blue}{\frac{1}{2} \cdot n0\_i} - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right)\right) \]

   15. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \left(\frac{1}{2} \cdot n0\_i + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot n0\_i}\right)\right)\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \left(\frac{1}{2} \cdot n0\_i + \frac{-1}{6} \cdot n0\_i\right)\right)\right)\right)\right)\right) \]

   17. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot n0\_i\right), \color{blue}{\left(\frac{-1}{6} \cdot n0\_i\right)}\right)\right)\right)\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{+.f32}\left(\left(n0\_i \cdot \frac{1}{2}\right), \left(\color{blue}{\frac{-1}{6}} \cdot n0\_i\right)\right)\right)\right)\right)\right)\right) \]

   19. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{2}\right), \left(\color{blue}{\frac{-1}{6}} \cdot n0\_i\right)\right)\right)\right)\right)\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{2}\right), \left(n0\_i \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   21. *-lowering-*.f32 99.0%

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{2}\right), \mathsf{*.f32}\left(n0\_i, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 99.0%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n1\_i \cdot 0.16666666666666666 + \left(n0\_i \cdot 0.5 + n0\_i \cdot -0.16666666666666666\right)\right)\right)} \]
9. Add Preprocessing

Alternative 2: 70.8% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n0\_i \cdot \left(1 - u\right)\\ \mathbf{if}\;n0\_i \leq -4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n0\_i \leq 1.500000029312222 \cdot 10^{-25}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (* n0_i (- 1.0 u))))
   (if (<= n0_i -4.999999918875795e-18)
     t_0
     (if (<= n0_i 1.500000029312222e-25) (* u n1_i) t_0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = n0_i * (1.0f - u);
	float tmp;
	if (n0_i <= -4.999999918875795e-18f) {
		tmp = t_0;
	} else if (n0_i <= 1.500000029312222e-25f) {
		tmp = u * n1_i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: t_0
    real(4) :: tmp
    t_0 = n0_i * (1.0e0 - u)
    if (n0_i <= (-4.999999918875795e-18)) then
        tmp = t_0
    else if (n0_i <= 1.500000029312222e-25) then
        tmp = u * n1_i
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(n0_i * Float32(Float32(1.0) - u))
	tmp = Float32(0.0)
	if (n0_i <= Float32(-4.999999918875795e-18))
		tmp = t_0;
	elseif (n0_i <= Float32(1.500000029312222e-25))
		tmp = Float32(u * n1_i);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	t_0 = n0_i * (single(1.0) - u);
	tmp = single(0.0);
	if (n0_i <= single(-4.999999918875795e-18))
		tmp = t_0;
	elseif (n0_i <= single(1.500000029312222e-25))
		tmp = u * n1_i;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.99999992e-18 or 1.50000003e-25 < n0_i

   1. Initial program 97.9%

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

   3. Taylor expanded in n0_i around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\sin \left(normAngle \cdot \left(1 - u\right)\right), \color{blue}{\left(\frac{n0\_i}{\sin normAngle}\right)}\right) \]

      4. sin-lowering-sin.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sin.f32}\left(\left(normAngle \cdot \left(1 - u\right)\right)\right), \left(\frac{\color{blue}{n0\_i}}{\sin normAngle}\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(normAngle, \left(1 - u\right)\right)\right), \left(\frac{n0\_i}{\sin normAngle}\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{\_.f32}\left(1, u\right)\right)\right), \left(\frac{n0\_i}{\sin normAngle}\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{\_.f32}\left(1, u\right)\right)\right), \mathsf{/.f32}\left(n0\_i, \color{blue}{\sin normAngle}\right)\right) \]

      8. sin-lowering-sin.f32 78.6%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(normAngle, \mathsf{\_.f32}\left(1, u\right)\right)\right), \mathsf{/.f32}\left(n0\_i, \mathsf{sin.f32}\left(normAngle\right)\right)\right) \]

   5. Simplified 78.6%

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

   6. Taylor expanded in normAngle around 0

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

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(n0\_i, \color{blue}{\left(1 - u\right)}\right) \]

      2. --lowering--.f32 78.5%

         \[\leadsto \mathsf{*.f32}\left(n0\_i, \mathsf{\_.f32}\left(1, \color{blue}{u}\right)\right) \]

   8. Simplified 78.5%

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

   if -4.99999992e-18 < n0_i < 1.50000003e-25

   1. Initial program 97.0%

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

   3. Taylor expanded in normAngle around 0

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

   5. Simplified 98.8%

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

   6. Taylor expanded in n0_i around -inf

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

   8. Simplified 98.6%

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

   9. Taylor expanded in normAngle around 0

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

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

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

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\left(u - \left(1 + \frac{n1\_i \cdot u}{n0\_i}\right)\right), \color{blue}{\left(\mathsf{neg}\left(n0\_i\right)\right)}\right) \]

       5. --lowering--.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(u, \left(1 + \frac{n1\_i \cdot u}{n0\_i}\right)\right), \left(\mathsf{neg}\left(\color{blue}{n0\_i}\right)\right)\right) \]

       6. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(u, \mathsf{+.f32}\left(1, \left(\frac{n1\_i \cdot u}{n0\_i}\right)\right)\right), \left(\mathsf{neg}\left(n0\_i\right)\right)\right) \]

       7. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(n1\_i \cdot u\right), n0\_i\right)\right)\right), \left(\mathsf{neg}\left(n0\_i\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(u \cdot n1\_i\right), n0\_i\right)\right)\right), \left(\mathsf{neg}\left(n0\_i\right)\right)\right) \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, n1\_i\right), n0\_i\right)\right)\right), \left(\mathsf{neg}\left(n0\_i\right)\right)\right) \]

       10. neg-lowering-neg.f32 98.1%

           \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, n1\_i\right), n0\_i\right)\right)\right), \mathsf{neg.f32}\left(n0\_i\right)\right) \]

   11. Simplified 98.1%

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

   12. Taylor expanded in n1_i around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f32 65.6%

          \[\leadsto \mathsf{*.f32}\left(u, \color{blue}{n1\_i}\right) \]

   14. Simplified 65.6%

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

Alternative 3: 61.1% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;n0\_i\\ \mathbf{elif}\;n0\_i \leq 1.500000029312222 \cdot 10^{-25}:\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -4.999999918875795e-18)
   n0_i
   (if (<= n0_i 1.500000029312222e-25) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -4.999999918875795e-18f) {
		tmp = n0_i;
	} else if (n0_i <= 1.500000029312222e-25f) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-4.999999918875795e-18)) then
        tmp = n0_i
    else if (n0_i <= 1.500000029312222e-25) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-4.999999918875795e-18))
		tmp = n0_i;
	elseif (n0_i <= Float32(1.500000029312222e-25))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-4.999999918875795e-18))
		tmp = n0_i;
	elseif (n0_i <= single(1.500000029312222e-25))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.99999992e-18 or 1.50000003e-25 < n0_i

   1. Initial program 97.9%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{n0\_i} \]
   4. Step-by-step derivation

   5. Simplified 62.7%

      \[\leadsto \color{blue}{n0\_i} \]

   if -4.99999992e-18 < n0_i < 1.50000003e-25

   1. Initial program 97.0%

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

   3. Taylor expanded in normAngle around 0

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

   5. Simplified 98.8%

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

   6. Taylor expanded in n0_i around -inf

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

   8. Simplified 98.6%

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

   9. Taylor expanded in normAngle around 0

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

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

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

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\left(u - \left(1 + \frac{n1\_i \cdot u}{n0\_i}\right)\right), \color{blue}{\left(\mathsf{neg}\left(n0\_i\right)\right)}\right) \]

       5. --lowering--.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(u, \left(1 + \frac{n1\_i \cdot u}{n0\_i}\right)\right), \left(\mathsf{neg}\left(\color{blue}{n0\_i}\right)\right)\right) \]

       6. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(u, \mathsf{+.f32}\left(1, \left(\frac{n1\_i \cdot u}{n0\_i}\right)\right)\right), \left(\mathsf{neg}\left(n0\_i\right)\right)\right) \]

       7. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(n1\_i \cdot u\right), n0\_i\right)\right)\right), \left(\mathsf{neg}\left(n0\_i\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(u \cdot n1\_i\right), n0\_i\right)\right)\right), \left(\mathsf{neg}\left(n0\_i\right)\right)\right) \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, n1\_i\right), n0\_i\right)\right)\right), \left(\mathsf{neg}\left(n0\_i\right)\right)\right) \]

       10. neg-lowering-neg.f32 98.1%

           \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, n1\_i\right), n0\_i\right)\right)\right), \mathsf{neg.f32}\left(n0\_i\right)\right) \]

   11. Simplified 98.1%

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

   12. Taylor expanded in n1_i around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f32 65.6%

          \[\leadsto \mathsf{*.f32}\left(u, \color{blue}{n1\_i}\right) \]

   14. Simplified 65.6%

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

Alternative 4: 98.1% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.8%

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

6. Taylor expanded in u around 0

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

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \color{blue}{\left(u \cdot \left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)}\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)}\right)\right) \]

   3. associate-+r+ N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \left(\left(n1\_i + -1 \cdot n0\_i\right) + \color{blue}{{normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i + -1 \cdot n0\_i\right), \color{blue}{\left({normAngle}^{2} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)}\right)\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i + \left(\mathsf{neg}\left(n0\_i\right)\right)\right), \left({normAngle}^{\color{blue}{2}} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(n1\_i - n0\_i\right), \left(\color{blue}{{normAngle}^{2}} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   7. --lowering--.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \left(\color{blue}{{normAngle}^{2}} \cdot \left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\left({normAngle}^{2}\right), \color{blue}{\left(\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right) - \frac{1}{6} \cdot n0\_i\right)}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\left(normAngle \cdot normAngle\right), \left(\color{blue}{\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right)} - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\color{blue}{\left(\frac{1}{6} \cdot n1\_i + \frac{1}{2} \cdot n0\_i\right)} - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right) \]

   11. associate--l+ N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \left(\frac{1}{6} \cdot n1\_i + \color{blue}{\left(\frac{1}{2} \cdot n0\_i - \frac{1}{6} \cdot n0\_i\right)}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\left(\frac{1}{6} \cdot n1\_i\right), \color{blue}{\left(\frac{1}{2} \cdot n0\_i - \frac{1}{6} \cdot n0\_i\right)}\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\left(n1\_i \cdot \frac{1}{6}\right), \left(\color{blue}{\frac{1}{2} \cdot n0\_i} - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \left(\color{blue}{\frac{1}{2} \cdot n0\_i} - \frac{1}{6} \cdot n0\_i\right)\right)\right)\right)\right)\right) \]

   15. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \left(\frac{1}{2} \cdot n0\_i + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot n0\_i}\right)\right)\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \left(\frac{1}{2} \cdot n0\_i + \frac{-1}{6} \cdot n0\_i\right)\right)\right)\right)\right)\right) \]

   17. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot n0\_i\right), \color{blue}{\left(\frac{-1}{6} \cdot n0\_i\right)}\right)\right)\right)\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{+.f32}\left(\left(n0\_i \cdot \frac{1}{2}\right), \left(\color{blue}{\frac{-1}{6}} \cdot n0\_i\right)\right)\right)\right)\right)\right)\right) \]

   19. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{2}\right), \left(\color{blue}{\frac{-1}{6}} \cdot n0\_i\right)\right)\right)\right)\right)\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{2}\right), \left(n0\_i \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   21. *-lowering-*.f32 99.0%

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(n1\_i, n0\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(normAngle, normAngle\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n1\_i, \frac{1}{6}\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(n0\_i, \frac{1}{2}\right), \mathsf{*.f32}\left(n0\_i, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 99.0%

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

9. Taylor expanded in normAngle around 0

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

    1. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \color{blue}{\left(u \cdot \left(n1\_i - n0\_i\right)\right)}\right) \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \color{blue}{\left(n1\_i - n0\_i\right)}\right)\right) \]

    3. --lowering--.f32 98.5%

       \[\leadsto \mathsf{+.f32}\left(n0\_i, \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(n1\_i, \color{blue}{n0\_i}\right)\right)\right) \]

11. Simplified 98.5%

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

Alternative 5: 81.5% accurate, 84.2× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot n1\_i \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in normAngle around 0

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

5. Simplified 98.8%

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

6. Taylor expanded in u around 0

   \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, n1\_i\right), \color{blue}{n0\_i}\right) \]
7. Step-by-step derivation

8. Simplified 84.8%

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

9. Final simplification 84.8%

   \[\leadsto n0\_i + u \cdot n1\_i \]
10. Add Preprocessing

Alternative 6: 47.7% accurate, 421.0× speedup

Math

\[\begin{array}{l} \\ n0\_i \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i;
end

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{n0\_i} \]
4. Step-by-step derivation

5. Simplified 45.6%

   \[\leadsto \color{blue}{n0\_i} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(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)))