Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 97.2%

Time: 4.2s

Alternatives: 7

Speedup: 1.0×

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

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: 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

Derivation

1. Initial program 96.9%

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

Alternative 2: 69.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ t_1 := \left(\sin \left(u \cdot normAngle\right) \cdot t\_0\right) \cdot n1\_i\\ \mathbf{if}\;n1\_i \leq -2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n1\_i \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t\_0\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sin normAngle)))
        (t_1 (* (* (sin (* u normAngle)) t_0) n1_i)))
   (if (<= n1_i -2.00000009162741e-18)
     t_1
     (if (<= n1_i 4.9999998413276127e-20)
       (* (* (sin (* (- 1.0 u) normAngle)) t_0) n0_i)
       t_1))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = 1.0f / sinf(normAngle);
	float t_1 = (sinf((u * normAngle)) * t_0) * n1_i;
	float tmp;
	if (n1_i <= -2.00000009162741e-18f) {
		tmp = t_1;
	} else if (n1_i <= 4.9999998413276127e-20f) {
		tmp = (sinf(((1.0f - u) * normAngle)) * t_0) * n0_i;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(4) :: tmp
    t_0 = 1.0e0 / sin(normangle)
    t_1 = (sin((u * normangle)) * t_0) * n1_i
    if (n1_i <= (-2.00000009162741e-18)) then
        tmp = t_1
    else if (n1_i <= 4.9999998413276127e-20) then
        tmp = (sin(((1.0e0 - u) * normangle)) * t_0) * n0_i
    else
        tmp = t_1
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(1.0) / sin(normAngle))
	t_1 = Float32(Float32(sin(Float32(u * normAngle)) * t_0) * n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-2.00000009162741e-18))
		tmp = t_1;
	elseif (n1_i <= Float32(4.9999998413276127e-20))
		tmp = Float32(Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * t_0) * n0_i);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	t_0 = single(1.0) / sin(normAngle);
	t_1 = (sin((u * normAngle)) * t_0) * n1_i;
	tmp = single(0.0);
	if (n1_i <= single(-2.00000009162741e-18))
		tmp = t_1;
	elseif (n1_i <= single(4.9999998413276127e-20))
		tmp = (sin(((single(1.0) - u) * normAngle)) * t_0) * n0_i;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -2.00000009e-18 or 4.99999984e-20 < n1_i

   1. Initial program 96.1%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   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) + \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) + \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) + {normAngle}^{2} \cdot \left(\left(\frac{-1}{5040} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{7}\right) + \frac{-1}{5040} \cdot \left(n1\_i \cdot {u}^{7}\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right) + \left(\frac{-1}{5040} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \left(\frac{-1}{5040} \cdot \left(n1\_i \cdot u\right) + \left(\frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \left(\frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\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. Applied rewrites 9.0%

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

   6. Taylor expanded in normAngle around 0

      \[\leadsto normAngle \cdot \color{blue}{\left(\left(1 + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot {\left(1 - u\right)}^{3} + \frac{1}{120} \cdot \left({normAngle}^{2} \cdot {\left(1 - u\right)}^{5}\right)\right)\right) - u\right)} \]

   8. Applied rewrites 60.5%

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

   if -2.00000009e-18 < n1_i < 4.99999984e-20

   1. Initial program 97.6%

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

   3. Taylor expanded in normAngle around 0

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

   5. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 37.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (sinf((u * normAngle)) * (1.0f / sinf(normAngle))) * 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 = (sin((u * normangle)) * (1.0e0 / sin(normangle))) * n1_i
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in 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) + \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) + \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) + {normAngle}^{2} \cdot \left(\left(\frac{-1}{5040} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{7}\right) + \frac{-1}{5040} \cdot \left(n1\_i \cdot {u}^{7}\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right) + \left(\frac{-1}{5040} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \left(\frac{-1}{5040} \cdot \left(n1\_i \cdot u\right) + \left(\frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \left(\frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\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. Applied rewrites 10.1%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto normAngle \cdot \color{blue}{\left(\left(1 + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot {\left(1 - u\right)}^{3} + \frac{1}{120} \cdot \left({normAngle}^{2} \cdot {\left(1 - u\right)}^{5}\right)\right)\right) - u\right)} \]

8. Applied rewrites 38.4%

   \[\leadsto \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot \color{blue}{n1\_i} \]
9. Add Preprocessing

Alternative 4: 10.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in 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) + \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) + \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) + {normAngle}^{2} \cdot \left(\left(\frac{-1}{5040} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{7}\right) + \frac{-1}{5040} \cdot \left(n1\_i \cdot {u}^{7}\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right) + \left(\frac{-1}{5040} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \left(\frac{-1}{5040} \cdot \left(n1\_i \cdot u\right) + \left(\frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \left(\frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\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. Applied rewrites 10.1%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto normAngle \cdot \color{blue}{\left(\left(1 + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot {\left(1 - u\right)}^{3} + \frac{1}{120} \cdot \left({normAngle}^{2} \cdot {\left(1 - u\right)}^{5}\right)\right)\right) - u\right)} \]

8. Applied rewrites 38.4%

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

9. Taylor expanded in normAngle around 0

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

11. Applied rewrites 11.3%

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

12. Taylor expanded in normAngle around 0

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

14. Applied rewrites 11.3%

    \[\leadsto \sin normAngle \cdot n1\_i \]
15. Add Preprocessing

Alternative 5: 10.7% accurate, 32.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(1 - u\right) \cdot normAngle\right) \cdot n1\_i \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((1.0f - u) * normAngle) * 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 = ((1.0e0 - u) * normangle) * n1_i
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in 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) + \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) + {normAngle}^{2} \cdot \left(\left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) + \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) + {normAngle}^{2} \cdot \left(\left(\frac{-1}{5040} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{7}\right) + \frac{-1}{5040} \cdot \left(n1\_i \cdot {u}^{7}\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{1}{120} \cdot \left(n1\_i \cdot {u}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right) + \left(\frac{-1}{5040} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \left(\frac{-1}{5040} \cdot \left(n1\_i \cdot u\right) + \left(\frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n1\_i \cdot {u}^{3}\right) - \frac{-1}{6} \cdot \left(n1\_i \cdot u\right)\right) + \left(\frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right) + \frac{1}{120} \cdot \left(n1\_i \cdot u\right)\right)\right)\right)\right)\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. Applied rewrites 10.1%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto normAngle \cdot \color{blue}{\left(\left(1 + {normAngle}^{2} \cdot \left(\frac{-1}{6} \cdot {\left(1 - u\right)}^{3} + \frac{1}{120} \cdot \left({normAngle}^{2} \cdot {\left(1 - u\right)}^{5}\right)\right)\right) - u\right)} \]

8. Applied rewrites 38.4%

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

9. Taylor expanded in normAngle around 0

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

11. Applied rewrites 11.3%

    \[\leadsto \left(\left(1 - u\right) \cdot normAngle\right) \cdot n1\_i \]
12. Add Preprocessing

Alternative 6: 9.7% accurate, 76.5× speedup

Math

\[\begin{array}{l} \\ u \cdot normAngle \end{array} \]

FPCore

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

C

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

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 = u * normangle
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in 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. Applied rewrites 57.2%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \color{blue}{{normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + {normAngle}^{2} \cdot \left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right)} \]

8. Applied rewrites 8.1%

   \[\leadsto 1 - \color{blue}{u} \]

9. Taylor expanded in normAngle around 0

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

11. Applied rewrites 10.3%

    \[\leadsto u \cdot \color{blue}{normAngle} \]
12. Add Preprocessing

Alternative 7: 8.2% accurate, 114.8× speedup

Math

\[\begin{array}{l} \\ 1 - u \end{array} \]

FPCore

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

C

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

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 = 1.0e0 - u
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in 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. Applied rewrites 57.2%

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

6. Taylor expanded in normAngle around 0

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \color{blue}{{normAngle}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + {normAngle}^{2} \cdot \left(\frac{1}{120} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{5}\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{6} \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right) + \frac{1}{120} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(n0\_i \cdot \left(1 - u\right)\right)\right)} \]

8. Applied rewrites 8.1%

   \[\leadsto 1 - \color{blue}{u} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(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)))