Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 98.9%

Time: 14.3s

Alternatives: 8

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: 97.3% 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

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, 1.3× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (normAngle * (u * ((n1_i / sinf(normAngle)) - ((n0_i * cosf(normAngle)) / sinf(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 = n0_i + (normangle * (u * ((n1_i / sin(normangle)) - ((n0_i * cos(normangle)) / sin(normangle)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in u around 0 90.1%

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

4. Taylor expanded in normAngle around inf 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 98.8% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \left(normAngle \cdot normAngle\right) \cdot \left(n0\_i \cdot 0.3333333333333333 + n1\_i \cdot 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (+
    (- n1_i n0_i)
    (*
     (* normAngle normAngle)
     (+ (* n0_i 0.3333333333333333) (* n1_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) * ((n0_i * 0.3333333333333333f) + (n1_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) * ((n0_i * 0.3333333333333333e0) + (n1_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(n0_i * Float32(0.3333333333333333)) + Float32(n1_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) * ((n0_i * single(0.3333333333333333)) + (n1_i * single(0.16666666666666666))))));
end

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in u around 0 90.1%

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

4. Taylor expanded in normAngle around 0 98.4%

   \[\leadsto n0\_i + u \cdot \color{blue}{\left(n1\_i + \left(-1 \cdot n0\_i + {normAngle}^{2} \cdot \left(-1 \cdot \left(-0.5 \cdot n0\_i - -0.16666666666666666 \cdot n0\_i\right) - -0.16666666666666666 \cdot n1\_i\right)\right)\right)} \]
5. Step-by-step derivation

   1. associate-+r+ 98.4%

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

   2. mul-1-neg 98.4%

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

   3. unsub-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. mul-1-neg 98.4%

      \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(\color{blue}{\left(-\left(-0.5 \cdot n0\_i - -0.16666666666666666 \cdot n0\_i\right)\right)} + \left(--0.16666666666666666 \cdot n1\_i\right)\right)\right) \]

   6. distribute-rgt-out-- 98.4%

      \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(\left(-\color{blue}{n0\_i \cdot \left(-0.5 - -0.16666666666666666\right)}\right) + \left(--0.16666666666666666 \cdot n1\_i\right)\right)\right) \]

   7. distribute-rgt-neg-in 98.4%

      \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(\color{blue}{n0\_i \cdot \left(-\left(-0.5 - -0.16666666666666666\right)\right)} + \left(--0.16666666666666666 \cdot n1\_i\right)\right)\right) \]

   8. metadata-eval 98.4%

      \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(n0\_i \cdot \left(-\color{blue}{-0.3333333333333333}\right) + \left(--0.16666666666666666 \cdot n1\_i\right)\right)\right) \]

   9. metadata-eval 98.4%

      \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(n0\_i \cdot \color{blue}{0.3333333333333333} + \left(--0.16666666666666666 \cdot n1\_i\right)\right)\right) \]

   10. *-commutative 98.4%

       \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(n0\_i \cdot 0.3333333333333333 + \left(-\color{blue}{n1\_i \cdot -0.16666666666666666}\right)\right)\right) \]

   11. distribute-rgt-neg-in 98.4%

       \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(n0\_i \cdot 0.3333333333333333 + \color{blue}{n1\_i \cdot \left(--0.16666666666666666\right)}\right)\right) \]

   12. metadata-eval 98.4%

       \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(n0\_i \cdot 0.3333333333333333 + n1\_i \cdot \color{blue}{0.16666666666666666}\right)\right) \]

6. Simplified 98.4%

   \[\leadsto n0\_i + u \cdot \color{blue}{\left(\left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(n0\_i \cdot 0.3333333333333333 + n1\_i \cdot 0.16666666666666666\right)\right)} \]
7. Step-by-step derivation

   1. unpow2 98.4%

      \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(n0\_i \cdot 0.3333333333333333 + n1\_i \cdot 0.16666666666666666\right)\right) \]

8. Applied egg-rr 98.4%

   \[\leadsto n0\_i + u \cdot \left(\left(n1\_i - n0\_i\right) + \color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(n0\_i \cdot 0.3333333333333333 + n1\_i \cdot 0.16666666666666666\right)\right) \]
9. Add Preprocessing

Alternative 3: 70.7% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -3.99999992980668 \cdot 10^{-13} \lor \neg \left(n1\_i \leq 2.000000033724767 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -3.99999992980668e-13) (not (<= n1_i 2.000000033724767e-16)))
   (* u n1_i)
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -3.99999992980668e-13f) || !(n1_i <= 2.000000033724767e-16f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i * (1.0f - u);
	}
	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 ((n1_i <= (-3.99999992980668e-13)) .or. (.not. (n1_i <= 2.000000033724767e-16))) then
        tmp = u * n1_i
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-3.99999992980668e-13)) || !(n1_i <= Float32(2.000000033724767e-16)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-3.99999992980668e-13)) || ~((n1_i <= single(2.000000033724767e-16))))
		tmp = u * n1_i;
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -3.99999993e-13 or 2.00000003e-16 < n1_i

   1. Initial program 96.7%

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

   3. Taylor expanded in n0_i around 0 61.6%

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

      1. associate-/l* 72.0%

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

      2. *-commutative 72.0%

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

   5. Simplified 72.0%

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

   6. Taylor expanded in normAngle around 0 71.2%

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

      1. *-commutative 71.2%

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

   8. Simplified 71.2%

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

   if -3.99999993e-13 < n1_i < 2.00000003e-16

   1. Initial program 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 73.8%

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

      3. *-commutative 73.8%

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

      4. associate-*l* 70.8%

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

      5. distribute-lft-out 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in normAngle around 0 69.5%

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

   6. Taylor expanded in u around 0 69.2%

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

      1. *-commutative 69.2%

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

   8. Simplified 69.2%

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

   9. Taylor expanded in normAngle around 0 97.9%

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

       1. +-commutative 97.9%

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

       2. *-commutative 97.9%

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

       3. fma-define 98.0%

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

   11. Simplified 98.0%

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

   12. Taylor expanded in n1_i around 0 76.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -3.99999992980668 \cdot 10^{-13} \lor \neg \left(n1\_i \leq 2.000000033724767 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.6% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -3.99999992980668 \cdot 10^{-13} \lor \neg \left(n1\_i \leq 2.000000033724767 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -3.99999992980668e-13) (not (<= n1_i 2.000000033724767e-16)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -3.99999992980668e-13f) || !(n1_i <= 2.000000033724767e-16f)) {
		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 ((n1_i <= (-3.99999992980668e-13)) .or. (.not. (n1_i <= 2.000000033724767e-16))) 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 ((n1_i <= Float32(-3.99999992980668e-13)) || !(n1_i <= Float32(2.000000033724767e-16)))
		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 ((n1_i <= single(-3.99999992980668e-13)) || ~((n1_i <= single(2.000000033724767e-16))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -3.99999993e-13 or 2.00000003e-16 < n1_i

   1. Initial program 96.7%

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

   3. Taylor expanded in n0_i around 0 61.6%

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

      1. associate-/l* 72.0%

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

      2. *-commutative 72.0%

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

   5. Simplified 72.0%

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

   6. Taylor expanded in normAngle around 0 71.2%

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

      1. *-commutative 71.2%

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

   8. Simplified 71.2%

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

   if -3.99999993e-13 < n1_i < 2.00000003e-16

   1. Initial program 98.3%

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

   3. Taylor expanded in u around 0 59.7%

      \[\leadsto \color{blue}{n0\_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -3.99999992980668 \cdot 10^{-13} \lor \neg \left(n1\_i \leq 2.000000033724767 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% 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 97.7%

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

3. Taylor expanded in u around 0 90.1%

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

4. Taylor expanded in normAngle around 0 97.4%

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

   1. mul-1-neg 97.4%

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

   2. unsub-neg 97.4%

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

6. Simplified 97.4%

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

Alternative 6: 81.7% accurate, 60.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. +-commutative 97.7%

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

   2. fma-define 97.8%

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

   3. associate-*r/ 97.9%

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

   4. *-rgt-identity 97.9%

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

   5. *-commutative 97.9%

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

   6. associate-*r* 80.9%

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

   7. associate-*r/ 81.0%

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

   8. *-rgt-identity 81.0%

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

3. Simplified 81.0%

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

   1. *-commutative 81.0%

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

   2. associate-*l/ 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in u around 0 84.0%

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

8. Taylor expanded in normAngle around 0 82.9%

   \[\leadsto \color{blue}{n0\_i + n1\_i \cdot u} \]
9. Step-by-step derivation

   1. *-commutative 82.9%

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

10. Simplified 82.9%

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

11. Taylor expanded in n1_i around inf 82.9%

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

    1. +-commutative 82.9%

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

13. Simplified 82.9%

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

14. Final simplification 82.9%

    \[\leadsto n1\_i \cdot \left(u + \frac{n0\_i}{n1\_i}\right) \]
15. Add Preprocessing

Alternative 7: 81.8% 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 97.7%

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

   1. +-commutative 97.7%

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

   2. fma-define 97.8%

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

   3. associate-*r/ 97.9%

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

   4. *-rgt-identity 97.9%

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

   5. *-commutative 97.9%

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

   6. associate-*r* 80.9%

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

   7. associate-*r/ 81.0%

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

   8. *-rgt-identity 81.0%

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

3. Simplified 81.0%

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

   1. *-commutative 81.0%

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

   2. associate-*l/ 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in u around 0 84.0%

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

8. Taylor expanded in normAngle around 0 82.9%

   \[\leadsto \color{blue}{n0\_i + n1\_i \cdot u} \]
9. Step-by-step derivation

   1. *-commutative 82.9%

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

10. Simplified 82.9%

    \[\leadsto \color{blue}{n0\_i + u \cdot n1\_i} \]
11. Add Preprocessing

Alternative 8: 47.0% 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 97.7%

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

3. Taylor expanded in u around 0 45.5%

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

Reproduce

herbie shell --seed 2024191 
(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)))