Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 98.8%

Time: 17.5s

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: 97.1% 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: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{n0_i}{\frac{\sin normAngle}{\sin \left(normAngle \cdot \left(1 - u\right)\right)}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0 96.4%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\frac{normAngle \cdot u}{\sin normAngle}} \cdot n1_i \]
4. Step-by-step derivation

   1. associate-/l* 98.7%

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\frac{normAngle}{\frac{\sin normAngle}{u}}} \cdot n1_i \]

5. Simplified 98.7%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\frac{normAngle}{\frac{\sin normAngle}{u}}} \cdot n1_i \]
6. Step-by-step derivation

   1. *-commutative 98.7%

      \[\leadsto \color{blue}{n0_i \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

   2. associate-*r* 82.2%

      \[\leadsto \color{blue}{\left(n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

   3. div-inv 82.4%

      \[\leadsto \color{blue}{\frac{n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

   4. associate-/l* 98.9%

      \[\leadsto \color{blue}{\frac{n0_i}{\frac{\sin normAngle}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

   5. *-commutative 98.9%

      \[\leadsto \frac{n0_i}{\frac{\sin normAngle}{\sin \color{blue}{\left(normAngle \cdot \left(1 - u\right)\right)}}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

7. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\frac{n0_i}{\frac{\sin normAngle}{\sin \left(normAngle \cdot \left(1 - u\right)\right)}}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

8. Final simplification 98.9%

   \[\leadsto \frac{n0_i}{\frac{\sin normAngle}{\sin \left(normAngle \cdot \left(1 - u\right)\right)}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]
9. Add Preprocessing

Alternative 2: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i + \sin \left(normAngle \cdot \left(1 - u\right)\right) \cdot \frac{n0_i}{\sin normAngle} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0 96.4%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\frac{normAngle \cdot u}{\sin normAngle}} \cdot n1_i \]
4. Step-by-step derivation

   1. associate-/l* 98.7%

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\frac{normAngle}{\frac{\sin normAngle}{u}}} \cdot n1_i \]

5. Simplified 98.7%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\frac{normAngle}{\frac{\sin normAngle}{u}}} \cdot n1_i \]

6. Taylor expanded in u around inf 82.4%

   \[\leadsto \color{blue}{\frac{n0_i \cdot \sin \left(normAngle \cdot \left(1 - u\right)\right)}{\sin normAngle}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]
7. Step-by-step derivation

   1. *-commutative 82.4%

      \[\leadsto \frac{\color{blue}{\sin \left(normAngle \cdot \left(1 - u\right)\right) \cdot n0_i}}{\sin normAngle} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

   2. associate-*r/ 98.8%

      \[\leadsto \color{blue}{\sin \left(normAngle \cdot \left(1 - u\right)\right) \cdot \frac{n0_i}{\sin normAngle}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

8. Simplified 98.8%

   \[\leadsto \color{blue}{\sin \left(normAngle \cdot \left(1 - u\right)\right) \cdot \frac{n0_i}{\sin normAngle}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

9. Final simplification 98.8%

   \[\leadsto \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i + \sin \left(normAngle \cdot \left(1 - u\right)\right) \cdot \frac{n0_i}{\sin normAngle} \]
10. Add Preprocessing

Alternative 3: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i + \frac{n0_i}{\frac{1}{1 - u} + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot \left(u + -1\right) + 0.16666666666666666 \cdot \frac{-1}{1 - u}\right)} \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((normAngle / (sinf(normAngle) / u)) * n1_i) + (n0_i / ((1.0f / (1.0f - u)) + (powf(normAngle, 2.0f) * ((-0.16666666666666666f * (u + -1.0f)) + (0.16666666666666666f * (-1.0f / (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 = ((normangle / (sin(normangle) / u)) * n1_i) + (n0_i / ((1.0e0 / (1.0e0 - u)) + ((normangle ** 2.0e0) * (((-0.16666666666666666e0) * (u + (-1.0e0))) + (0.16666666666666666e0 * ((-1.0e0) / (1.0e0 - u)))))))
end function

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = ((normAngle / (sin(normAngle) / u)) * n1_i) + (n0_i / ((single(1.0) / (single(1.0) - u)) + ((normAngle ^ single(2.0)) * ((single(-0.16666666666666666) * (u + single(-1.0))) + (single(0.16666666666666666) * (single(-1.0) / (single(1.0) - u)))))));
end

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0 96.4%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\frac{normAngle \cdot u}{\sin normAngle}} \cdot n1_i \]
4. Step-by-step derivation

   1. associate-/l* 98.7%

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\frac{normAngle}{\frac{\sin normAngle}{u}}} \cdot n1_i \]

5. Simplified 98.7%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\frac{normAngle}{\frac{\sin normAngle}{u}}} \cdot n1_i \]
6. Step-by-step derivation

   1. *-commutative 98.7%

      \[\leadsto \color{blue}{n0_i \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right)} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

   2. associate-*r* 82.2%

      \[\leadsto \color{blue}{\left(n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right) \cdot \frac{1}{\sin normAngle}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

   3. div-inv 82.4%

      \[\leadsto \color{blue}{\frac{n0_i \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

   4. associate-/l* 98.9%

      \[\leadsto \color{blue}{\frac{n0_i}{\frac{\sin normAngle}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

   5. *-commutative 98.9%

      \[\leadsto \frac{n0_i}{\frac{\sin normAngle}{\sin \color{blue}{\left(normAngle \cdot \left(1 - u\right)\right)}}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

7. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\frac{n0_i}{\frac{\sin normAngle}{\sin \left(normAngle \cdot \left(1 - u\right)\right)}}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

8. Taylor expanded in normAngle around 0 98.5%

   \[\leadsto \frac{n0_i}{\color{blue}{-1 \cdot \left({normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot \left(1 - u\right) + 0.16666666666666666 \cdot \frac{1}{1 - u}\right)\right) + \frac{1}{1 - u}}} + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

9. Final simplification 98.5%

   \[\leadsto \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i + \frac{n0_i}{\frac{1}{1 - u} + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot \left(u + -1\right) + 0.16666666666666666 \cdot \frac{-1}{1 - u}\right)} \]
10. Add Preprocessing

Alternative 4: 98.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i + n0_i \cdot \left(1 - u\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Add Preprocessing

3. Taylor expanded in u around 0 96.4%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\frac{normAngle \cdot u}{\sin normAngle}} \cdot n1_i \]
4. Step-by-step derivation

   1. associate-/l* 98.7%

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\frac{normAngle}{\frac{\sin normAngle}{u}}} \cdot n1_i \]

5. Simplified 98.7%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\frac{normAngle}{\frac{\sin normAngle}{u}}} \cdot n1_i \]

6. Taylor expanded in normAngle around 0 98.1%

   \[\leadsto \color{blue}{\left(1 - u\right)} \cdot n0_i + \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i \]

7. Final simplification 98.1%

   \[\leadsto \frac{normAngle}{\frac{\sin normAngle}{u}} \cdot n1_i + n0_i \cdot \left(1 - u\right) \]
8. Add Preprocessing

Alternative 5: 98.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. *-commutative 96.5%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. associate-*l* 80.2%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. *-commutative 80.2%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

   4. associate-*l* 73.6%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   5. distribute-lft-out 73.5%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Simplified 73.5%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]
4. Add Preprocessing

5. Taylor expanded in normAngle around 0 97.1%

   \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

6. Taylor expanded in u around 0 97.3%

   \[\leadsto \color{blue}{n0_i + u \cdot \left(n1_i + -1 \cdot n0_i\right)} \]
7. Step-by-step derivation

   1. +-commutative 97.3%

      \[\leadsto \color{blue}{u \cdot \left(n1_i + -1 \cdot n0_i\right) + n0_i} \]

   2. fma-def 97.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1_i + -1 \cdot n0_i, n0_i\right)} \]

   3. mul-1-neg 97.4%

      \[\leadsto \mathsf{fma}\left(u, n1_i + \color{blue}{\left(-n0_i\right)}, n0_i\right) \]

   4. unsub-neg 97.4%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1_i - n0_i}, n0_i\right) \]

8. Simplified 97.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right)} \]

9. Final simplification 97.4%

   \[\leadsto \mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \]
10. Add Preprocessing

Alternative 6: 98.0% 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 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. *-commutative 96.5%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. associate-*l* 80.2%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. *-commutative 80.2%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

   4. associate-*l* 73.6%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   5. distribute-lft-out 73.5%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Simplified 73.5%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]
4. Add Preprocessing

5. Taylor expanded in normAngle around 0 97.1%

   \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

6. Taylor expanded in u around -inf 97.3%

   \[\leadsto \color{blue}{n0_i + -1 \cdot \left(u \cdot \left(n0_i + -1 \cdot n1_i\right)\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 97.3%

      \[\leadsto n0_i + \color{blue}{\left(-u \cdot \left(n0_i + -1 \cdot n1_i\right)\right)} \]

   2. unsub-neg 97.3%

      \[\leadsto \color{blue}{n0_i - u \cdot \left(n0_i + -1 \cdot n1_i\right)} \]

   3. mul-1-neg 97.3%

      \[\leadsto n0_i - u \cdot \left(n0_i + \color{blue}{\left(-n1_i\right)}\right) \]

   4. unsub-neg 97.3%

      \[\leadsto n0_i - u \cdot \color{blue}{\left(n0_i - n1_i\right)} \]

8. Simplified 97.3%

   \[\leadsto \color{blue}{n0_i - u \cdot \left(n0_i - n1_i\right)} \]

9. Final simplification 97.3%

   \[\leadsto n0_i + u \cdot \left(n1_i - n0_i\right) \]
10. Add Preprocessing

Alternative 7: 82.2% 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 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. fma-def 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i\right)} \]

   2. associate-*l* 96.3%

      \[\leadsto \mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \color{blue}{\sin \left(u \cdot normAngle\right) \cdot \left(\frac{1}{\sin normAngle} \cdot n1_i\right)}\right) \]

3. Simplified 96.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0_i, \sin \left(u \cdot normAngle\right) \cdot \left(\frac{1}{\sin normAngle} \cdot n1_i\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in u around 0 81.9%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, n0_i, \sin \left(u \cdot normAngle\right) \cdot \left(\frac{1}{\sin normAngle} \cdot n1_i\right)\right) \]

6. Taylor expanded in normAngle around 0 83.4%

   \[\leadsto \color{blue}{n0_i + n1_i \cdot u} \]
7. Step-by-step derivation

   1. *-commutative 83.4%

      \[\leadsto n0_i + \color{blue}{u \cdot n1_i} \]

8. Simplified 83.4%

   \[\leadsto \color{blue}{n0_i + u \cdot n1_i} \]

9. Final simplification 83.4%

   \[\leadsto n0_i + u \cdot n1_i \]
10. Add Preprocessing

Alternative 8: 38.1% accurate, 140.3× speedup

Math

\[\begin{array}{l} \\ u \cdot n1_i \end{array} \]

FPCore

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

C

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

Fortran

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 * n1_i
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.5%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
2. Step-by-step derivation

   1. *-commutative 96.5%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   2. associate-*l* 80.2%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

   3. *-commutative 80.2%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1_i \]

   4. associate-*l* 73.6%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

   5. distribute-lft-out 73.5%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]

3. Simplified 73.5%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0_i + \sin \left(u \cdot normAngle\right) \cdot n1_i\right)} \]
4. Add Preprocessing

5. Taylor expanded in normAngle around 0 97.1%

   \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right) + n1_i \cdot u} \]

6. Taylor expanded in n0_i around 0 40.1%

   \[\leadsto \color{blue}{n1_i \cdot u} \]
7. Step-by-step derivation

   1. *-commutative 40.1%

      \[\leadsto \color{blue}{u \cdot n1_i} \]

8. Simplified 40.1%

   \[\leadsto \color{blue}{u \cdot n1_i} \]

9. Final simplification 40.1%

   \[\leadsto u \cdot n1_i \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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)))