Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.0%

Time: 14.9s

Alternatives: 9

Speedup: 46.8×

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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(Float32(Float32(normAngle / sin(normAngle)) * u), n1_i, Float32(Float32(n0_i / sin(normAngle)) * sin(Float32(normAngle * Float32(Float32(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 u around 0 96.9%

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

   1. associate-/l* 98.3%

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

5. Simplified 98.3%

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

   1. +-commutative 98.3%

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

   2. fma-define 98.5%

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

   3. clear-num 98.3%

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

   4. un-div-inv 98.5%

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

   5. un-div-inv 99.0%

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

   6. *-commutative 99.0%

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

   7. *-commutative 99.0%

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

   8. clear-num 98.9%

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

   9. un-div-inv 98.9%

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

   10. *-commutative 98.9%

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

7. Applied egg-rr 98.9%

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

   1. associate-/r/ 99.1%

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

   2. associate-/r/ 99.1%

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

   3. *-commutative 99.1%

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

9. Simplified 99.1%

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

Alternative 2: 98.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n1_i * (normAngle / (sin(normAngle) / u))) + (n0_i * (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 u around 0 96.9%

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

   1. associate-/l* 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in normAngle around 0 98.4%

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

   1. clear-num 98.2%

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

   2. un-div-inv 98.4%

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

8. Applied egg-rr 98.4%

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

9. Final simplification 98.4%

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

Alternative 3: 98.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n1_i * (normAngle * (u / sin(normAngle)))) + (n0_i * (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 u around 0 96.9%

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

   1. associate-/l* 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in normAngle around 0 98.4%

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

7. Final simplification 98.4%

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

Alternative 4: 97.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(n0_i, Float32(Float32(1.0) - u), Float32(u * 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 97.5%

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

   1. fma-define 97.6%

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

   2. *-commutative 97.6%

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

5. Simplified 97.6%

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

Alternative 5: 86.3% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.4249999320343114 \cdot 10^{-14} \lor \neg \left(n0\_i \leq 3.99999992980668 \cdot 10^{-13}\right):\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -2.4249999320343114e-14)
         (not (<= n0_i 3.99999992980668e-13)))
   (- n0_i (* u n0_i))
   (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -2.4249999320343114e-14f) || !(n0_i <= 3.99999992980668e-13f)) {
		tmp = n0_i - (u * n0_i);
	} else {
		tmp = n0_i + (u * n1_i);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n0_i <= (-2.4249999320343114e-14)) .or. (.not. (n0_i <= 3.99999992980668e-13))) then
        tmp = n0_i - (u * n0_i)
    else
        tmp = n0_i + (u * n1_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-2.4249999320343114e-14)) || !(n0_i <= Float32(3.99999992980668e-13)))
		tmp = Float32(n0_i - Float32(u * n0_i));
	else
		tmp = Float32(n0_i + Float32(u * n1_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-2.4249999320343114e-14)) || ~((n0_i <= single(3.99999992980668e-13))))
		tmp = n0_i - (u * n0_i);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -2.42499993e-14 or 3.99999993e-13 < n0_i

   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 n0_i around inf 88.7%

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

      1. associate-/l* 94.9%

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

   5. Simplified 94.9%

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

   6. Taylor expanded in normAngle around 0 94.4%

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

      1. sub-neg 94.4%

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

      2. distribute-rgt-in 94.8%

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

      3. *-un-lft-identity 94.8%

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

   8. Applied egg-rr 94.8%

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

   if -2.42499993e-14 < n0_i < 3.99999993e-13

   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 81.7%

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

   4. Taylor expanded in normAngle around 0 82.7%

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

      1. *-commutative 82.7%

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

   6. Simplified 82.7%

      \[\leadsto \color{blue}{n0\_i + u \cdot n1\_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.4249999320343114 \cdot 10^{-14} \lor \neg \left(n0\_i \leq 3.99999992980668 \cdot 10^{-13}\right):\\ \;\;\;\;n0\_i - u \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.2% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.4249999320343114 \cdot 10^{-14} \lor \neg \left(n0\_i \leq 3.99999992980668 \cdot 10^{-13}\right):\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -2.4249999320343114e-14)
         (not (<= n0_i 3.99999992980668e-13)))
   (* n0_i (- 1.0 u))
   (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -2.4249999320343114e-14f) || !(n0_i <= 3.99999992980668e-13f)) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = n0_i + (u * n1_i);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n0_i <= (-2.4249999320343114e-14)) .or. (.not. (n0_i <= 3.99999992980668e-13))) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = n0_i + (u * n1_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-2.4249999320343114e-14)) || !(n0_i <= Float32(3.99999992980668e-13)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(n0_i + Float32(u * n1_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-2.4249999320343114e-14)) || ~((n0_i <= single(3.99999992980668e-13))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -2.42499993e-14 or 3.99999993e-13 < n0_i

   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 n0_i around inf 88.7%

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

      1. associate-/l* 94.9%

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

   5. Simplified 94.9%

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

   6. Taylor expanded in normAngle around 0 94.4%

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

   if -2.42499993e-14 < n0_i < 3.99999993e-13

   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 81.7%

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

   4. Taylor expanded in normAngle around 0 82.7%

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

      1. *-commutative 82.7%

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

   6. Simplified 82.7%

      \[\leadsto \color{blue}{n0\_i + u \cdot n1\_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0\_i \leq -2.4249999320343114 \cdot 10^{-14} \lor \neg \left(n0\_i \leq 3.99999992980668 \cdot 10^{-13}\right):\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0\_i + u \cdot n1\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 46.8× speedup

Math

\[\begin{array}{l} \\ u \cdot n1\_i + n0\_i \cdot \left(1 - u\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (u * n1_i) + (n0_i * (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 97.5%

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

4. Final simplification 97.5%

   \[\leadsto u \cdot n1\_i + n0\_i \cdot \left(1 - u\right) \]
5. Add Preprocessing

Alternative 8: 58.2% accurate, 84.2× speedup

Math

\[\begin{array}{l} \\ n0\_i \cdot \left(1 - u\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i * (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 n0_i around inf 49.8%

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

   1. associate-/l* 59.5%

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

5. Simplified 59.5%

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

6. Taylor expanded in normAngle around 0 59.1%

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

Alternative 9: 46.7% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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 u around 0 46.8%

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

Reproduce

herbie shell --seed 2024160 
(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)))