Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 100.0%

Time: 19.4s

Alternatives: 10

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: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \langle \left( \frac{n0_i \cdot \sin \left(normAngle - normAngle \cdot u\right) + n1_i \cdot \sin \left(normAngle \cdot u\right)}{\sin normAngle} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (/
    (+
     (* n0_i (sin (- normAngle (* normAngle u))))
     (* n1_i (sin (* normAngle u))))
    (sin normAngle)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	double tmp = ((((double) n0_i) * sin((((double) normAngle) - (((double) normAngle) * ((double) u))))) + (((double) n1_i) * sin((((double) normAngle) * ((double) u))))) / sin(normAngle);
	return (float) 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(8) :: tmp
    tmp = ((real(n0_i, 8) * sin((real(normangle, 8) - (real(normangle, 8) * real(u, 8))))) + (real(n1_i, 8) * sin((real(normangle, 8) * real(u, 8))))) / sin(normangle)
    code = real(tmp, 4)
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float64(Float64(Float64(Float64(n0_i) * sin(Float64(Float64(normAngle) - Float64(Float64(normAngle) * Float64(u))))) + Float64(Float64(n1_i) * sin(Float64(Float64(normAngle) * Float64(u))))) / sin(normAngle))
	return Float32(tmp)
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = ((double(n0_i) * sin((double(normAngle) - (double(normAngle) * double(u))))) + (double(n1_i) * sin((double(normAngle) * double(u))))) / sin(normAngle);
	tmp_2 = single(tmp);
end

Derivation

1. Initial program 100.0%

   \[\langle \frac{\mathsf{fma}\left(\sin \left(normAngle - u \cdot normAngle\right), n0_i, \sin \left(u \cdot normAngle\right) \cdot n1_i\right)}{\sin normAngle} \rangle_{\text{binary64}} \]

2. Taylor expanded in normAngle around 0 100.0%

   \[\leadsto \langle \frac{\color{blue}{n0_i \cdot \sin \left(normAngle - normAngle \cdot u\right) + n1_i \cdot \sin \left(normAngle \cdot u\right)}}{\sin normAngle} \rangle_{\text{binary64}} \]

3. Final simplification 100.0%

   \[\leadsto \langle \frac{n0_i \cdot \sin \left(normAngle - normAngle \cdot u\right) + n1_i \cdot \sin \left(normAngle \cdot u\right)}{\sin normAngle} \rangle_{\text{binary64}} \]

Alternative 2: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ n0_i \cdot \left(1 - \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left(\left(1 - u\right) - {\left(1 - u\right)}^{3}\right)\right)\right)\right) + n1_i \cdot \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (*
   n0_i
   (-
    1.0
    (+
     u
     (*
      (* normAngle normAngle)
      (* -0.16666666666666666 (- (- 1.0 u) (pow (- 1.0 u) 3.0)))))))
  (*
   n1_i
   (+
    u
    (* (* normAngle normAngle) (* -0.16666666666666666 (- (pow u 3.0) u)))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n0_i * (1.0f - (u + ((normAngle * normAngle) * (-0.16666666666666666f * ((1.0f - u) - powf((1.0f - u), 3.0f))))))) + (n1_i * (u + ((normAngle * normAngle) * (-0.16666666666666666f * (powf(u, 3.0f) - 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 + ((normangle * normangle) * ((-0.16666666666666666e0) * ((1.0e0 - u) - ((1.0e0 - u) ** 3.0e0))))))) + (n1_i * (u + ((normangle * normangle) * ((-0.16666666666666666e0) * ((u ** 3.0e0) - u)))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(n0_i * Float32(Float32(1.0) - Float32(u + Float32(Float32(normAngle * normAngle) * Float32(Float32(-0.16666666666666666) * Float32(Float32(Float32(1.0) - u) - (Float32(Float32(1.0) - u) ^ Float32(3.0)))))))) + Float32(n1_i * Float32(u + Float32(Float32(normAngle * normAngle) * Float32(Float32(-0.16666666666666666) * Float32((u ^ Float32(3.0)) - u))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n0_i * (single(1.0) - (u + ((normAngle * normAngle) * (single(-0.16666666666666666) * ((single(1.0) - u) - ((single(1.0) - u) ^ single(3.0)))))))) + (n1_i * (u + ((normAngle * normAngle) * (single(-0.16666666666666666) * ((u ^ single(3.0)) - 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. Taylor expanded in normAngle around 0 97.8%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\left(u + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {u}^{3} - -0.16666666666666666 \cdot u\right)\right)} \cdot n1_i \]
3. Step-by-step derivation

   1. unpow2 97.8%

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(u + \color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(-0.16666666666666666 \cdot {u}^{3} - -0.16666666666666666 \cdot u\right)\right) \cdot n1_i \]

   2. distribute-lft-out-- 97.8%

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)}\right) \cdot n1_i \]

4. Simplified 97.8%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right)} \cdot n1_i \]

5. Taylor expanded in normAngle around 0 98.5%

   \[\leadsto \color{blue}{\left(\left(1 + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right)\right) - u\right)} \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]
6. Step-by-step derivation

   1. associate--l+ 98.5%

      \[\leadsto \color{blue}{\left(1 + \left({normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right) - u\right)\right)} \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

   2. unpow2 98.5%

      \[\leadsto \left(1 + \left(\color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right) - u\right)\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

   3. distribute-lft-out-- 98.5%

      \[\leadsto \left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right)\right)} - u\right)\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

7. Simplified 98.5%

   \[\leadsto \color{blue}{\left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right)\right) - u\right)\right)} \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

8. Final simplification 98.5%

   \[\leadsto n0_i \cdot \left(1 - \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left(\left(1 - u\right) - {\left(1 - u\right)}^{3}\right)\right)\right)\right) + n1_i \cdot \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \]

Alternative 3: 98.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ n0_i \cdot \left(1 - \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left(\left(1 - u\right) - {\left(1 - u\right)}^{3}\right)\right)\right)\right) + n1_i \cdot \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (*
   n0_i
   (-
    1.0
    (+
     u
     (*
      (* normAngle normAngle)
      (* -0.16666666666666666 (- (- 1.0 u) (pow (- 1.0 u) 3.0)))))))
  (* n1_i (+ u (* (* normAngle normAngle) (* u 0.16666666666666666))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n0_i * (1.0f - (u + ((normAngle * normAngle) * (-0.16666666666666666f * ((1.0f - u) - powf((1.0f - u), 3.0f))))))) + (n1_i * (u + ((normAngle * normAngle) * (u * 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 * (1.0e0 - (u + ((normangle * normangle) * ((-0.16666666666666666e0) * ((1.0e0 - u) - ((1.0e0 - u) ** 3.0e0))))))) + (n1_i * (u + ((normangle * normangle) * (u * 0.16666666666666666e0))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(n0_i * Float32(Float32(1.0) - Float32(u + Float32(Float32(normAngle * normAngle) * Float32(Float32(-0.16666666666666666) * Float32(Float32(Float32(1.0) - u) - (Float32(Float32(1.0) - u) ^ Float32(3.0)))))))) + Float32(n1_i * Float32(u + Float32(Float32(normAngle * normAngle) * Float32(u * Float32(0.16666666666666666))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n0_i * (single(1.0) - (u + ((normAngle * normAngle) * (single(-0.16666666666666666) * ((single(1.0) - u) - ((single(1.0) - u) ^ single(3.0)))))))) + (n1_i * (u + ((normAngle * normAngle) * (u * single(0.16666666666666666)))));
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. Taylor expanded in normAngle around 0 97.8%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\left(u + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {u}^{3} - -0.16666666666666666 \cdot u\right)\right)} \cdot n1_i \]
3. Step-by-step derivation

   1. unpow2 97.8%

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(u + \color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(-0.16666666666666666 \cdot {u}^{3} - -0.16666666666666666 \cdot u\right)\right) \cdot n1_i \]

   2. distribute-lft-out-- 97.8%

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)}\right) \cdot n1_i \]

4. Simplified 97.8%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right)} \cdot n1_i \]

5. Taylor expanded in normAngle around 0 98.5%

   \[\leadsto \color{blue}{\left(\left(1 + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right)\right) - u\right)} \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]
6. Step-by-step derivation

   1. associate--l+ 98.5%

      \[\leadsto \color{blue}{\left(1 + \left({normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right) - u\right)\right)} \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

   2. unpow2 98.5%

      \[\leadsto \left(1 + \left(\color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right) - u\right)\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

   3. distribute-lft-out-- 98.5%

      \[\leadsto \left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right)\right)} - u\right)\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

7. Simplified 98.5%

   \[\leadsto \color{blue}{\left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right)\right) - u\right)\right)} \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

8. Taylor expanded in u around 0 98.5%

   \[\leadsto \left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right)\right) - u\right)\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(0.16666666666666666 \cdot u\right)}\right) \cdot n1_i \]

9. Final simplification 98.5%

   \[\leadsto n0_i \cdot \left(1 - \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left(\left(1 - u\right) - {\left(1 - u\right)}^{3}\right)\right)\right)\right) + n1_i \cdot \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot 0.16666666666666666\right)\right) \]

Alternative 4: 98.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ n1_i \cdot \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) + n0_i \cdot \left(1 - \left(u - \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot 0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (*
   n1_i
   (+
    u
    (* (* normAngle normAngle) (* -0.16666666666666666 (- (pow u 3.0) u)))))
  (* n0_i (- 1.0 (- u (* (* normAngle normAngle) (* u 0.3333333333333333)))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n1_i * (u + ((normAngle * normAngle) * (-0.16666666666666666f * (powf(u, 3.0f) - u))))) + (n0_i * (1.0f - (u - ((normAngle * normAngle) * (u * 0.3333333333333333f)))));
}

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 + ((normangle * normangle) * ((-0.16666666666666666e0) * ((u ** 3.0e0) - u))))) + (n0_i * (1.0e0 - (u - ((normangle * normangle) * (u * 0.3333333333333333e0)))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(n1_i * Float32(u + Float32(Float32(normAngle * normAngle) * Float32(Float32(-0.16666666666666666) * Float32((u ^ Float32(3.0)) - u))))) + Float32(n0_i * Float32(Float32(1.0) - Float32(u - Float32(Float32(normAngle * normAngle) * Float32(u * Float32(0.3333333333333333)))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n1_i * (u + ((normAngle * normAngle) * (single(-0.16666666666666666) * ((u ^ single(3.0)) - u))))) + (n0_i * (single(1.0) - (u - ((normAngle * normAngle) * (u * single(0.3333333333333333))))));
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. Taylor expanded in normAngle around 0 97.8%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\left(u + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {u}^{3} - -0.16666666666666666 \cdot u\right)\right)} \cdot n1_i \]
3. Step-by-step derivation

   1. unpow2 97.8%

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(u + \color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(-0.16666666666666666 \cdot {u}^{3} - -0.16666666666666666 \cdot u\right)\right) \cdot n1_i \]

   2. distribute-lft-out-- 97.8%

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)}\right) \cdot n1_i \]

4. Simplified 97.8%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right)} \cdot n1_i \]

5. Taylor expanded in normAngle around 0 98.5%

   \[\leadsto \color{blue}{\left(\left(1 + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right)\right) - u\right)} \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]
6. Step-by-step derivation

   1. associate--l+ 98.5%

      \[\leadsto \color{blue}{\left(1 + \left({normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right) - u\right)\right)} \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

   2. unpow2 98.5%

      \[\leadsto \left(1 + \left(\color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(-0.16666666666666666 \cdot {\left(1 - u\right)}^{3} - -0.16666666666666666 \cdot \left(1 - u\right)\right) - u\right)\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

   3. distribute-lft-out-- 98.5%

      \[\leadsto \left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right)\right)} - u\right)\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

7. Simplified 98.5%

   \[\leadsto \color{blue}{\left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right)\right) - u\right)\right)} \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

8. Taylor expanded in u around 0 98.3%

   \[\leadsto \left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot u\right)} - u\right)\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]
9. Step-by-step derivation

   1. *-commutative 98.3%

      \[\leadsto \left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(u \cdot 0.3333333333333333\right)} - u\right)\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

10. Simplified 98.3%

    \[\leadsto \left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(u \cdot 0.3333333333333333\right)} - u\right)\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

11. Final simplification 98.3%

    \[\leadsto n1_i \cdot \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) + n0_i \cdot \left(1 - \left(u - \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot 0.3333333333333333\right)\right)\right) \]

Alternative 5: 98.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ n1_i \cdot \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) + n0_i \cdot \left(1 - u\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (*
   n1_i
   (+
    u
    (* (* normAngle normAngle) (* -0.16666666666666666 (- (pow u 3.0) u)))))
  (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n1_i * (u + ((normAngle * normAngle) * (-0.16666666666666666f * (powf(u, 3.0f) - 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 * (u + ((normangle * normangle) * ((-0.16666666666666666e0) * ((u ** 3.0e0) - u))))) + (n0_i * (1.0e0 - u))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(n1_i * Float32(u + Float32(Float32(normAngle * normAngle) * Float32(Float32(-0.16666666666666666) * Float32((u ^ Float32(3.0)) - u))))) + Float32(n0_i * Float32(Float32(1.0) - u)))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n1_i * (u + ((normAngle * normAngle) * (single(-0.16666666666666666) * ((u ^ single(3.0)) - 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. Taylor expanded in normAngle around 0 97.8%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\left(u + {normAngle}^{2} \cdot \left(-0.16666666666666666 \cdot {u}^{3} - -0.16666666666666666 \cdot u\right)\right)} \cdot n1_i \]
3. Step-by-step derivation

   1. unpow2 97.8%

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(u + \color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(-0.16666666666666666 \cdot {u}^{3} - -0.16666666666666666 \cdot u\right)\right) \cdot n1_i \]

   2. distribute-lft-out-- 97.8%

      \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)}\right) \cdot n1_i \]

4. Simplified 97.8%

   \[\leadsto \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \color{blue}{\left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right)} \cdot n1_i \]

5. Taylor expanded in normAngle around 0 98.1%

   \[\leadsto \color{blue}{\left(1 - u\right)} \cdot n0_i + \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) \cdot n1_i \]

6. Final simplification 98.1%

   \[\leadsto n1_i \cdot \left(u + \left(normAngle \cdot normAngle\right) \cdot \left(-0.16666666666666666 \cdot \left({u}^{3} - u\right)\right)\right) + n0_i \cdot \left(1 - u\right) \]

Alternative 6: 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 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*r* 77.5%

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

   3. *-commutative 77.5%

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

   4. associate-*l* 71.9%

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

   5. *-commutative 71.9%

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

   6. distribute-rgt-out 72.0%

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

   7. associate-*l/ 72.2%

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

3. Simplified 72.2%

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

4. Taylor expanded in normAngle around 0 97.1%

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

   1. flip-- 96.8%

      \[\leadsto n0_i \cdot \color{blue}{\frac{1 \cdot 1 - u \cdot u}{1 + u}} + n1_i \cdot u \]

   2. associate-*r/ 96.8%

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

   3. metadata-eval 96.8%

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

   4. +-commutative 96.8%

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

6. Applied egg-rr 96.8%

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

   1. associate-/l* 96.7%

      \[\leadsto \color{blue}{\frac{n0_i}{\frac{u + 1}{1 - u \cdot u}}} + n1_i \cdot u \]

8. Simplified 96.7%

   \[\leadsto \color{blue}{\frac{n0_i}{\frac{u + 1}{1 - u \cdot u}}} + n1_i \cdot u \]

9. Taylor expanded in u around 0 97.3%

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

    1. +-commutative 97.3%

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

    2. sub-neg 97.3%

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

    3. neg-mul-1 97.3%

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

    4. fma-def 97.4%

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

    5. neg-mul-1 97.4%

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

    6. sub-neg 97.4%

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

11. Simplified 97.4%

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

12. Final simplification 97.4%

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

Alternative 7: 59.8% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 7.199999872485958 \cdot 10^{-22}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -4.999999980020986e-13)
   (* u n1_i)
   (if (<= n1_i 7.199999872485958e-22) n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -4.999999980020986e-13f) {
		tmp = u * n1_i;
	} else if (n1_i <= 7.199999872485958e-22f) {
		tmp = n0_i;
	} else {
		tmp = 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 (n1_i <= (-4.999999980020986e-13)) then
        tmp = u * n1_i
    else if (n1_i <= 7.199999872485958e-22) then
        tmp = n0_i
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-4.999999980020986e-13))
		tmp = Float32(u * n1_i);
	elseif (n1_i <= Float32(7.199999872485958e-22))
		tmp = n0_i;
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n1_i <= single(-4.999999980020986e-13))
		tmp = u * n1_i;
	elseif (n1_i <= single(7.199999872485958e-22))
		tmp = n0_i;
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.99999998e-13 or 7.19999987e-22 < n1_i

   1. Initial program 95.6%

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

      1. *-commutative 95.6%

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

      2. associate-*r* 87.6%

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

      3. *-commutative 87.6%

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

      4. associate-*l* 80.7%

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

      5. *-commutative 80.7%

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

      6. distribute-rgt-out 80.9%

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

      7. associate-*l/ 81.0%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in normAngle around 0 96.7%

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

   5. Taylor expanded in n0_i around 0 64.9%

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

      1. *-commutative 64.9%

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

   7. Simplified 64.9%

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

   if -4.99999998e-13 < n1_i < 7.19999987e-22

   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%

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

      2. associate-*r* 70.8%

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

      3. *-commutative 70.8%

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

      4. associate-*l* 66.1%

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

      5. *-commutative 66.1%

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

      6. distribute-rgt-out 66.1%

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

      7. associate-*l/ 66.3%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in u around 0 62.5%

      \[\leadsto \color{blue}{n0_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{elif}\;n1_i \leq 7.199999872485958 \cdot 10^{-22}:\\ \;\;\;\;n0_i\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 8: 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 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*r* 77.5%

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

   3. *-commutative 77.5%

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

   4. associate-*l* 71.9%

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

   5. *-commutative 71.9%

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

   6. distribute-rgt-out 72.0%

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

   7. associate-*l/ 72.2%

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

3. Simplified 72.2%

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

4. Taylor expanded in normAngle around 0 97.1%

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

   1. flip-- 96.8%

      \[\leadsto n0_i \cdot \color{blue}{\frac{1 \cdot 1 - u \cdot u}{1 + u}} + n1_i \cdot u \]

   2. associate-*r/ 96.8%

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

   3. metadata-eval 96.8%

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

   4. +-commutative 96.8%

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

6. Applied egg-rr 96.8%

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

   1. associate-/l* 96.7%

      \[\leadsto \color{blue}{\frac{n0_i}{\frac{u + 1}{1 - u \cdot u}}} + n1_i \cdot u \]

8. Simplified 96.7%

   \[\leadsto \color{blue}{\frac{n0_i}{\frac{u + 1}{1 - u \cdot u}}} + n1_i \cdot u \]

9. Taylor expanded in u around 0 97.3%

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

10. Final simplification 97.3%

    \[\leadsto n0_i + u \cdot \left(n1_i - n0_i\right) \]

Alternative 9: 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 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. Step-by-step derivation

   1. *-commutative 96.9%

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

   2. associate-*r* 77.5%

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

   3. *-commutative 77.5%

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

   4. associate-*l* 71.9%

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

   5. *-commutative 71.9%

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

   6. distribute-rgt-out 72.0%

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

   7. associate-*l/ 72.2%

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

3. Simplified 72.2%

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

4. Taylor expanded in normAngle around 0 97.1%

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

5. Taylor expanded in u around 0 80.3%

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

6. Final simplification 80.3%

   \[\leadsto n0_i + u \cdot n1_i \]

Alternative 10: 47.5% 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. Step-by-step derivation

   1. *-commutative 96.9%

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

   2. associate-*r* 77.5%

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

   3. *-commutative 77.5%

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

   4. associate-*l* 71.9%

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

   5. *-commutative 71.9%

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

   6. distribute-rgt-out 72.0%

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

   7. associate-*l/ 72.2%

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

3. Simplified 72.2%

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

4. Taylor expanded in u around 0 47.0%

   \[\leadsto \color{blue}{n0_i} \]

5. Final simplification 47.0%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023297 
(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)))