Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 98.8%

Time: 16.7s

Alternatives: 9

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: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t_0\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot t_0\right) \cdot n1_i \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sin normAngle))))
   (+
    (* (* (sin (* (- 1.0 u) normAngle)) t_0) n0_i)
    (* (* (sin (* u normAngle)) t_0) n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = 1.0f / sinf(normAngle);
	return ((sinf(((1.0f - u) * normAngle)) * t_0) * n0_i) + ((sinf((u * normAngle)) * t_0) * n1_i);
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: t_0
    t_0 = 1.0e0 / sin(normangle)
    code = ((sin(((1.0e0 - u) * normangle)) * t_0) * n0_i) + ((sin((u * normangle)) * t_0) * n1_i)
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(1.0) / sin(normAngle))
	return Float32(Float32(Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * t_0) * n0_i) + Float32(Float32(sin(Float32(u * normAngle)) * t_0) * n1_i))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	t_0 = single(1.0) / sin(normAngle);
	tmp = ((sin(((single(1.0) - u) * normAngle)) * t_0) * n0_i) + ((sin((u * normAngle)) * t_0) * n1_i);
end

Alternative 1: 98.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ n0_i - \left(u \cdot \left(n0_i - n1_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\left(n0_i \cdot -0.5 - n0_i \cdot -0.16666666666666666\right) + n1_i \cdot -0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (-
  n0_i
  (+
   (* u (- n0_i n1_i))
   (*
    (pow normAngle 2.0)
    (*
     u
     (+
      (- (* n0_i -0.5) (* n0_i -0.16666666666666666))
      (* n1_i -0.16666666666666666)))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i - ((u * (n0_i - n1_i)) + (powf(normAngle, 2.0f) * (u * (((n0_i * -0.5f) - (n0_i * -0.16666666666666666f)) + (n1_i * -0.16666666666666666f)))));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i - ((u * (n0_i - n1_i)) + ((normangle ** 2.0e0) * (u * (((n0_i * (-0.5e0)) - (n0_i * (-0.16666666666666666e0))) + (n1_i * (-0.16666666666666666e0))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i - Float32(Float32(u * Float32(n0_i - n1_i)) + Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(Float32(Float32(n0_i * Float32(-0.5)) - Float32(n0_i * Float32(-0.16666666666666666))) + Float32(n1_i * Float32(-0.16666666666666666)))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i - ((u * (n0_i - n1_i)) + ((normAngle ^ single(2.0)) * (u * (((n0_i * single(-0.5)) - (n0_i * single(-0.16666666666666666))) + (n1_i * single(-0.16666666666666666))))));
end

Derivation

1. Initial program 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-*l* 79.8%

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

   3. *-commutative 79.8%

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

   4. associate-*l* 72.8%

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

   5. distribute-lft-out 72.8%

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

3. Simplified 72.8%

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

5. Taylor expanded in u around 0 88.6%

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

6. Taylor expanded in normAngle around 0 99.0%

   \[\leadsto n0_i + \color{blue}{\left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-1 \cdot \left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) - -0.16666666666666666 \cdot n1_i\right)\right)\right)} \]

7. Final simplification 99.0%

   \[\leadsto n0_i - \left(u \cdot \left(n0_i - n1_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\left(n0_i \cdot -0.5 - n0_i \cdot -0.16666666666666666\right) + n1_i \cdot -0.16666666666666666\right)\right)\right) \]
8. Add Preprocessing

Alternative 2: 98.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-*l* 79.8%

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

   3. *-commutative 79.8%

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

   4. associate-*l* 72.8%

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

   5. distribute-lft-out 72.8%

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

3. Simplified 72.8%

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

5. Taylor expanded in u around 0 88.6%

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

6. Taylor expanded in normAngle around 0 99.0%

   \[\leadsto n0_i + \color{blue}{\left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-1 \cdot \left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) - -0.16666666666666666 \cdot n1_i\right)\right)\right)} \]

7. Taylor expanded in n0_i around 0 98.7%

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

   1. associate-*r* 98.7%

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

   2. *-commutative 98.7%

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

   3. associate-*l* 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

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

Alternative 3: 98.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((n1_i - n0_i) - (powf(normAngle, 2.0f) * (n1_i * -0.16666666666666666f))));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i + (u * ((n1_i - n0_i) - ((normangle ** 2.0e0) * (n1_i * (-0.16666666666666666e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-*l* 79.8%

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

   3. *-commutative 79.8%

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

   4. associate-*l* 72.8%

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

   5. distribute-lft-out 72.8%

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

3. Simplified 72.8%

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

5. Taylor expanded in u around 0 88.6%

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

6. Taylor expanded in normAngle around 0 99.0%

   \[\leadsto n0_i + \color{blue}{\left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-1 \cdot \left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) - -0.16666666666666666 \cdot n1_i\right)\right)\right)} \]

7. Taylor expanded in n0_i around 0 98.7%

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

   1. associate-*r* 98.7%

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

   2. *-commutative 98.7%

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

   3. associate-*l* 98.7%

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

9. Simplified 98.7%

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

10. Taylor expanded in u around -inf 98.7%

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

    1. mul-1-neg 98.7%

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

    2. unsub-neg 98.7%

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

    3. +-commutative 98.7%

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

    4. mul-1-neg 98.7%

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

    5. unsub-neg 98.7%

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

    6. associate-*r* 98.7%

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

    7. *-commutative 98.7%

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

    8. *-commutative 98.7%

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

    9. mul-1-neg 98.7%

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

    10. sub-neg 98.7%

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

12. Simplified 98.7%

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

13. Final simplification 98.7%

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

Alternative 4: 98.2% 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 96.7%

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

3. Taylor expanded in normAngle around 0 97.9%

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

4. Taylor expanded in normAngle around 0 97.9%

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

   1. +-commutative 97.9%

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

   2. *-commutative 97.9%

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

   3. sub-neg 97.9%

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

   4. mul-1-neg 97.9%

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

   5. distribute-lft-in 97.8%

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

   6. *-rgt-identity 97.8%

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

   7. mul-1-neg 97.8%

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

   8. distribute-rgt-neg-in 97.8%

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

   9. mul-1-neg 97.8%

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

   10. +-commutative 97.8%

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

   11. associate-+l+ 97.9%

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

   12. *-commutative 97.9%

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

   13. associate-*r* 97.9%

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

   14. distribute-rgt-in 97.9%

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

   15. fma-def 98.1%

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

   16. mul-1-neg 98.1%

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

   17. unsub-neg 98.1%

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

6. Simplified 98.1%

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

7. Final simplification 98.1%

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

Alternative 5: 71.2% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -4.00000012549758 \cdot 10^{-22} \lor \neg \left(n0_i \leq 9.999999887266023 \cdot 10^{-27}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -4.00000012549758e-22) (not (<= n0_i 9.999999887266023e-27)))
   (* n0_i (- 1.0 u))
   (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -4.00000012549758e-22f) || !(n0_i <= 9.999999887266023e-27f)) {
		tmp = n0_i * (1.0f - u);
	} 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 ((n0_i <= (-4.00000012549758e-22)) .or. (.not. (n0_i <= 9.999999887266023e-27))) then
        tmp = n0_i * (1.0e0 - u)
    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 ((n0_i <= Float32(-4.00000012549758e-22)) || !(n0_i <= Float32(9.999999887266023e-27)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	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 ((n0_i <= single(-4.00000012549758e-22)) || ~((n0_i <= single(9.999999887266023e-27))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.00000013e-22 or 9.99999989e-27 < n0_i

   1. Initial program 97.4%

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

      1. *-commutative 97.4%

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

      2. associate-*l* 80.1%

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

      3. *-commutative 80.1%

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

      4. associate-*l* 78.7%

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

      5. distribute-lft-out 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in normAngle around 0 98.0%

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

      1. add-sqr-sqrt 47.2%

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

      2. sqrt-unprod 84.4%

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

      3. pow2 84.4%

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

      4. *-commutative 84.4%

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

   7. Applied egg-rr 84.4%

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

      1. unpow2 84.4%

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

      2. rem-sqrt-square 85.0%

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

   9. Simplified 85.0%

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

   10. Taylor expanded in n0_i around inf 77.6%

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

   if -4.00000013e-22 < n0_i < 9.99999989e-27

   1. Initial program 95.7%

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

   3. Taylor expanded in normAngle around 0 97.5%

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

   4. Taylor expanded in u around inf 73.6%

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

      1. *-commutative 73.6%

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

   6. Simplified 73.6%

      \[\leadsto \color{blue}{u \cdot n1_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -4.00000012549758 \cdot 10^{-22} \lor \neg \left(n0_i \leq 9.999999887266023 \cdot 10^{-27}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.9% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -4.00000012549758 \cdot 10^{-22}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 9.999999887266023 \cdot 10^{-27}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -4.00000012549758e-22)
   n0_i
   (if (<= n0_i 9.999999887266023e-27) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -4.00000012549758e-22f) {
		tmp = n0_i;
	} else if (n0_i <= 9.999999887266023e-27f) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if (n0_i <= (-4.00000012549758e-22)) then
        tmp = n0_i
    else if (n0_i <= 9.999999887266023e-27) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-4.00000012549758e-22))
		tmp = n0_i;
	elseif (n0_i <= Float32(9.999999887266023e-27))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-4.00000012549758e-22))
		tmp = n0_i;
	elseif (n0_i <= single(9.999999887266023e-27))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.00000013e-22 or 9.99999989e-27 < n0_i

   1. Initial program 97.4%

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

   3. Taylor expanded in normAngle around 0 98.2%

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

   4. Taylor expanded in u around 0 62.1%

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

   if -4.00000013e-22 < n0_i < 9.99999989e-27

   1. Initial program 95.7%

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

   3. Taylor expanded in normAngle around 0 97.5%

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

   4. Taylor expanded in u around inf 73.6%

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

      1. *-commutative 73.6%

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

   6. Simplified 73.6%

      \[\leadsto \color{blue}{u \cdot n1_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -4.00000012549758 \cdot 10^{-22}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 9.999999887266023 \cdot 10^{-27}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-*l* 79.8%

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

   3. *-commutative 79.8%

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

   4. associate-*l* 72.8%

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

   5. distribute-lft-out 72.8%

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

3. Simplified 72.8%

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

5. Taylor expanded in normAngle around 0 97.9%

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

6. Taylor expanded in u around -inf 97.9%

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

   1. mul-1-neg 97.9%

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

   2. unsub-neg 97.9%

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

   3. mul-1-neg 97.9%

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

   4. unsub-neg 97.9%

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

8. Simplified 97.9%

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

9. Final simplification 97.9%

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

Alternative 8: 81.6% 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 96.7%

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

3. Taylor expanded in normAngle around 0 97.9%

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

4. Taylor expanded in u around 0 83.3%

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

5. Final simplification 83.3%

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

Alternative 9: 47.4% 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 96.7%

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

3. Taylor expanded in normAngle around 0 97.9%

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

4. Taylor expanded in u around 0 46.1%

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

5. Final simplification 46.1%

   \[\leadsto n0_i \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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)))