Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 99.2%

Time: 14.6s

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: 97.2% 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: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * fma(Float32(-1.0), Float32(Float32(n0_i / sin(normAngle)) * Float32(normAngle * cos(normAngle))), Float32(normAngle * Float32(n1_i / sin(normAngle))))))
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. fma-def 97.0%

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

   2. associate-*r/ 97.1%

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

   3. *-rgt-identity 97.1%

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

   4. associate-*r/ 97.3%

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

   5. *-rgt-identity 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in u around 0 87.1%

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

   1. fma-def 87.1%

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

   2. associate-/l* 90.4%

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

   3. associate-/r/ 90.3%

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

   4. associate-/l* 99.1%

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

   5. associate-/r/ 99.1%

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

6. Simplified 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + ((powf(normAngle, 2.0f) * (u * (((n0_i * -0.16666666666666666f) - (n0_i * -0.5f)) - (n1_i * -0.16666666666666666f)))) + (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 + (((normangle ** 2.0e0) * (u * (((n0_i * (-0.16666666666666666e0)) - (n0_i * (-0.5e0))) - (n1_i * (-0.16666666666666666e0))))) + (u * (n1_i - n0_i)))
end function

Julia

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

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (((normAngle ^ single(2.0)) * (u * (((n0_i * single(-0.16666666666666666)) - (n0_i * single(-0.5))) - (n1_i * single(-0.16666666666666666))))) + (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. fma-def 97.0%

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

   2. associate-*r/ 97.1%

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

   3. *-rgt-identity 97.1%

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

   4. associate-*r/ 97.3%

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

   5. *-rgt-identity 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in u around 0 87.1%

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

   1. fma-def 87.1%

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

   2. associate-/l* 90.4%

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

   3. associate-/r/ 90.3%

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

   4. associate-/l* 99.1%

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

   5. associate-/r/ 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in normAngle around 0 99.1%

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

8. Final simplification 99.1%

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

Alternative 3: 98.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ n0_i + \left({normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333\right)\right) + u \cdot \left(n1_i - n0_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (+
   (* (pow normAngle 2.0) (* u (* n0_i 0.3333333333333333)))
   (* u (- n1_i n0_i)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + ((powf(normAngle, 2.0f) * (u * (n0_i * 0.3333333333333333f))) + (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 + (((normangle ** 2.0e0) * (u * (n0_i * 0.3333333333333333e0))) + (u * (n1_i - n0_i)))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(n0_i * Float32(0.3333333333333333)))) + Float32(u * Float32(n1_i - n0_i))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (((normAngle ^ single(2.0)) * (u * (n0_i * single(0.3333333333333333)))) + (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. fma-def 97.0%

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

   2. associate-*r/ 97.1%

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

   3. *-rgt-identity 97.1%

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

   4. associate-*r/ 97.3%

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

   5. *-rgt-identity 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in u around 0 87.1%

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

   1. fma-def 87.1%

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

   2. associate-/l* 90.4%

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

   3. associate-/r/ 90.3%

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

   4. associate-/l* 99.1%

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

   5. associate-/r/ 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in normAngle around 0 99.1%

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

8. Taylor expanded in n0_i around inf 98.8%

   \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(n0_i \cdot u\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r* 98.8%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(\left(0.3333333333333333 \cdot n0_i\right) \cdot u\right)}\right) \]

   2. *-commutative 98.8%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \left(\color{blue}{\left(n0_i \cdot 0.3333333333333333\right)} \cdot u\right)\right) \]

   3. *-commutative 98.8%

      \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(n0_i \cdot 0.3333333333333333\right)\right)}\right) \]

10. Simplified 98.8%

    \[\leadsto n0_i + \left(u \cdot \left(n1_i + -1 \cdot n0_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(n0_i \cdot 0.3333333333333333\right)\right)}\right) \]

11. Taylor expanded in n1_i around 0 98.7%

    \[\leadsto n0_i + \left(\color{blue}{\left(-1 \cdot \left(n0_i \cdot u\right) + n1_i \cdot u\right)} + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333\right)\right)\right) \]
12. Step-by-step derivation

    1. +-commutative 98.7%

       \[\leadsto n0_i + \left(\color{blue}{\left(n1_i \cdot u + -1 \cdot \left(n0_i \cdot u\right)\right)} + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333\right)\right)\right) \]

    2. mul-1-neg 98.7%

       \[\leadsto n0_i + \left(\left(n1_i \cdot u + \color{blue}{\left(-n0_i \cdot u\right)}\right) + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333\right)\right)\right) \]

    3. sub-neg 98.7%

       \[\leadsto n0_i + \left(\color{blue}{\left(n1_i \cdot u - n0_i \cdot u\right)} + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333\right)\right)\right) \]

    4. distribute-rgt-out-- 98.8%

       \[\leadsto n0_i + \left(\color{blue}{u \cdot \left(n1_i - n0_i\right)} + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333\right)\right)\right) \]

13. Simplified 98.8%

    \[\leadsto n0_i + \left(\color{blue}{u \cdot \left(n1_i - n0_i\right)} + {normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333\right)\right)\right) \]

14. Final simplification 98.8%

    \[\leadsto n0_i + \left({normAngle}^{2} \cdot \left(u \cdot \left(n0_i \cdot 0.3333333333333333\right)\right) + u \cdot \left(n1_i - n0_i\right)\right) \]

Alternative 4: 98.3% 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. fma-def 97.0%

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

   2. associate-*r/ 97.1%

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

   3. *-rgt-identity 97.1%

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

   4. associate-*r/ 97.3%

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

   5. *-rgt-identity 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in normAngle around 0 98.3%

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

   1. *-commutative 98.3%

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

   2. flip-- 98.2%

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

   3. associate-*l/ 98.2%

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

   4. metadata-eval 98.2%

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

   5. pow1 98.2%

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

   6. pow1 98.2%

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

   7. pow-sqr 98.2%

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

   8. metadata-eval 98.2%

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

   9. +-commutative 98.2%

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

6. Applied egg-rr 98.2%

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

   1. +-commutative 98.2%

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

   2. *-commutative 98.2%

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

   3. fma-def 98.2%

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

   4. *-commutative 98.2%

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

   5. associate-*r/ 98.3%

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

   6. metadata-eval 98.3%

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

   7. unpow2 98.3%

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

   8. +-commutative 98.3%

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

   9. flip-- 98.4%

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

8. Applied egg-rr 98.4%

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

9. Taylor expanded in u around 0 98.6%

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

    1. +-commutative 98.6%

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

    2. fma-def 98.7%

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

    3. mul-1-neg 98.7%

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

    4. unsub-neg 98.7%

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

11. Simplified 98.7%

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

12. Final simplification 98.7%

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

Alternative 5: 70.6% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -2.499999990010493 \cdot 10^{-13} \lor \neg \left(n1_i \leq 5.000000018137469 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -2.499999990010493e-13)
         (not (<= n1_i 5.000000018137469e-16)))
   (* u n1_i)
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -2.499999990010493e-13f) || !(n1_i <= 5.000000018137469e-16f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-2.499999990010493e-13)) .or. (.not. (n1_i <= 5.000000018137469e-16))) then
        tmp = u * n1_i
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-2.499999990010493e-13)) || !(n1_i <= Float32(5.000000018137469e-16)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-2.499999990010493e-13)) || ~((n1_i <= single(5.000000018137469e-16))))
		tmp = u * n1_i;
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -2.49999999e-13 or 5.00000002e-16 < n1_i

   1. Initial program 94.9%

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

      1. fma-def 94.8%

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

      2. associate-*r/ 94.9%

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

      3. *-rgt-identity 94.9%

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

      4. associate-*r/ 95.1%

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

      5. *-rgt-identity 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in normAngle around 0 98.3%

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

   5. Taylor expanded in n0_i around 0 65.4%

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

   if -2.49999999e-13 < n1_i < 5.00000002e-16

   1. Initial program 98.1%

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

      1. fma-def 98.2%

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

      2. associate-*r/ 98.4%

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

      3. *-rgt-identity 98.4%

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

      4. associate-*r/ 98.4%

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

      5. *-rgt-identity 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in normAngle around 0 98.4%

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

      1. fma-def 98.5%

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

      2. *-commutative 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in n0_i around inf 77.6%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -2.499999990010493 \cdot 10^{-13} \lor \neg \left(n1_i \leq 5.000000018137469 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 6: 86.5% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999943633011 \cdot 10^{-27} \lor \neg \left(n1_i \leq 4.0000000126843074 \cdot 10^{-28}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -4.999999943633011e-27)
         (not (<= n1_i 4.0000000126843074e-28)))
   (+ n0_i (* u n1_i))
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -4.999999943633011e-27f) || !(n1_i <= 4.0000000126843074e-28f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-4.999999943633011e-27)) .or. (.not. (n1_i <= 4.0000000126843074e-28))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-4.999999943633011e-27)) || !(n1_i <= Float32(4.0000000126843074e-28)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-4.999999943633011e-27)) || ~((n1_i <= single(4.0000000126843074e-28))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.99999994e-27 or 4.00000001e-28 < n1_i

   1. Initial program 96.3%

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

      1. fma-def 96.3%

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

      2. associate-*r/ 96.4%

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

      3. *-rgt-identity 96.4%

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

      4. associate-*r/ 96.6%

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

      5. *-rgt-identity 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in u around 0 82.6%

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

   5. Taylor expanded in normAngle around 0 84.7%

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

   if -4.99999994e-27 < n1_i < 4.00000001e-28

   1. Initial program 98.4%

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

      1. fma-def 98.6%

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

      2. associate-*r/ 98.7%

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

      3. *-rgt-identity 98.7%

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

      4. associate-*r/ 98.7%

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

      5. *-rgt-identity 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in normAngle around 0 97.8%

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

      1. fma-def 97.9%

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

      2. *-commutative 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in n0_i around inf 90.5%

      \[\leadsto \color{blue}{n0_i \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999943633011 \cdot 10^{-27} \lor \neg \left(n1_i \leq 4.0000000126843074 \cdot 10^{-28}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 7: 86.6% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999943633011 \cdot 10^{-27} \lor \neg \left(n1_i \leq 4.0000000126843074 \cdot 10^{-28}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -4.999999943633011e-27)
         (not (<= n1_i 4.0000000126843074e-28)))
   (+ n0_i (* u n1_i))
   (- n0_i (* n0_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -4.999999943633011e-27f) || !(n1_i <= 4.0000000126843074e-28f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i - (n0_i * u);
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-4.999999943633011e-27)) .or. (.not. (n1_i <= 4.0000000126843074e-28))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i - (n0_i * u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-4.999999943633011e-27)) || !(n1_i <= Float32(4.0000000126843074e-28)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i - Float32(n0_i * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-4.999999943633011e-27)) || ~((n1_i <= single(4.0000000126843074e-28))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i - (n0_i * u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.99999994e-27 or 4.00000001e-28 < n1_i

   1. Initial program 96.3%

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

      1. fma-def 96.3%

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

      2. associate-*r/ 96.4%

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

      3. *-rgt-identity 96.4%

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

      4. associate-*r/ 96.6%

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

      5. *-rgt-identity 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in u around 0 82.6%

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

   5. Taylor expanded in normAngle around 0 84.7%

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

   if -4.99999994e-27 < n1_i < 4.00000001e-28

   1. Initial program 98.4%

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

      1. fma-def 98.6%

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

      2. associate-*r/ 98.7%

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

      3. *-rgt-identity 98.7%

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

      4. associate-*r/ 98.7%

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

      5. *-rgt-identity 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in normAngle around 0 97.8%

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

      1. fma-def 97.9%

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

      2. *-commutative 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in n0_i around inf 90.5%

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

      1. sub-neg 90.5%

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

      2. distribute-rgt-in 90.8%

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

      3. *-un-lft-identity 90.8%

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

   9. Applied egg-rr 90.8%

      \[\leadsto \color{blue}{n0_i + \left(-u\right) \cdot n0_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -4.999999943633011 \cdot 10^{-27} \lor \neg \left(n1_i \leq 4.0000000126843074 \cdot 10^{-28}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \]

Alternative 8: 60.1% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -2.499999990010493 \cdot 10^{-13} \lor \neg \left(n1_i \leq 5.000000018137469 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -2.499999990010493e-13)
         (not (<= n1_i 5.000000018137469e-16)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -2.499999990010493e-13f) || !(n1_i <= 5.000000018137469e-16f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    real(4) :: tmp
    if ((n1_i <= (-2.499999990010493e-13)) .or. (.not. (n1_i <= 5.000000018137469e-16))) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-2.499999990010493e-13)) || !(n1_i <= Float32(5.000000018137469e-16)))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-2.499999990010493e-13)) || ~((n1_i <= single(5.000000018137469e-16))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -2.49999999e-13 or 5.00000002e-16 < n1_i

   1. Initial program 94.9%

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

      1. fma-def 94.8%

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

      2. associate-*r/ 94.9%

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

      3. *-rgt-identity 94.9%

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

      4. associate-*r/ 95.1%

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

      5. *-rgt-identity 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in normAngle around 0 98.3%

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

   5. Taylor expanded in n0_i around 0 65.4%

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

   if -2.49999999e-13 < n1_i < 5.00000002e-16

   1. Initial program 98.1%

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

      1. fma-def 98.2%

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

      2. associate-*r/ 98.4%

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

      3. *-rgt-identity 98.4%

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

      4. associate-*r/ 98.4%

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

      5. *-rgt-identity 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in u around 0 57.6%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -2.499999990010493 \cdot 10^{-13} \lor \neg \left(n1_i \leq 5.000000018137469 \cdot 10^{-16}\right):\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 9: 98.2% 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. fma-def 97.0%

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

   2. associate-*r/ 97.1%

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

   3. *-rgt-identity 97.1%

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

   4. associate-*r/ 97.3%

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

   5. *-rgt-identity 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in normAngle around 0 98.3%

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

   1. fma-def 98.4%

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

   2. *-commutative 98.4%

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

6. Simplified 98.4%

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

7. Taylor expanded in u around -inf 98.6%

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

   1. mul-1-neg 98.6%

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

   2. unsub-neg 98.6%

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

   3. mul-1-neg 98.6%

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

   4. unsub-neg 98.6%

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 10: 47.6% 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. fma-def 97.0%

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

   2. associate-*r/ 97.1%

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

   3. *-rgt-identity 97.1%

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

   4. associate-*r/ 97.3%

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

   5. *-rgt-identity 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in u around 0 46.4%

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

5. Final simplification 46.4%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023336 
(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)))