Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.2% → 98.7%

Time: 19.5s

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: 98.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left(n0\_i - n0\_i \cdot u\right) + \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.5 \cdot \left(n0\_i \cdot u\right) + n0\_i \cdot 0.5\right) + -0.16666666666666666 \cdot \left(n0\_i - n1\_i\right)\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*l* 81.9%

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

   3. *-commutative 81.9%

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

   4. associate-*l* 74.5%

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

   5. distribute-lft-out 74.5%

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

3. Simplified 74.5%

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

5. Taylor expanded in normAngle around 0 99.0%

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right)\right)\right)} \]

6. Taylor expanded in u around 0 98.9%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(\left(-0.5 \cdot \left(n0\_i \cdot u\right) + 0.5 \cdot n0\_i\right) - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)}\right) \]

7. Taylor expanded in u around 0 99.2%

   \[\leadsto \color{blue}{\left(n0\_i + -1 \cdot \left(n0\_i \cdot u\right)\right)} + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.5 \cdot \left(n0\_i \cdot u\right) + 0.5 \cdot n0\_i\right) - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. associate-*r* 99.2%

      \[\leadsto \left(n0\_i + \color{blue}{\left(-1 \cdot n0\_i\right) \cdot u}\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.5 \cdot \left(n0\_i \cdot u\right) + 0.5 \cdot n0\_i\right) - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

   2. neg-mul-1 99.2%

      \[\leadsto \left(n0\_i + \color{blue}{\left(-n0\_i\right)} \cdot u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.5 \cdot \left(n0\_i \cdot u\right) + 0.5 \cdot n0\_i\right) - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

9. Simplified 99.2%

   \[\leadsto \color{blue}{\left(n0\_i + \left(-n0\_i\right) \cdot u\right)} + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.5 \cdot \left(n0\_i \cdot u\right) + 0.5 \cdot n0\_i\right) - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

10. Final simplification 99.2%

    \[\leadsto \left(n0\_i - n0\_i \cdot u\right) + \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.5 \cdot \left(n0\_i \cdot u\right) + n0\_i \cdot 0.5\right) + -0.16666666666666666 \cdot \left(n0\_i - n1\_i\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 2: 98.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.5 \cdot \left(n0\_i \cdot u\right) + n0\_i \cdot 0.5\right) + -0.16666666666666666 \cdot \left(n0\_i - n1\_i\right)\right)\right)\right) + n0\_i \cdot \left(1 - u\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (+
   (* u n1_i)
   (*
    (pow normAngle 2.0)
    (*
     u
     (+
      (+ (* -0.5 (* n0_i u)) (* n0_i 0.5))
      (* -0.16666666666666666 (- n0_i n1_i))))))
  (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((u * n1_i) + (powf(normAngle, 2.0f) * (u * (((-0.5f * (n0_i * u)) + (n0_i * 0.5f)) + (-0.16666666666666666f * (n0_i - n1_i)))))) + (n0_i * (1.0f - u));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = ((u * n1_i) + ((normangle ** 2.0e0) * (u * ((((-0.5e0) * (n0_i * u)) + (n0_i * 0.5e0)) + ((-0.16666666666666666e0) * (n0_i - n1_i)))))) + (n0_i * (1.0e0 - u))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(Float32(u * n1_i) + Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(Float32(Float32(Float32(-0.5) * Float32(n0_i * u)) + Float32(n0_i * Float32(0.5))) + Float32(Float32(-0.16666666666666666) * Float32(n0_i - n1_i)))))) + Float32(n0_i * Float32(Float32(1.0) - u)))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = ((u * n1_i) + ((normAngle ^ single(2.0)) * (u * (((single(-0.5) * (n0_i * u)) + (n0_i * single(0.5))) + (single(-0.16666666666666666) * (n0_i - n1_i)))))) + (n0_i * (single(1.0) - u));
end

Derivation

1. Initial program 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*l* 81.9%

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

   3. *-commutative 81.9%

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

   4. associate-*l* 74.5%

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

   5. distribute-lft-out 74.5%

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

3. Simplified 74.5%

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

5. Taylor expanded in normAngle around 0 99.0%

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right)\right)\right)} \]

6. Taylor expanded in u around 0 98.9%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(\left(-0.5 \cdot \left(n0\_i \cdot u\right) + 0.5 \cdot n0\_i\right) - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)}\right) \]

7. Final simplification 98.9%

   \[\leadsto \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.5 \cdot \left(n0\_i \cdot u\right) + n0\_i \cdot 0.5\right) + -0.16666666666666666 \cdot \left(n0\_i - n1\_i\right)\right)\right)\right) + n0\_i \cdot \left(1 - u\right) \]
8. Add Preprocessing

Alternative 3: 98.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ n0\_i \cdot \left(1 - u\right) + \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(u \cdot \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* n0_i (- 1.0 u))
  (+
   (* u n1_i)
   (*
    (pow normAngle 2.0)
    (* u (+ (* n1_i 0.16666666666666666) (* n0_i 0.3333333333333333)))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n0_i * (1.0f - u)) + ((u * n1_i) + (powf(normAngle, 2.0f) * (u * ((n1_i * 0.16666666666666666f) + (n0_i * 0.3333333333333333f)))));
}

Fortran

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = (n0_i * (1.0e0 - u)) + ((u * n1_i) + ((normangle ** 2.0e0) * (u * ((n1_i * 0.16666666666666666e0) + (n0_i * 0.3333333333333333e0)))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(n0_i * Float32(Float32(1.0) - u)) + Float32(Float32(u * n1_i) + Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(Float32(n1_i * Float32(0.16666666666666666)) + Float32(n0_i * Float32(0.3333333333333333)))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n0_i * (single(1.0) - u)) + ((u * n1_i) + ((normAngle ^ single(2.0)) * (u * ((n1_i * single(0.16666666666666666)) + (n0_i * single(0.3333333333333333))))));
end

Derivation

1. Initial program 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*l* 81.9%

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

   3. *-commutative 81.9%

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

   4. associate-*l* 74.5%

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

   5. distribute-lft-out 74.5%

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

3. Simplified 74.5%

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

5. Taylor expanded in normAngle around 0 99.0%

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right)\right)\right)} \]

6. Taylor expanded in u around 0 99.0%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(\left(0.5 \cdot n0\_i + u \cdot \left(-0.5 \cdot n0\_i + u \cdot \left(-0.16666666666666666 \cdot n1\_i + 0.16666666666666666 \cdot n0\_i\right)\right)\right) - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)}\right) \]

7. Taylor expanded in u around 0 98.9%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \color{blue}{\left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)}\right)\right) \]
8. Step-by-step derivation

   1. +-commutative 98.9%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \color{blue}{\left(-1 \cdot n0\_i + n1\_i\right)}\right)\right)\right) \]

   2. distribute-lft-in 98.9%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(0.5 \cdot n0\_i - \color{blue}{\left(-0.16666666666666666 \cdot \left(-1 \cdot n0\_i\right) + -0.16666666666666666 \cdot n1\_i\right)}\right)\right)\right) \]

   3. associate-*r* 98.9%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(0.5 \cdot n0\_i - \left(\color{blue}{\left(-0.16666666666666666 \cdot -1\right) \cdot n0\_i} + -0.16666666666666666 \cdot n1\_i\right)\right)\right)\right) \]

   4. metadata-eval 98.9%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(0.5 \cdot n0\_i - \left(\color{blue}{0.16666666666666666} \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)\right)\right)\right) \]

   5. +-commutative 98.9%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(0.5 \cdot n0\_i - \color{blue}{\left(-0.16666666666666666 \cdot n1\_i + 0.16666666666666666 \cdot n0\_i\right)}\right)\right)\right) \]

   6. associate--r+ 98.9%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \color{blue}{\left(\left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot n1\_i\right) - 0.16666666666666666 \cdot n0\_i\right)}\right)\right) \]

   7. fma-neg 98.9%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\color{blue}{\mathsf{fma}\left(0.5, n0\_i, --0.16666666666666666 \cdot n1\_i\right)} - 0.16666666666666666 \cdot n0\_i\right)\right)\right) \]

   8. distribute-lft-neg-in 98.9%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\mathsf{fma}\left(0.5, n0\_i, \color{blue}{\left(--0.16666666666666666\right) \cdot n1\_i}\right) - 0.16666666666666666 \cdot n0\_i\right)\right)\right) \]

   9. metadata-eval 98.9%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\mathsf{fma}\left(0.5, n0\_i, \color{blue}{0.16666666666666666} \cdot n1\_i\right) - 0.16666666666666666 \cdot n0\_i\right)\right)\right) \]

   10. fma-define 98.9%

       \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\color{blue}{\left(0.5 \cdot n0\_i + 0.16666666666666666 \cdot n1\_i\right)} - 0.16666666666666666 \cdot n0\_i\right)\right)\right) \]

   11. +-commutative 98.9%

       \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\color{blue}{\left(0.16666666666666666 \cdot n1\_i + 0.5 \cdot n0\_i\right)} - 0.16666666666666666 \cdot n0\_i\right)\right)\right) \]

   12. associate--l+ 98.9%

       \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \color{blue}{\left(0.16666666666666666 \cdot n1\_i + \left(0.5 \cdot n0\_i - 0.16666666666666666 \cdot n0\_i\right)\right)}\right)\right) \]

   13. distribute-rgt-out-- 98.9%

       \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(0.16666666666666666 \cdot n1\_i + \color{blue}{n0\_i \cdot \left(0.5 - 0.16666666666666666\right)}\right)\right)\right) \]

   14. metadata-eval 98.9%

       \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(0.16666666666666666 \cdot n1\_i + n0\_i \cdot \color{blue}{0.3333333333333333}\right)\right)\right) \]

9. Simplified 98.9%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \color{blue}{\left(0.16666666666666666 \cdot n1\_i + n0\_i \cdot 0.3333333333333333\right)}\right)\right) \]

10. Final simplification 98.9%

    \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(u \cdot \left(n1\_i \cdot 0.16666666666666666 + n0\_i \cdot 0.3333333333333333\right)\right)\right) \]
11. Add Preprocessing

Alternative 4: 98.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ n0\_i \cdot \left(1 - u\right) + \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(u \cdot \left(n1\_i \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*l* 81.9%

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

   3. *-commutative 81.9%

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

   4. associate-*l* 74.5%

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

   5. distribute-lft-out 74.5%

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

3. Simplified 74.5%

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

5. Taylor expanded in normAngle around 0 99.0%

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right)\right)\right)} \]

6. Taylor expanded in u around 0 98.9%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(\left(-0.5 \cdot \left(n0\_i \cdot u\right) + 0.5 \cdot n0\_i\right) - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)}\right) \]

7. Taylor expanded in n0_i around 0 98.6%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \color{blue}{\left(0.16666666666666666 \cdot n1\_i\right)}\right)\right) \]

8. Final simplification 98.6%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(u \cdot \left(n1\_i \cdot 0.16666666666666666\right)\right)\right) \]
9. Add Preprocessing

Alternative 5: 70.0% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -5.0000000843119176 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 2.999999901276418 \cdot 10^{-11}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -5.0000000843119176e-17)
         (not (<= n1_i 2.999999901276418e-11)))
   (* 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 <= -5.0000000843119176e-17f) || !(n1_i <= 2.999999901276418e-11f)) {
		tmp = 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 <= (-5.0000000843119176e-17)) .or. (.not. (n1_i <= 2.999999901276418e-11))) then
        tmp = 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(-5.0000000843119176e-17)) || !(n1_i <= Float32(2.999999901276418e-11)))
		tmp = 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(-5.0000000843119176e-17)) || ~((n1_i <= single(2.999999901276418e-11))))
		tmp = 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 < -5.00000008e-17 or 2.9999999e-11 < n1_i

   1. Initial program 95.5%

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

      1. fma-define 95.6%

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

      2. associate-*r/ 95.7%

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

      3. *-rgt-identity 95.7%

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

      4. associate-*r/ 96.0%

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

      5. *-rgt-identity 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in n0_i around 0 54.7%

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

      1. associate-/l* 65.2%

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

   7. Simplified 65.2%

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

   8. Taylor expanded in normAngle around 0 66.0%

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

   if -5.00000008e-17 < n1_i < 2.9999999e-11

   1. Initial program 97.8%

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

      1. fma-define 98.0%

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

      2. associate-*r/ 98.1%

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

      3. *-rgt-identity 98.1%

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

      4. associate-*r/ 98.2%

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

      5. *-rgt-identity 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in normAngle around 0 98.3%

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

      1. fma-define 98.5%

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

      2. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in n0_i around inf 76.5%

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

   9. Taylor expanded in u around 0 76.8%

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

       1. associate-*r* 99.3%

          \[\leadsto \left(n0\_i + \color{blue}{\left(-1 \cdot n0\_i\right) \cdot u}\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.5 \cdot \left(n0\_i \cdot u\right) + 0.5 \cdot n0\_i\right) - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

       2. neg-mul-1 99.3%

          \[\leadsto \left(n0\_i + \color{blue}{\left(-n0\_i\right)} \cdot u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\left(-0.5 \cdot \left(n0\_i \cdot u\right) + 0.5 \cdot n0\_i\right) - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

   11. Simplified 76.8%

       \[\leadsto \color{blue}{n0\_i + \left(-n0\_i\right) \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -5.0000000843119176 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 2.999999901276418 \cdot 10^{-11}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.8% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -5.0000000843119176 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 2.999999901276418 \cdot 10^{-11}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -5.0000000843119176e-17)
         (not (<= n1_i 2.999999901276418e-11)))
   (* 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 <= -5.0000000843119176e-17f) || !(n1_i <= 2.999999901276418e-11f)) {
		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 <= (-5.0000000843119176e-17)) .or. (.not. (n1_i <= 2.999999901276418e-11))) 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(-5.0000000843119176e-17)) || !(n1_i <= Float32(2.999999901276418e-11)))
		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(-5.0000000843119176e-17)) || ~((n1_i <= single(2.999999901276418e-11))))
		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 < -5.00000008e-17 or 2.9999999e-11 < n1_i

   1. Initial program 95.5%

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

      1. fma-define 95.6%

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

      2. associate-*r/ 95.7%

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

      3. *-rgt-identity 95.7%

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

      4. associate-*r/ 96.0%

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

      5. *-rgt-identity 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in n0_i around 0 54.7%

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

      1. associate-/l* 65.2%

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

   7. Simplified 65.2%

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

   8. Taylor expanded in normAngle around 0 66.0%

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

   if -5.00000008e-17 < n1_i < 2.9999999e-11

   1. Initial program 97.8%

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

      1. fma-define 98.0%

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

      2. associate-*r/ 98.1%

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

      3. *-rgt-identity 98.1%

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

      4. associate-*r/ 98.2%

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

      5. *-rgt-identity 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in normAngle around 0 98.3%

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

      1. fma-define 98.5%

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

      2. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in n0_i around inf 76.5%

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -5.0000000843119176 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 2.999999901276418 \cdot 10^{-11}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.1% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -5.0000000843119176 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 5.499999993481436 \cdot 10^{-18}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -5.0000000843119176e-17)
         (not (<= n1_i 5.499999993481436e-18)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -5.0000000843119176e-17f) || !(n1_i <= 5.499999993481436e-18f)) {
		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 <= (-5.0000000843119176e-17)) .or. (.not. (n1_i <= 5.499999993481436e-18))) 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(-5.0000000843119176e-17)) || !(n1_i <= Float32(5.499999993481436e-18)))
		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(-5.0000000843119176e-17)) || ~((n1_i <= single(5.499999993481436e-18))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -5.00000008e-17 or 5.49999999e-18 < n1_i

   1. Initial program 96.2%

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

      1. fma-define 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/ 96.8%

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

      5. *-rgt-identity 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in n0_i around 0 49.9%

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

      1. associate-/l* 61.2%

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

   7. Simplified 61.2%

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

   8. Taylor expanded in normAngle around 0 61.9%

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

   if -5.00000008e-17 < n1_i < 5.49999999e-18

   1. Initial program 97.6%

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

      1. fma-define 97.8%

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

      2. associate-*r/ 98.0%

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

      3. *-rgt-identity 98.0%

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

      4. associate-*r/ 98.0%

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

      5. *-rgt-identity 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in u around 0 63.0%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -5.0000000843119176 \cdot 10^{-17} \lor \neg \left(n1\_i \leq 5.499999993481436 \cdot 10^{-18}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.3% accurate, 46.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.0%

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

   1. fma-define 97.1%

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

   2. associate-*r/ 97.2%

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

   3. *-rgt-identity 97.2%

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

   4. associate-*r/ 97.4%

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

   5. *-rgt-identity 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in normAngle around 0 97.8%

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

   1. fma-define 97.9%

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

   2. *-commutative 97.9%

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

7. Simplified 97.9%

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

8. Taylor expanded in u around inf 97.6%

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

   1. +-commutative 97.6%

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

   2. +-commutative 97.6%

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

   3. associate-+l+ 97.8%

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

   4. +-commutative 97.8%

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

   5. neg-mul-1 97.8%

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

   6. unsub-neg 97.8%

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

10. Simplified 97.8%

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

11. Final simplification 97.8%

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

Alternative 9: 97.8% 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 97.0%

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

   1. fma-define 97.1%

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

   2. associate-*r/ 97.2%

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

   3. *-rgt-identity 97.2%

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

   4. associate-*r/ 97.4%

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

   5. *-rgt-identity 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in normAngle around 0 97.8%

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

   1. fma-define 97.9%

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

   2. *-commutative 97.9%

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

7. Simplified 97.9%

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

8. Taylor expanded in u around 0 98.2%

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

9. Final simplification 98.2%

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

Alternative 10: 46.5% accurate, 421.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.0%

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

   1. fma-define 97.1%

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

   2. associate-*r/ 97.2%

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

   3. *-rgt-identity 97.2%

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

   4. associate-*r/ 97.4%

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

   5. *-rgt-identity 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in u around 0 45.9%

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

Reproduce

herbie shell --seed 2024085 
(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)))