Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.3% → 99.1%

Time: 10.3s

Alternatives: 7

Speedup: 27.0×

Specification

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\mathsf{PI}\left(\right)}{2}\right) \land \left(-1 \leq n0\_i \land n0\_i \leq 1\right)\right) \land \left(-1 \leq n1\_i \land n1\_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t\_0\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot t\_0\right) \cdot n1\_i \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t\_0\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot t\_0\right) \cdot n1\_i \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in u around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-cos.f32 N/A

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

   10. lower-sin.f32 98.9

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

8. Applied rewrites 98.9%

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

10. Applied rewrites 98.9%

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

12. Applied rewrites 98.9%

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

13. Final simplification 98.9%

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

Alternative 2: 98.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in u around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sin.f32 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in normAngle around 0

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

   1. lower--.f32 98.7

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

8. Applied rewrites 98.7%

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

9. Final simplification 98.7%

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

Alternative 3: 85.6% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-n0\_i, u, n1\_i \cdot u\right) + n0\_i\\ \mathbf{if}\;n1\_i \leq -1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n1\_i \leq 5.999999920033662 \cdot 10^{-24}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (+ (fma (- n0_i) u (* n1_i u)) n0_i)))
   (if (<= n1_i -1.9999999774532045e-26)
     t_0
     (if (<= n1_i 5.999999920033662e-24) (* (- 1.0 u) n0_i) t_0))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = fmaf(-n0_i, u, (n1_i * u)) + n0_i;
	float tmp;
	if (n1_i <= -1.9999999774532045e-26f) {
		tmp = t_0;
	} else if (n1_i <= 5.999999920033662e-24f) {
		tmp = (1.0f - u) * n0_i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(fma(Float32(-n0_i), u, Float32(n1_i * u)) + n0_i)
	tmp = Float32(0.0)
	if (n1_i <= Float32(-1.9999999774532045e-26))
		tmp = t_0;
	elseif (n1_i <= Float32(5.999999920033662e-24))
		tmp = Float32(Float32(Float32(1.0) - u) * n0_i);
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.99999998e-26 or 5.99999992e-24 < 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. Add Preprocessing

   3. Taylor expanded in normAngle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 53.1

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

   5. Applied rewrites 52.6%

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

   7. Applied rewrites 96.0%

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

   9. Applied rewrites 96.1%

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

   11. Applied rewrites 84.9%

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

   if -1.99999998e-26 < n1_i < 5.99999992e-24

   1. Initial program 97.9%

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

   3. Taylor expanded in normAngle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 12.5

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

   5. Applied rewrites 12.4%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 98.4%

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

   10. Taylor expanded in n0_i around -inf

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

   12. Applied rewrites 91.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(-n0\_i, u, n1\_i \cdot u\right) + n0\_i\\ \mathbf{elif}\;n1\_i \leq 5.999999920033662 \cdot 10^{-24}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-n0\_i, u, n1\_i \cdot u\right) + n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.2% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -4.000000014509975 \cdot 10^{-15}:\\ \;\;\;\;n1\_i \cdot u\\ \mathbf{elif}\;n1\_i \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\left(1 - u\right) \cdot n0\_i\\ \mathbf{else}:\\ \;\;\;\;n1\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n1_i -4.000000014509975e-15)
   (* n1_i u)
   (if (<= n1_i 5.000000229068525e-19) (* (- 1.0 u) n0_i) (* n1_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n1_i <= -4.000000014509975e-15f) {
		tmp = n1_i * u;
	} else if (n1_i <= 5.000000229068525e-19f) {
		tmp = (1.0f - u) * n0_i;
	} else {
		tmp = n1_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 <= (-4.000000014509975e-15)) then
        tmp = n1_i * u
    else if (n1_i <= 5.000000229068525e-19) then
        tmp = (1.0e0 - u) * n0_i
    else
        tmp = n1_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(-4.000000014509975e-15))
		tmp = Float32(n1_i * u);
	elseif (n1_i <= Float32(5.000000229068525e-19))
		tmp = Float32(Float32(Float32(1.0) - u) * n0_i);
	else
		tmp = Float32(n1_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(-4.000000014509975e-15))
		tmp = n1_i * u;
	elseif (n1_i <= single(5.000000229068525e-19))
		tmp = (single(1.0) - u) * n0_i;
	else
		tmp = n1_i * u;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -4.00000001e-15 or 5.00000023e-19 < n1_i

   1. Initial program 96.1%

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

   3. Taylor expanded in n0_i around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-sin.f32 63.6

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

   5. Applied rewrites 63.6%

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

   6. Taylor expanded in normAngle around 0

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

   8. Applied rewrites 63.1%

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

   if -4.00000001e-15 < n1_i < 5.00000023e-19

   1. Initial program 97.3%

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

   3. Taylor expanded in normAngle around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 20.0

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

   5. Applied rewrites 19.5%

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

   7. Applied rewrites 97.6%

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

   9. Applied rewrites 97.8%

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

   10. Taylor expanded in n0_i around -inf

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

   12. Applied rewrites 79.8%

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

Alternative 5: 97.9% accurate, 27.0× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n0_i - (n0_i * u)) + (n1_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 = (n0_i - (n0_i * u)) + (n1_i * u)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in normAngle around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 38.2

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

5. Applied rewrites 37.9%

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

7. Applied rewrites 96.9%

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

9. Applied rewrites 96.9%

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

11. Applied rewrites 96.9%

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

12. Final simplification 96.9%

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

Alternative 6: 97.7% accurate, 27.0× speedup

Math

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

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((1.0f - u) * n0_i) + (n1_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 = ((1.0e0 - u) * n0_i) + (n1_i * u)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in normAngle around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 38.2

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

5. Applied rewrites 37.9%

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

7. Applied rewrites 96.9%

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

8. Final simplification 96.9%

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

Alternative 7: 38.4% accurate, 76.5× speedup

Math

\[\begin{array}{l} \\ n1\_i \cdot u \end{array} \]

FPCore

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

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n1_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 = n1_i * u
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in n0_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-sin.f32 38.6

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

5. Applied rewrites 38.6%

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

6. Taylor expanded in normAngle around 0

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

8. Applied rewrites 38.2%

   \[\leadsto n1\_i \cdot \color{blue}{u} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024285 
(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)))