UniformSampleCone, x

Percentage Accurate: 57.0% → 99.0%

Time: 14.1s

Alternatives: 24

Speedup: 9.8×

Specification

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot ux\right)\right) \cdot ux\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (fma
    (fma -2.0 maxCos 2.0)
    ux
    (* (* (- 1.0 maxCos) (* (- maxCos 1.0) ux)) ux)))
  (cos (* PI (* 2.0 uy)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf(fmaf(-2.0f, maxCos, 2.0f), ux, (((1.0f - maxCos) * ((maxCos - 1.0f) * ux)) * ux))) * cosf((((float) M_PI) * (2.0f * uy)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(fma(fma(Float32(-2.0), maxCos, Float32(2.0)), ux, Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(maxCos - Float32(1.0)) * ux)) * ux))) * cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

5. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]
6. Step-by-step derivation

7. Applied rewrites 99.1%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), \color{blue}{ux}, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

8. Final simplification 99.1%

   \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot ux\right)\right) \cdot ux\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \]
9. Add Preprocessing

Alternative 2: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot ux + \left(1 - ux\right)\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot maxCos, ux, ux \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* maxCos ux) (- 1.0 ux))))
   (if (<=
        (* (sqrt (- 1.0 (* t_0 t_0))) (cos (* PI (* 2.0 uy))))
        0.019999999552965164)
     (sqrt (fma (* -2.0 maxCos) ux (* ux 2.0)))
     (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (maxCos * ux) + (1.0f - ux);
	float tmp;
	if ((sqrtf((1.0f - (t_0 * t_0))) * cosf((((float) M_PI) * (2.0f * uy)))) <= 0.019999999552965164f) {
		tmp = sqrtf(fmaf((-2.0f * maxCos), ux, (ux * 2.0f)));
	} else {
		tmp = sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(maxCos * ux) + Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) * cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))) <= Float32(0.019999999552965164))
		tmp = sqrt(fma(Float32(Float32(-2.0) * maxCos), ux, Float32(ux * Float32(2.0))));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))) < 0.0199999996

   1. Initial program 38.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      2. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

      10. lower-fma.f32 31.6

          \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

   5. Applied rewrites 31.6%

      \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

   6. Taylor expanded in ux around 0

      \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 72.6%

      \[\leadsto \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.6%

       \[\leadsto \sqrt{\mathsf{fma}\left(-2 \cdot maxCos, ux, 2 \cdot ux\right)} \]

   if 0.0199999996 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))))

   1. Initial program 88.9%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      2. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

      10. lower-fma.f32 78.7

          \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.3%

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(maxCos \cdot ux + \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot maxCos, ux, ux \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot ux + \left(1 - ux\right)\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* maxCos ux) (- 1.0 ux))))
   (if (<=
        (* (sqrt (- 1.0 (* t_0 t_0))) (cos (* PI (* 2.0 uy))))
        0.019999999552965164)
     (sqrt (* (fma -2.0 maxCos 2.0) ux))
     (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (maxCos * ux) + (1.0f - ux);
	float tmp;
	if ((sqrtf((1.0f - (t_0 * t_0))) * cosf((((float) M_PI) * (2.0f * uy)))) <= 0.019999999552965164f) {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
	} else {
		tmp = sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(maxCos * ux) + Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) * cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))) <= Float32(0.019999999552965164))
		tmp = sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))) < 0.0199999996

   1. Initial program 38.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      2. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

      10. lower-fma.f32 31.6

          \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

   5. Applied rewrites 31.6%

      \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

   6. Taylor expanded in ux around 0

      \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 72.6%

      \[\leadsto \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \]

   if 0.0199999996 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))))

   1. Initial program 88.9%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      2. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

      10. lower-fma.f32 78.7

          \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.3%

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(maxCos \cdot ux + \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot ux\right), ux, \mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (fma
    (* (- 1.0 maxCos) (* (- maxCos 1.0) ux))
    ux
    (* (fma -2.0 maxCos 2.0) ux)))
  (cos (* PI (* 2.0 uy)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf(((1.0f - maxCos) * ((maxCos - 1.0f) * ux)), ux, (fmaf(-2.0f, maxCos, 2.0f) * ux))) * cosf((((float) M_PI) * (2.0f * uy)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(fma(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(maxCos - Float32(1.0)) * ux)), ux, Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux))) * cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

5. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]
6. Step-by-step derivation

7. Applied rewrites 99.1%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right), \color{blue}{ux}, \mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux\right)} \]

8. Final simplification 99.1%

   \[\leadsto \sqrt{\mathsf{fma}\left(\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot ux\right), ux, \mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \]
9. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\left(1 - maxCos\right) \cdot ux, maxCos - 1, \mathsf{fma}\left(-2, maxCos, 2\right)\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* (+ uy uy) PI))
  (sqrt
   (* (fma (* (- 1.0 maxCos) ux) (- maxCos 1.0) (fma -2.0 maxCos 2.0)) ux))))

C

float code(float ux, float uy, float maxCos) {
	return cosf(((uy + uy) * ((float) M_PI))) * sqrtf((fmaf(((1.0f - maxCos) * ux), (maxCos - 1.0f), fmaf(-2.0f, maxCos, 2.0f)) * ux));
}

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(Float32(uy + uy) * Float32(pi))) * sqrt(Float32(fma(Float32(Float32(Float32(1.0) - maxCos) * ux), Float32(maxCos - Float32(1.0)), fma(Float32(-2.0), maxCos, Float32(2.0))) * ux)))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

5. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]
6. Step-by-step derivation

   1. lift-cos.f32 N/A

      \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   2. lift-*.f32 N/A

      \[\leadsto \cos \color{blue}{\left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   3. lift-*.f32 N/A

      \[\leadsto \cos \left(\color{blue}{\left(uy \cdot 2\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   4. lift-PI.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   5. *-commutative N/A

      \[\leadsto \cos \left(\color{blue}{\left(2 \cdot uy\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   6. lift-PI.f32 N/A

      \[\leadsto \cos \left(\left(2 \cdot uy\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   7. associate-*l* N/A

      \[\leadsto \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   8. cos-2 N/A

      \[\leadsto \color{blue}{\left(\cos \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(uy \cdot \mathsf{PI}\left(\right)\right) - \sin \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   9. lower--.f32 N/A

      \[\leadsto \color{blue}{\left(\cos \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(uy \cdot \mathsf{PI}\left(\right)\right) - \sin \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   10. lower-*.f32 N/A

       \[\leadsto \left(\color{blue}{\cos \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(uy \cdot \mathsf{PI}\left(\right)\right)} - \sin \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   11. lower-cos.f32 N/A

       \[\leadsto \left(\color{blue}{\cos \left(uy \cdot \mathsf{PI}\left(\right)\right)} \cdot \cos \left(uy \cdot \mathsf{PI}\left(\right)\right) - \sin \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   12. lower-*.f32 N/A

       \[\leadsto \left(\cos \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)} \cdot \cos \left(uy \cdot \mathsf{PI}\left(\right)\right) - \sin \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   13. lower-cos.f32 N/A

       \[\leadsto \left(\cos \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\cos \left(uy \cdot \mathsf{PI}\left(\right)\right)} - \sin \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   14. lower-*.f32 N/A

       \[\leadsto \left(\cos \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)} - \sin \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   15. lower-*.f32 N/A

       \[\leadsto \left(\cos \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(uy \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\sin \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   16. lower-sin.f32 N/A

       \[\leadsto \left(\cos \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(uy \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\sin \left(uy \cdot \mathsf{PI}\left(\right)\right)} \cdot \sin \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   17. lower-*.f32 N/A

       \[\leadsto \left(\cos \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(uy \cdot \mathsf{PI}\left(\right)\right) - \sin \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)} \cdot \sin \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   18. lower-sin.f32 N/A

       \[\leadsto \left(\cos \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(uy \cdot \mathsf{PI}\left(\right)\right) - \sin \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sin \left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   19. lower-*.f32 98.9

       \[\leadsto \left(\cos \left(uy \cdot \pi\right) \cdot \cos \left(uy \cdot \pi\right) - \sin \left(uy \cdot \pi\right) \cdot \sin \color{blue}{\left(uy \cdot \pi\right)}\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

7. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\left(\cos \left(uy \cdot \pi\right) \cdot \cos \left(uy \cdot \pi\right) - \sin \left(uy \cdot \pi\right) \cdot \sin \left(uy \cdot \pi\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(\cos \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(uy \cdot \mathsf{PI}\left(\right)\right) - \sin \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \cdot \left(\cos \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(uy \cdot \mathsf{PI}\left(\right)\right) - \sin \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   3. lower-*.f32 98.9

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \cdot \left(\cos \left(uy \cdot \pi\right) \cdot \cos \left(uy \cdot \pi\right) - \sin \left(uy \cdot \pi\right) \cdot \sin \left(uy \cdot \pi\right)\right)} \]

9. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\left(1 - maxCos\right) \cdot ux, maxCos - 1, \mathsf{fma}\left(-2, maxCos, 2\right)\right) \cdot ux} \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)} \]

10. Final simplification 99.0%

    \[\leadsto \cos \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\left(1 - maxCos\right) \cdot ux, maxCos - 1, \mathsf{fma}\left(-2, maxCos, 2\right)\right) \cdot ux} \]
11. Add Preprocessing

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, 2, -2\right) \cdot ux, maxCos, \left(2 - ux\right) \cdot ux\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt (fma (* (fma ux 2.0 -2.0) ux) maxCos (* (- 2.0 ux) ux)))
  (cos (* PI (* 2.0 uy)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf((fmaf(ux, 2.0f, -2.0f) * ux), maxCos, ((2.0f - ux) * ux))) * cosf((((float) M_PI) * (2.0f * uy)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(fma(Float32(fma(ux, Float32(2.0), Float32(-2.0)) * ux), maxCos, Float32(Float32(Float32(2.0) - ux) * ux))) * cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

5. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{maxCos \cdot \left(ux \cdot \left(2 \cdot ux - 2\right)\right) + \color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]
7. Step-by-step derivation

8. Applied rewrites 98.4%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, 2, -2\right) \cdot ux, \color{blue}{maxCos}, \left(2 - ux\right) \cdot ux\right)} \]

9. Final simplification 98.4%

   \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, 2, -2\right) \cdot ux, maxCos, \left(2 - ux\right) \cdot ux\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \]
10. Add Preprocessing

Alternative 7: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.010499999858438969:\\ \;\;\;\;\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot ux\right)\right) \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.010499999858438969)
   (*
    (fma (* (* uy uy) -2.0) (* PI PI) 1.0)
    (sqrt
     (fma
      (fma -2.0 maxCos 2.0)
      ux
      (* (* (- 1.0 maxCos) (* (- maxCos 1.0) ux)) ux))))
   (* (sqrt (* (- 2.0 ux) ux)) (cos (* PI (* 2.0 uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.010499999858438969f) {
		tmp = fmaf(((uy * uy) * -2.0f), (((float) M_PI) * ((float) M_PI)), 1.0f) * sqrtf(fmaf(fmaf(-2.0f, maxCos, 2.0f), ux, (((1.0f - maxCos) * ((maxCos - 1.0f) * ux)) * ux)));
	} else {
		tmp = sqrtf(((2.0f - ux) * ux)) * cosf((((float) M_PI) * (2.0f * uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.010499999858438969))
		tmp = Float32(fma(Float32(Float32(uy * uy) * Float32(-2.0)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)) * sqrt(fma(fma(Float32(-2.0), maxCos, Float32(2.0)), ux, Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(maxCos - Float32(1.0)) * ux)) * ux))));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(2.0) - ux) * ux)) * cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0104999999

   1. Initial program 55.1%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

      2. lower-*.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), \color{blue}{ux}, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   8. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{uy}^{2} \cdot -2}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{uy}^{2} \cdot -2}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(uy \cdot uy\right)} \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

      7. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(uy \cdot uy\right)} \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

      10. lower-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

      11. lower-PI.f32 99.6

          \[\leadsto \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \color{blue}{\pi}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   if 0.0104999999 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 54.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

      2. lower-*.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   5. Applied rewrites 97.6%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -1 \cdot ux\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.3%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - ux\right) \cdot \color{blue}{ux}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.010499999858438969:\\ \;\;\;\;\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot ux\right)\right) \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(-ux\right) \cdot ux\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt (fma (fma -2.0 maxCos 2.0) ux (* (- ux) ux)))
  (cos (* PI (* 2.0 uy)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf(fmaf(-2.0f, maxCos, 2.0f), ux, (-ux * ux))) * cosf((((float) M_PI) * (2.0f * uy)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(fma(fma(Float32(-2.0), maxCos, Float32(2.0)), ux, Float32(Float32(-ux) * ux))) * cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

5. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]
6. Step-by-step derivation

7. Applied rewrites 99.1%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), \color{blue}{ux}, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

8. Taylor expanded in maxCos around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(-1 \cdot ux\right) \cdot ux\right)} \]
9. Step-by-step derivation

10. Applied rewrites 97.9%

    \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(-ux\right) \cdot ux\right)} \]

11. Final simplification 97.9%

    \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(-ux\right) \cdot ux\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \]
12. Add Preprocessing

Alternative 9: 75.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1\right) - ux\\ t_1 := maxCos \cdot ux + \left(1 - ux\right)\\ \mathbf{if}\;1 - t\_1 \cdot t\_1 \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot maxCos, ux, ux \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- (fma maxCos ux 1.0) ux)) (t_1 (+ (* maxCos ux) (- 1.0 ux))))
   (if (<= (- 1.0 (* t_1 t_1)) 0.00039999998989515007)
     (sqrt (fma (* -2.0 maxCos) ux (* ux 2.0)))
     (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = fmaf(maxCos, ux, 1.0f) - ux;
	float t_1 = (maxCos * ux) + (1.0f - ux);
	float tmp;
	if ((1.0f - (t_1 * t_1)) <= 0.00039999998989515007f) {
		tmp = sqrtf(fmaf((-2.0f * maxCos), ux, (ux * 2.0f)));
	} else {
		tmp = sqrtf((1.0f - (t_0 * t_0)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(fma(maxCos, ux, Float32(1.0)) - ux)
	t_1 = Float32(Float32(maxCos * ux) + Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(t_1 * t_1)) <= Float32(0.00039999998989515007))
		tmp = sqrt(fma(Float32(Float32(-2.0) * maxCos), ux, Float32(ux * Float32(2.0))));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))) < 3.9999999e-4

   1. Initial program 36.2%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      2. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

      10. lower-fma.f32 33.0

          \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

   5. Applied rewrites 33.0%

      \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

   6. Taylor expanded in ux around 0

      \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \]
   9. Step-by-step derivation

   10. Applied rewrites 76.5%

       \[\leadsto \sqrt{\mathsf{fma}\left(-2 \cdot maxCos, ux, 2 \cdot ux\right)} \]

   if 3.9999999e-4 < (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))

   1. Initial program 88.1%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      2. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

      10. lower-fma.f32 71.3

          \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(maxCos \cdot ux + \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right) \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot maxCos, ux, ux \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, -2, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot ux\right)\right) \cdot ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (fma (* (* (* uy uy) PI) PI) -2.0 1.0)
  (sqrt
   (fma
    (fma -2.0 maxCos 2.0)
    ux
    (* (* (- 1.0 maxCos) (* (- maxCos 1.0) ux)) ux)))))

C

float code(float ux, float uy, float maxCos) {
	return fmaf((((uy * uy) * ((float) M_PI)) * ((float) M_PI)), -2.0f, 1.0f) * sqrtf(fmaf(fmaf(-2.0f, maxCos, 2.0f), ux, (((1.0f - maxCos) * ((maxCos - 1.0f) * ux)) * ux)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(fma(Float32(Float32(Float32(uy * uy) * Float32(pi)) * Float32(pi)), Float32(-2.0), Float32(1.0)) * sqrt(fma(fma(Float32(-2.0), maxCos, Float32(2.0)), ux, Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(maxCos - Float32(1.0)) * ux)) * ux))))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

5. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]

6. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   2. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot {uy}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   9. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   10. lower-PI.f32 87.3

       \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \color{blue}{\pi}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

8. Applied rewrites 87.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
9. Step-by-step derivation

10. Applied rewrites 87.3%

    \[\leadsto \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, \color{blue}{-2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
11. Step-by-step derivation

12. Applied rewrites 87.4%

    \[\leadsto \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, -2, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), \color{blue}{ux}, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

13. Final simplification 87.4%

    \[\leadsto \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, -2, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot ux\right)\right) \cdot ux\right)} \]
14. Add Preprocessing

Alternative 11: 88.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot ux\right)\right) \cdot ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (fma (* (* uy uy) -2.0) (* PI PI) 1.0)
  (sqrt
   (fma
    (fma -2.0 maxCos 2.0)
    ux
    (* (* (- 1.0 maxCos) (* (- maxCos 1.0) ux)) ux)))))

C

float code(float ux, float uy, float maxCos) {
	return fmaf(((uy * uy) * -2.0f), (((float) M_PI) * ((float) M_PI)), 1.0f) * sqrtf(fmaf(fmaf(-2.0f, maxCos, 2.0f), ux, (((1.0f - maxCos) * ((maxCos - 1.0f) * ux)) * ux)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(fma(Float32(Float32(uy * uy) * Float32(-2.0)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)) * sqrt(fma(fma(Float32(-2.0), maxCos, Float32(2.0)), ux, Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(maxCos - Float32(1.0)) * ux)) * ux))))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

5. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]
6. Step-by-step derivation

7. Applied rewrites 99.1%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), \color{blue}{ux}, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

8. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   2. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{uy}^{2} \cdot -2}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{uy}^{2} \cdot -2}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(uy \cdot uy\right)} \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(uy \cdot uy\right)} \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   9. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   10. lower-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   11. lower-PI.f32 87.4

       \[\leadsto \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \color{blue}{\pi}, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

10. Applied rewrites 87.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

11. Final simplification 87.4%

    \[\leadsto \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot ux\right)\right) \cdot ux\right)} \]
12. Add Preprocessing

Alternative 12: 88.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(maxCos - 1, \left(\left(1 - maxCos\right) \cdot ux\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux\right)} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (fma
    (- maxCos 1.0)
    (* (* (- 1.0 maxCos) ux) ux)
    (* (fma maxCos -2.0 2.0) ux)))
  (fma (* (* uy uy) -2.0) (* PI PI) 1.0)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf((maxCos - 1.0f), (((1.0f - maxCos) * ux) * ux), (fmaf(maxCos, -2.0f, 2.0f) * ux))) * fmaf(((uy * uy) * -2.0f), (((float) M_PI) * ((float) M_PI)), 1.0f);
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(fma(Float32(maxCos - Float32(1.0)), Float32(Float32(Float32(Float32(1.0) - maxCos) * ux) * ux), Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) * ux))) * fma(Float32(Float32(uy * uy) * Float32(-2.0)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

5. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]

6. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   2. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot {uy}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   9. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   10. lower-PI.f32 87.3

       \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \color{blue}{\pi}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

8. Applied rewrites 87.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
9. Step-by-step derivation

10. Applied rewrites 87.4%

    \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(maxCos - 1, \color{blue}{\left(\left(1 - maxCos\right) \cdot ux\right) \cdot ux}, \mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux\right)} \]

11. Final simplification 87.4%

    \[\leadsto \sqrt{\mathsf{fma}\left(maxCos - 1, \left(\left(1 - maxCos\right) \cdot ux\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux\right)} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right) \]
12. Add Preprocessing

Alternative 13: 88.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \cdot \mathsf{fma}\left(\left(\pi \cdot uy\right) \cdot \left(\pi \cdot uy\right), -2, 1\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (* (fma (- 1.0 maxCos) (* (- maxCos 1.0) ux) (fma maxCos -2.0 2.0)) ux))
  (fma (* (* PI uy) (* PI uy)) -2.0 1.0)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((fmaf((1.0f - maxCos), ((maxCos - 1.0f) * ux), fmaf(maxCos, -2.0f, 2.0f)) * ux)) * fmaf(((((float) M_PI) * uy) * (((float) M_PI) * uy)), -2.0f, 1.0f);
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(fma(Float32(Float32(1.0) - maxCos), Float32(Float32(maxCos - Float32(1.0)) * ux), fma(maxCos, Float32(-2.0), Float32(2.0))) * ux)) * fma(Float32(Float32(Float32(pi) * uy) * Float32(Float32(pi) * uy)), Float32(-2.0), Float32(1.0)))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

5. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]

6. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   2. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot {uy}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   9. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   10. lower-PI.f32 87.3

       \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \color{blue}{\pi}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

8. Applied rewrites 87.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
9. Step-by-step derivation

10. Applied rewrites 87.3%

    \[\leadsto \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, \color{blue}{-2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
11. Step-by-step derivation

12. Applied rewrites 87.3%

    \[\leadsto \mathsf{fma}\left(\left(\pi \cdot uy\right) \cdot \left(\pi \cdot uy\right), -2, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

13. Final simplification 87.3%

    \[\leadsto \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \cdot \mathsf{fma}\left(\left(\pi \cdot uy\right) \cdot \left(\pi \cdot uy\right), -2, 1\right) \]
14. Add Preprocessing

Alternative 14: 88.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (* (fma (- 1.0 maxCos) (* (- maxCos 1.0) ux) (fma maxCos -2.0 2.0)) ux))
  (fma (* (* uy uy) -2.0) (* PI PI) 1.0)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((fmaf((1.0f - maxCos), ((maxCos - 1.0f) * ux), fmaf(maxCos, -2.0f, 2.0f)) * ux)) * fmaf(((uy * uy) * -2.0f), (((float) M_PI) * ((float) M_PI)), 1.0f);
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(fma(Float32(Float32(1.0) - maxCos), Float32(Float32(maxCos - Float32(1.0)) * ux), fma(maxCos, Float32(-2.0), Float32(2.0))) * ux)) * fma(Float32(Float32(uy * uy) * Float32(-2.0)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

5. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]

6. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   2. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot {uy}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   9. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   10. lower-PI.f32 87.3

       \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \color{blue}{\pi}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

8. Applied rewrites 87.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

9. Final simplification 87.3%

   \[\leadsto \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right) \]
10. Add Preprocessing

Alternative 15: 87.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, ux, -2\right) \cdot ux, maxCos, \left(2 - ux\right) \cdot ux\right)} \cdot \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, -2, 1\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt (fma (* (fma 2.0 ux -2.0) ux) maxCos (* (- 2.0 ux) ux)))
  (fma (* (* (* uy uy) PI) PI) -2.0 1.0)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf((fmaf(2.0f, ux, -2.0f) * ux), maxCos, ((2.0f - ux) * ux))) * fmaf((((uy * uy) * ((float) M_PI)) * ((float) M_PI)), -2.0f, 1.0f);
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(fma(Float32(fma(Float32(2.0), ux, Float32(-2.0)) * ux), maxCos, Float32(Float32(Float32(2.0) - ux) * ux))) * fma(Float32(Float32(Float32(uy * uy) * Float32(pi)) * Float32(pi)), Float32(-2.0), Float32(1.0)))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

5. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]

6. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   2. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot {uy}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   9. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   10. lower-PI.f32 87.3

       \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \color{blue}{\pi}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

8. Applied rewrites 87.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
9. Step-by-step derivation

10. Applied rewrites 87.3%

    \[\leadsto \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, \color{blue}{-2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

11. Taylor expanded in maxCos around 0

    \[\leadsto \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), -2, 1\right) \cdot \sqrt{maxCos \cdot \left(ux \cdot \left(2 \cdot ux - 2\right)\right) + \color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]
12. Step-by-step derivation

13. Applied rewrites 87.0%

    \[\leadsto \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, -2, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, ux, -2\right) \cdot ux, \color{blue}{maxCos}, \left(2 - ux\right) \cdot ux\right)} \]

14. Final simplification 87.0%

    \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, ux, -2\right) \cdot ux, maxCos, \left(2 - ux\right) \cdot ux\right)} \cdot \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, -2, 1\right) \]
15. Add Preprocessing

Alternative 16: 87.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, ux, -2\right) \cdot ux, maxCos, \left(2 - ux\right) \cdot ux\right)} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt (fma (* (fma 2.0 ux -2.0) ux) maxCos (* (- 2.0 ux) ux)))
  (fma (* (* uy uy) -2.0) (* PI PI) 1.0)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf((fmaf(2.0f, ux, -2.0f) * ux), maxCos, ((2.0f - ux) * ux))) * fmaf(((uy * uy) * -2.0f), (((float) M_PI) * ((float) M_PI)), 1.0f);
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(fma(Float32(fma(Float32(2.0), ux, Float32(-2.0)) * ux), maxCos, Float32(Float32(Float32(2.0) - ux) * ux))) * fma(Float32(Float32(uy * uy) * Float32(-2.0)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

5. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]

6. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   2. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot {uy}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   9. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   10. lower-PI.f32 87.3

       \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \color{blue}{\pi}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

8. Applied rewrites 87.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

9. Taylor expanded in maxCos around 0

   \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{maxCos \cdot \left(ux \cdot \left(2 \cdot ux - 2\right)\right) + \color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 87.0%

    \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, ux, -2\right) \cdot ux, \color{blue}{maxCos}, \left(2 - ux\right) \cdot ux\right)} \]

12. Final simplification 87.0%

    \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(2, ux, -2\right) \cdot ux, maxCos, \left(2 - ux\right) \cdot ux\right)} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right) \]
13. Add Preprocessing

Alternative 17: 86.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, -2, 1\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot ux\right)\right) \cdot ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 9.999999747378752e-6)
   (* (fma (* (* (* uy uy) PI) PI) -2.0 1.0) (sqrt (* (- 2.0 ux) ux)))
   (*
    1.0
    (sqrt
     (fma
      (fma -2.0 maxCos 2.0)
      ux
      (* (* (- 1.0 maxCos) (* (- maxCos 1.0) ux)) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 9.999999747378752e-6f) {
		tmp = fmaf((((uy * uy) * ((float) M_PI)) * ((float) M_PI)), -2.0f, 1.0f) * sqrtf(((2.0f - ux) * ux));
	} else {
		tmp = 1.0f * sqrtf(fmaf(fmaf(-2.0f, maxCos, 2.0f), ux, (((1.0f - maxCos) * ((maxCos - 1.0f) * ux)) * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(9.999999747378752e-6))
		tmp = Float32(fma(Float32(Float32(Float32(uy * uy) * Float32(pi)) * Float32(pi)), Float32(-2.0), Float32(1.0)) * sqrt(Float32(Float32(Float32(2.0) - ux) * ux)));
	else
		tmp = Float32(Float32(1.0) * sqrt(fma(fma(Float32(-2.0), maxCos, Float32(2.0)), ux, Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(maxCos - Float32(1.0)) * ux)) * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999975e-6

   1. Initial program 56.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

      2. lower-*.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   5. Applied rewrites 99.0%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]

   6. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      2. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot {uy}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      9. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      10. lower-PI.f32 87.0

          \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \color{blue}{\pi}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   8. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
   9. Step-by-step derivation

   10. Applied rewrites 87.0%

       \[\leadsto \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, \color{blue}{-2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   11. Taylor expanded in maxCos around 0

       \[\leadsto \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), -2, 1\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -1 \cdot ux\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 86.8%

       \[\leadsto \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, -2, 1\right) \cdot \sqrt{\left(2 - ux\right) \cdot \color{blue}{ux}} \]

   if 9.99999975e-6 < maxCos

   1. Initial program 48.5%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

      2. lower-*.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   5. Applied rewrites 99.4%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.5%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), \color{blue}{ux}, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]

   8. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{1} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 82.6%

       \[\leadsto \color{blue}{1} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, -2, 1\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-2, maxCos, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(\left(maxCos - 1\right) \cdot ux\right)\right) \cdot ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 75.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux, ux - \mathsf{fma}\left(maxCos, ux, 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot maxCos, ux, ux \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (+ (* maxCos ux) (- 1.0 ux)) 0.9998499751091003)
   (sqrt (fma (- (fma maxCos ux 1.0) ux) (- ux (fma maxCos ux 1.0)) 1.0))
   (sqrt (fma (* -2.0 maxCos) ux (* ux 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (((maxCos * ux) + (1.0f - ux)) <= 0.9998499751091003f) {
		tmp = sqrtf(fmaf((fmaf(maxCos, ux, 1.0f) - ux), (ux - fmaf(maxCos, ux, 1.0f)), 1.0f));
	} else {
		tmp = sqrtf(fmaf((-2.0f * maxCos), ux, (ux * 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(maxCos * ux) + Float32(Float32(1.0) - ux)) <= Float32(0.9998499751091003))
		tmp = sqrt(fma(Float32(fma(maxCos, ux, Float32(1.0)) - ux), Float32(ux - fma(maxCos, ux, Float32(1.0))), Float32(1.0)));
	else
		tmp = sqrt(fma(Float32(Float32(-2.0) * maxCos), ux, Float32(ux * Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999849975

   1. Initial program 87.5%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      2. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

      10. lower-fma.f32 71.2

          \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

   5. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 71.7%

      \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux, -\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right), 1\right)} \]

   if 0.999849975 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 35.5%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      2. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

      10. lower-fma.f32 32.3

          \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

   5. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

   6. Taylor expanded in ux around 0

      \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.6%

      \[\leadsto \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \]
   9. Step-by-step derivation

   10. Applied rewrites 76.6%

       \[\leadsto \sqrt{\mathsf{fma}\left(-2 \cdot maxCos, ux, 2 \cdot ux\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux, ux - \mathsf{fma}\left(maxCos, ux, 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot maxCos, ux, ux \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 86.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, -2, 1\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-ux, \left(1 - maxCos\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 9.999999747378752e-6)
   (* (fma (* (* (* uy uy) PI) PI) -2.0 1.0) (sqrt (* (- 2.0 ux) ux)))
   (sqrt
    (*
     (fma (- ux) (* (- 1.0 maxCos) (- 1.0 maxCos)) (fma -2.0 maxCos 2.0))
     ux))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 9.999999747378752e-6f) {
		tmp = fmaf((((uy * uy) * ((float) M_PI)) * ((float) M_PI)), -2.0f, 1.0f) * sqrtf(((2.0f - ux) * ux));
	} else {
		tmp = sqrtf((fmaf(-ux, ((1.0f - maxCos) * (1.0f - maxCos)), fmaf(-2.0f, maxCos, 2.0f)) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(9.999999747378752e-6))
		tmp = Float32(fma(Float32(Float32(Float32(uy * uy) * Float32(pi)) * Float32(pi)), Float32(-2.0), Float32(1.0)) * sqrt(Float32(Float32(Float32(2.0) - ux) * ux)));
	else
		tmp = sqrt(Float32(fma(Float32(-ux), Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(1.0) - maxCos)), fma(Float32(-2.0), maxCos, Float32(2.0))) * ux));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999975e-6

   1. Initial program 56.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

      2. lower-*.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   5. Applied rewrites 99.0%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]

   6. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      2. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot {uy}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      9. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      10. lower-PI.f32 87.0

          \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \color{blue}{\pi}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   8. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
   9. Step-by-step derivation

   10. Applied rewrites 87.0%

       \[\leadsto \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, \color{blue}{-2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   11. Taylor expanded in maxCos around 0

       \[\leadsto \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), -2, 1\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -1 \cdot ux\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 86.8%

       \[\leadsto \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, -2, 1\right) \cdot \sqrt{\left(2 - ux\right) \cdot \color{blue}{ux}} \]

   if 9.99999975e-6 < maxCos

   1. Initial program 48.5%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      2. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

      10. lower-fma.f32 41.4

          \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

   5. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

   6. Taylor expanded in ux around 0

      \[\leadsto \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 82.3%

      \[\leadsto \sqrt{\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right) \cdot ux} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot \pi\right) \cdot \pi, -2, 1\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-ux, \left(1 - maxCos\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 86.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-ux, \left(1 - maxCos\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 9.999999747378752e-6)
   (* (fma (* (* uy uy) -2.0) (* PI PI) 1.0) (sqrt (* (- 2.0 ux) ux)))
   (sqrt
    (*
     (fma (- ux) (* (- 1.0 maxCos) (- 1.0 maxCos)) (fma -2.0 maxCos 2.0))
     ux))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 9.999999747378752e-6f) {
		tmp = fmaf(((uy * uy) * -2.0f), (((float) M_PI) * ((float) M_PI)), 1.0f) * sqrtf(((2.0f - ux) * ux));
	} else {
		tmp = sqrtf((fmaf(-ux, ((1.0f - maxCos) * (1.0f - maxCos)), fmaf(-2.0f, maxCos, 2.0f)) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(9.999999747378752e-6))
		tmp = Float32(fma(Float32(Float32(uy * uy) * Float32(-2.0)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)) * sqrt(Float32(Float32(Float32(2.0) - ux) * ux)));
	else
		tmp = sqrt(Float32(fma(Float32(-ux), Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(1.0) - maxCos)), fma(Float32(-2.0), maxCos, Float32(2.0))) * ux));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999975e-6

   1. Initial program 56.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

      2. lower-*.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right) \cdot ux}} \]

   5. Applied rewrites 99.0%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux}} \]

   6. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      2. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot {uy}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      9. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

      10. lower-PI.f32 87.0

          \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \color{blue}{\pi}, 1\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   8. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)} \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, \left(maxCos - 1\right) \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right)\right) \cdot ux} \]

   9. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -1 \cdot ux\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 86.8%

       \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\left(2 - ux\right) \cdot \color{blue}{ux}} \]

   if 9.99999975e-6 < maxCos

   1. Initial program 48.5%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      2. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

      10. lower-fma.f32 41.4

          \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

   5. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

   6. Taylor expanded in ux around 0

      \[\leadsto \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 82.3%

      \[\leadsto \sqrt{\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right) \cdot ux} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-ux, \left(1 - maxCos\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 79.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(-ux, \left(1 - maxCos\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt
  (* (fma (- ux) (* (- 1.0 maxCos) (- 1.0 maxCos)) (fma -2.0 maxCos 2.0)) ux)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((fmaf(-ux, ((1.0f - maxCos) * (1.0f - maxCos)), fmaf(-2.0f, maxCos, 2.0f)) * ux));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(fma(Float32(-ux), Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(1.0) - maxCos)), fma(Float32(-2.0), maxCos, Float32(2.0))) * ux))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. lower--.f32 N/A

      \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   6. +-commutative N/A

      \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   7. lower-fma.f32 N/A

      \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   8. lower--.f32 N/A

      \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

   9. +-commutative N/A

      \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

   10. lower-fma.f32 46.9

       \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

5. Applied rewrites 46.9%

   \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

6. Taylor expanded in ux around 0

   \[\leadsto \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]
7. Step-by-step derivation

8. Applied rewrites 79.7%

   \[\leadsto \sqrt{\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right) \cdot ux} \]

9. Final simplification 79.7%

   \[\leadsto \sqrt{\mathsf{fma}\left(-ux, \left(1 - maxCos\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right) \cdot ux} \]
10. Add Preprocessing

Alternative 22: 64.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (* (fma -2.0 maxCos 2.0) ux)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux))
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. lower--.f32 N/A

      \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   6. +-commutative N/A

      \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   7. lower-fma.f32 N/A

      \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   8. lower--.f32 N/A

      \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

   9. +-commutative N/A

      \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

   10. lower-fma.f32 46.9

       \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

5. Applied rewrites 46.9%

   \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

6. Taylor expanded in ux around 0

   \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
7. Step-by-step derivation

8. Applied rewrites 65.3%

   \[\leadsto \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \]
9. Add Preprocessing

Alternative 23: 62.2% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot 2} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (* ux 2.0)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * 2.0f));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * 2.0e0))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(2.0)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * single(2.0)));
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. lower--.f32 N/A

      \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   6. +-commutative N/A

      \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   7. lower-fma.f32 N/A

      \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   8. lower--.f32 N/A

      \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

   9. +-commutative N/A

      \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

   10. lower-fma.f32 46.9

       \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

5. Applied rewrites 46.9%

   \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

6. Taylor expanded in ux around 0

   \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
7. Step-by-step derivation

8. Applied rewrites 65.3%

   \[\leadsto \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \]

9. Taylor expanded in maxCos around 0

   \[\leadsto \sqrt{2 \cdot ux} \]
10. Step-by-step derivation

11. Applied rewrites 62.2%

    \[\leadsto \sqrt{2 \cdot ux} \]

12. Final simplification 62.2%

    \[\leadsto \sqrt{ux \cdot 2} \]
13. Add Preprocessing

Alternative 24: 6.6% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 - 1} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (- 1.0 1.0)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((1.0f - 1.0f));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((1.0e0 - 1.0e0))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(Float32(1.0) - Float32(1.0)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((single(1.0) - single(1.0)));
end

Derivation

1. Initial program 55.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. lower--.f32 N/A

      \[\leadsto \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{1 - \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   6. +-commutative N/A

      \[\leadsto \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   7. lower-fma.f32 N/A

      \[\leadsto \sqrt{1 - \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   8. lower--.f32 N/A

      \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]

   9. +-commutative N/A

      \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)} \]

   10. lower-fma.f32 46.9

       \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1\right)} - ux\right)} \]

5. Applied rewrites 46.9%

   \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

6. Taylor expanded in ux around 0

   \[\leadsto \sqrt{1 - 1} \]
7. Step-by-step derivation

8. Applied rewrites 6.6%

   \[\leadsto \sqrt{1 - 1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))