UniformSampleCone, x

Percentage Accurate: 57.6% → 99.1%

Time: 5.2s

Alternatives: 14

Speedup: 12.1×

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

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

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (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: 57.6% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (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: 99.1% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} t_0 := ux - maxCos \cdot ux\\ \sin \left(\mathsf{fma}\left(-2 \cdot \pi, uy, 0.5 \cdot \pi\right)\right) \cdot \sqrt{\left(t\_0 - 0\right) \cdot \left(-\left(t\_0 - 2\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (let* ((t_0 (- ux (* maxCos ux))))
  (*
   (sin (fma (* -2.0 PI) uy (* 0.5 PI)))
   (sqrt (* (- t_0 0.0) (- (- t_0 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux - (maxCos * ux);
	return sinf(fmaf((-2.0f * ((float) M_PI)), uy, (0.5f * ((float) M_PI)))) * sqrtf(((t_0 - 0.0f) * -(t_0 - 2.0f)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux - Float32(maxCos * ux))
	return Float32(sin(fma(Float32(Float32(-2.0) * Float32(pi)), uy, Float32(Float32(0.5) * Float32(pi)))) * sqrt(Float32(Float32(t_0 - Float32(0.0)) * Float32(-Float32(t_0 - Float32(2.0))))))
end

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

3. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]
4. Step-by-step derivation

5. Applied rewrites 99.1%

   \[\leadsto \sin \left(\mathsf{fma}\left(-2 \cdot \pi, uy, 0.5 \cdot \pi\right)\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]
6. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

3. Applied rewrites 98.9%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) - -1\right)} \]
4. Step-by-step derivation

5. Applied rewrites 98.9%

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

7. Applied rewrites 99.0%

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

Alternative 3: 99.0% accurate, 1.1× speedup

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

Math

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

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (let* ((t_0 (- ux (* maxCos ux))))
  (* (cos (* (* uy 2.0) PI)) (sqrt (* t_0 (- 2.0 t_0))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

3. Applied rewrites 98.9%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) - -1\right)} \]
4. Step-by-step derivation

5. Applied rewrites 98.9%

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

7. Applied rewrites 99.0%

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

Alternative 4: 99.0% accurate, 1.1× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

3. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]
4. Step-by-step derivation

5. Applied rewrites 99.0%

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

Alternative 5: 97.7% accurate, 1.1× speedup

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

Math

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

FPCore

(FPCore (ux uy maxCos)
  :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 (* (- ux (* maxCos ux)) (- (- 1.0 ux) -1.0)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

3. Applied rewrites 98.9%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) - -1\right)} \]
4. Step-by-step derivation

5. Applied rewrites 98.9%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 97.7%

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

Alternative 6: 93.0% accurate, 1.2× speedup

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

Math

\[\sin \left(\mathsf{fma}\left(-2 \cdot \pi, uy, 0.5 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \]

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (* (sin (fma (* -2.0 PI) uy (* 0.5 PI))) (sqrt (* ux (- 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	return sinf(fmaf((-2.0f * ((float) M_PI)), uy, (0.5f * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(fma(Float32(Float32(-2.0) * Float32(pi)), uy, Float32(Float32(0.5) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))))
end

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

3. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]
4. Step-by-step derivation

5. Applied rewrites 99.1%

   \[\leadsto \sin \left(\mathsf{fma}\left(-2 \cdot \pi, uy, 0.5 \cdot \pi\right)\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \sin \left(\mathsf{fma}\left(-2 \cdot \pi, uy, 0.5 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \]
7. Step-by-step derivation

8. Applied rewrites 93.0%

   \[\leadsto \sin \left(\mathsf{fma}\left(-2 \cdot \pi, uy, 0.5 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \]
9. Add Preprocessing

Alternative 7: 92.8% accurate, 1.3× speedup

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

Math

\[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \]

FPCore

(FPCore (ux uy maxCos)
  :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 (* ux (- 2.0 ux)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

3. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

4. Taylor expanded in maxCos around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \]
5. Step-by-step derivation

6. Applied rewrites 92.8%

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

Alternative 8: 80.4% accurate, 2.4× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 49.5%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 80.3%

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

9. Applied rewrites 80.4%

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

Alternative 9: 80.3% accurate, 2.7× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 49.5%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 80.3%

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

9. Applied rewrites 80.3%

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

Alternative 10: 80.3% accurate, 3.4× speedup

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

Math

\[\sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)} \]

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (sqrt (* (- ux (* maxCos ux)) (- (+ 2.0 (* maxCos ux)) ux))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

3. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

4. Taylor expanded in uy around 0

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

6. Applied rewrites 80.3%

   \[\leadsto \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)} \]
7. Add Preprocessing

Alternative 11: 79.4% accurate, 4.0× speedup

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

Math

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

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (sqrt (* ux (- (+ 2.0 (* -1.0 ux)) (* 2.0 maxCos)))))

C

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

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * ((2.0e0 + ((-1.0e0) * ux)) - (2.0e0 * maxcos))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 49.5%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 80.3%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 79.4%

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

Alternative 12: 76.1% accurate, 5.9× speedup

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

Math

\[\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)} \]

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (sqrt (* ux (+ 2.0 (* -1.0 ux)))))

C

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

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (2.0e0 + ((-1.0e0) * ux))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 49.5%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 80.3%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 76.1%

    \[\leadsto \sqrt{ux \cdot \left(2 + -1 \cdot ux\right)} \]
11. Add Preprocessing

Alternative 13: 64.9% accurate, 6.2× speedup

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

Math

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

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (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 57.6%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 49.5%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 64.9%

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

9. Applied rewrites 64.9%

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

Alternative 14: 62.3% accurate, 12.1× speedup

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

Math

\[\sqrt{ux + ux} \]

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (sqrt (+ ux ux)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux + ux));
}

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux + ux))
end function

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux + ux));
end

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 49.5%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 64.9%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 62.3%

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

12. Applied rewrites 62.3%

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

Reproduce

herbie shell --seed 2026035 +o sampling:rival3
(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)))))))