UniformSampleCone, x

Percentage Accurate: 57.4% → 99.1%

Time: 5.9s

Alternatives: 13

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: 57.4% 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, 0.9× 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

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

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 (- uy) (+ PI PI) (* 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(-uy, (((float) M_PI) + ((float) M_PI)), (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(-uy), Float32(Float32(pi) + Float32(pi)), 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 57.4%

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

3. Applied rewrites 98.9%

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

5. Applied rewrites 99.1%

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

Alternative 2: 99.1% 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

\[\sin \left(\mathsf{fma}\left(-uy, \pi + \pi, 0.5 \cdot \pi\right)\right) \cdot \sqrt{\left(\left(\mathsf{fma}\left(maxCos, ux, 1\right) + 1\right) - ux\right) \cdot \left(ux - maxCos \cdot ux\right)} \]

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)))
  (*
 (sin (fma (- uy) (+ PI PI) (* 0.5 PI)))
 (sqrt (* (- (+ (fma maxCos ux 1.0) 1.0) ux) (- ux (* maxCos ux))))))

C

float code(float ux, float uy, float maxCos) {
	return sinf(fmaf(-uy, (((float) M_PI) + ((float) M_PI)), (0.5f * ((float) M_PI)))) * sqrtf((((fmaf(maxCos, ux, 1.0f) + 1.0f) - ux) * (ux - (maxCos * ux))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(fma(Float32(-uy), Float32(Float32(pi) + Float32(pi)), Float32(Float32(0.5) * Float32(pi)))) * sqrt(Float32(Float32(Float32(fma(maxCos, ux, Float32(1.0)) + Float32(1.0)) - ux) * Float32(ux - Float32(maxCos * ux)))))
end

Derivation

1. Initial program 57.4%

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

3. Applied rewrites 98.9%

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

5. Applied rewrites 99.1%

   \[\leadsto \sin \left(\mathsf{fma}\left(-uy, \pi + \pi, 0.5 \cdot \pi\right)\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.1%

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

9. Applied rewrites 99.1%

   \[\leadsto \sin \left(\mathsf{fma}\left(-uy, \pi + \pi, 0.5 \cdot \pi\right)\right) \cdot \sqrt{\left(\left(\mathsf{fma}\left(maxCos, ux, 1\right) + 1\right) - ux\right) \cdot \left(ux - maxCos \cdot ux\right)} \]
10. Add Preprocessing

Alternative 3: 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

\[\sin \left(\mathsf{fma}\left(-uy, \pi + \pi, 1.5707963705062866\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(maxCos, ux, 2\right) - ux\right) \cdot \left(ux - maxCos \cdot ux\right)} \]

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)))
  (*
 (sin (fma (- uy) (+ PI PI) 1.5707963705062866))
 (sqrt (* (- (fma maxCos ux 2.0) ux) (- ux (* maxCos ux))))))

C

float code(float ux, float uy, float maxCos) {
	return sinf(fmaf(-uy, (((float) M_PI) + ((float) M_PI)), 1.5707963705062866f)) * sqrtf(((fmaf(maxCos, ux, 2.0f) - ux) * (ux - (maxCos * ux))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(fma(Float32(-uy), Float32(Float32(pi) + Float32(pi)), Float32(1.5707963705062866))) * sqrt(Float32(Float32(fma(maxCos, ux, Float32(2.0)) - ux) * Float32(ux - Float32(maxCos * ux)))))
end

Derivation

1. Initial program 57.4%

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

3. Applied rewrites 98.9%

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

5. Applied rewrites 99.1%

   \[\leadsto \sin \left(\mathsf{fma}\left(-uy, \pi + \pi, 0.5 \cdot \pi\right)\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.1%

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

8. Evaluated real constant 99.1%

   \[\leadsto \sin \left(\mathsf{fma}\left(-uy, \pi + \pi, 1.5707963705062866\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(maxCos, ux, 2\right) - ux\right) \cdot \left(ux - maxCos \cdot ux\right)} \]
9. Add Preprocessing

Alternative 4: 98.9% 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

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

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 (* (+ uy uy) PI))))

C

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

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(uy + uy) * Float32(pi))))
end

Derivation

1. Initial program 57.4%

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

3. Applied rewrites 98.9%

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

5. Applied rewrites 98.9%

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

Alternative 5: 97.5% 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

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

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 (* maxCos ux)) (- 2.0 ux)))))

C

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

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(2.0) - ux))))
end

MATLAB

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

Derivation

1. Initial program 57.4%

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

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

8. Applied rewrites 97.5%

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

Alternative 6: 96.3% 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

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(-2, uy \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

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)))
  (if (<= uy 0.0001500000071246177)
  (sqrt
   (*
    ux
    (-
     (+ 2.0 (* -1.0 (* ux (pow (- maxCos 1.0) 2.0))))
     (* 2.0 maxCos))))
  (* (sin (fma -2.0 (* uy PI) (* 0.5 PI))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0001500000071246177f) {
		tmp = sqrtf((ux * ((2.0f + (-1.0f * (ux * powf((maxCos - 1.0f), 2.0f)))) - (2.0f * maxCos))));
	} else {
		tmp = sinf(fmaf(-2.0f, (uy * ((float) M_PI)), (0.5f * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0001500000071246177))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(Float32(-1.0) * Float32(ux * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0))))) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = Float32(sin(fma(Float32(-2.0), Float32(uy * Float32(pi)), Float32(Float32(0.5) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 1.50000007e-4

   1. Initial program 57.4%

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

   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 49.1%

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

   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 80.0%

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

   if 1.50000007e-4 < uy

   1. Initial program 57.4%

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

   3. Applied rewrites 98.9%

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 92.5%

      \[\leadsto \sin \left(\mathsf{fma}\left(-2, uy \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 96.2% 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

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

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)))
  (if (<= uy 0.0001500000071246177)
  (sqrt
   (*
    ux
    (-
     (+ 2.0 (* -1.0 (* ux (pow (- maxCos 1.0) 2.0))))
     (* 2.0 maxCos))))
  (* (cos (* (* uy 2.0) PI)) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0001500000071246177f) {
		tmp = sqrtf((ux * ((2.0f + (-1.0f * (ux * powf((maxCos - 1.0f), 2.0f)))) - (2.0f * maxCos))));
	} else {
		tmp = cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0001500000071246177))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(Float32(-1.0) * Float32(ux * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0))))) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0001500000071246177))
		tmp = sqrt((ux * ((single(2.0) + (single(-1.0) * (ux * ((maxCos - single(1.0)) ^ single(2.0))))) - (single(2.0) * maxCos))));
	else
		tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt((ux * (single(2.0) - ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 1.50000007e-4

   1. Initial program 57.4%

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

   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 49.1%

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

   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 80.0%

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

   if 1.50000007e-4 < uy

   1. Initial program 57.4%

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

   3. Applied rewrites 98.9%

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

   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 92.4%

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

Alternative 8: 80.0% accurate, 1.7× 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

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

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 (pow (- maxCos 1.0) 2.0))))
   (* 2.0 maxCos)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * ((2.0f + (-1.0f * (ux * powf((maxCos - 1.0f), 2.0f)))) - (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 * ((maxcos - 1.0e0) ** 2.0e0)))) - (2.0e0 * maxcos))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

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 49.1%

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

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 80.0%

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

Alternative 9: 79.9% 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 57.4%

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in ux around 0

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

6. Applied rewrites 76.8%

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

7. Taylor expanded in uy around 0

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

9. Applied rewrites 79.9%

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

Alternative 10: 74.2% accurate, 1.7× 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

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9997000098228455:\\ \;\;\;\;\sqrt{\left(1 - \left(1 + -2 \cdot ux\right)\right) - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2, ux \cdot \left(-2 \cdot maxCos\right)\right)}\\ \end{array} \]

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 (+ (- 1.0 ux) (* ux maxCos))))
  (if (<= (* t_0 t_0) 0.9997000098228455)
    (sqrt (- (- 1.0 (+ 1.0 (* -2.0 ux))) (* ux ux)))
    (sqrt (fma ux 2.0 (* ux (* -2.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.9997000098228455f) {
		tmp = sqrtf(((1.0f - (1.0f + (-2.0f * ux))) - (ux * ux)));
	} else {
		tmp = sqrtf(fmaf(ux, 2.0f, (ux * (-2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_0) <= Float32(0.9997000098228455))
		tmp = sqrt(Float32(Float32(Float32(1.0) - Float32(Float32(1.0) + Float32(Float32(-2.0) * ux))) - Float32(ux * ux)));
	else
		tmp = sqrt(fma(ux, Float32(2.0), Float32(ux * Float32(Float32(-2.0) * maxCos))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.99970001

   1. Initial program 57.4%

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

   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 49.1%

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

   6. Applied rewrites 49.9%

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

   7. Taylor expanded in maxCos around 0

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

   9. Applied rewrites 48.3%

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

   if 0.99970001 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 57.4%

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

   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 49.1%

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

   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 64.8%

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

   9. Applied rewrites 64.8%

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

Alternative 11: 64.8% accurate, 4.9× 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

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

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 (* -2.0 maxCos)))))

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

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 49.1%

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

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 64.8%

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

9. Applied rewrites 64.8%

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

Alternative 12: 64.8% 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 57.4%

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

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 49.1%

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

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 64.8%

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

9. Applied rewrites 64.8%

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

Alternative 13: 62.0% 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 57.4%

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

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 49.1%

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

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 64.8%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 62.0%

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

12. Applied rewrites 62.0%

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

Reproduce

herbie shell --seed 2025359 
(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)))))))