UniformSampleCone, y

Percentage Accurate: 57.8% → 98.4%
Time: 17.9s
Alternatives: 12
Speedup: 1.5×

Specification

?
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t_0 \cdot t_0} \end{array} \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))
float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

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

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

\begin{array}{l}

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

Sampling outcomes in binary32 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 12 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 57.8% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t_0 \cdot t_0} \end{array} \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))
float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

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

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

\begin{array}{l}

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

Alternative 1: 98.4% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \sqrt[3]{{\sin \left(\pi \cdot \left(2 \cdot uy\right)\right)}^{3} \cdot {\left(\mathsf{fma}\left({ux}^{2}, \left(1 - maxCos\right) \cdot \left(maxCos + -1\right), ux \cdot \left(2 + maxCos \cdot -2\right)\right)\right)}^{1.5}} \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (cbrt
  (*
   (pow (sin (* PI (* 2.0 uy))) 3.0)
   (pow
    (fma
     (pow ux 2.0)
     (* (- 1.0 maxCos) (+ maxCos -1.0))
     (* ux (+ 2.0 (* maxCos -2.0))))
    1.5))))
float code(float ux, float uy, float maxCos) {
	return cbrtf((powf(sinf((((float) M_PI) * (2.0f * uy))), 3.0f) * powf(fmaf(powf(ux, 2.0f), ((1.0f - maxCos) * (maxCos + -1.0f)), (ux * (2.0f + (maxCos * -2.0f)))), 1.5f)));
}

function code(ux, uy, maxCos)
	return cbrt(Float32((sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) ^ Float32(3.0)) * (fma((ux ^ Float32(2.0)), Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))), Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))))) ^ Float32(1.5))))
end

\begin{array}{l}

\\
\sqrt[3]{{\sin \left(\pi \cdot \left(2 \cdot uy\right)\right)}^{3} \cdot {\left(\mathsf{fma}\left({ux}^{2}, \left(1 - maxCos\right) \cdot \left(maxCos + -1\right), ux \cdot \left(2 + maxCos \cdot -2\right)\right)\right)}^{1.5}}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 98.3% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) - ux \cdot \left(-1 + \left(maxCos + \left(maxCos + -1\right)\right)\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(uy \cdot \left(\pi \cdot 2\right)\right)\right)\right) \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (-
    (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0)))
    (* ux (+ -1.0 (+ maxCos (+ maxCos -1.0))))))
  (expm1 (log1p (sin (* uy (* PI 2.0)))))))
float code(float ux, float uy, float maxCos) {
	return sqrtf(((powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f))) - (ux * (-1.0f + (maxCos + (maxCos + -1.0f)))))) * expm1f(log1pf(sinf((uy * (((float) M_PI) * 2.0f)))));
}

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0)))) - Float32(ux * Float32(Float32(-1.0) + Float32(maxCos + Float32(maxCos + Float32(-1.0))))))) * expm1(log1p(sin(Float32(uy * Float32(Float32(pi) * Float32(2.0)))))))
end

\begin{array}{l}

\\
\sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) - ux \cdot \left(-1 + \left(maxCos + \left(maxCos + -1\right)\right)\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(uy \cdot \left(\pi \cdot 2\right)\right)\right)\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 98.3% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(uy \cdot \left(\pi \cdot 2\right)\right)\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) - ux \cdot \left(-1 + \left(maxCos + \left(maxCos + -1\right)\right)\right)} \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (log1p (expm1 (sin (* uy (* PI 2.0)))))
  (sqrt
   (-
    (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0)))
    (* ux (+ -1.0 (+ maxCos (+ maxCos -1.0))))))))
float code(float ux, float uy, float maxCos) {
	return log1pf(expm1f(sinf((uy * (((float) M_PI) * 2.0f))))) * sqrtf(((powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f))) - (ux * (-1.0f + (maxCos + (maxCos + -1.0f))))));
}

function code(ux, uy, maxCos)
	return Float32(log1p(expm1(sin(Float32(uy * Float32(Float32(pi) * Float32(2.0)))))) * sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0)))) - Float32(ux * Float32(Float32(-1.0) + Float32(maxCos + Float32(maxCos + Float32(-1.0))))))))
end

\begin{array}{l}

\\
\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(uy \cdot \left(\pi \cdot 2\right)\right)\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) - ux \cdot \left(-1 + \left(maxCos + \left(maxCos + -1\right)\right)\right)}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 4: 98.3% accurate, 0.8× speedup?

\[\begin{array}{l} \\ \sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)} \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* 2.0 (* PI uy)))
  (sqrt
   (+
    (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0)))
    (* ux (- 2.0 (* 2.0 maxCos)))))))
float code(float ux, float uy, float maxCos) {
	return sinf((2.0f * (((float) M_PI) * uy))) * sqrtf(((powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f))) + (ux * (2.0f - (2.0f * maxCos)))));
}

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

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

\begin{array}{l}

\\
\sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 5: 95.6% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0006000000284984708:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(\left(-ux\right) - -2\right)}\\ \end{array} \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0006000000284984708)
   (*
    2.0
    (*
     (* PI uy)
     (sqrt
      (+
       (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0)))
       (* ux (- 2.0 (* 2.0 maxCos)))))))
   (* (sin (* PI (* 2.0 uy))) (sqrt (* ux (- (- ux) -2.0))))))
float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.0006000000284984708f) {
		tmp = 2.0f * ((((float) M_PI) * uy) * sqrtf(((powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f))) + (ux * (2.0f - (2.0f * maxCos))))));
	} else {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * (-ux - -2.0f)));
	}
	return tmp;
}

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0006000000284984708))
		tmp = Float32(Float32(2.0) * Float32(Float32(Float32(pi) * uy) * sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0)))) + Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))))));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * Float32(Float32(-ux) - Float32(-2.0)))));
	end
	return tmp
end

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.0006000000284984708))
		tmp = single(2.0) * ((single(pi) * uy) * sqrt((((ux ^ single(2.0)) * ((single(1.0) - maxCos) * (maxCos + single(-1.0)))) + (ux * (single(2.0) - (single(2.0) * maxCos))))));
	else
		tmp = sin((single(pi) * (single(2.0) * uy))) * sqrt((ux * (-ux - single(-2.0))));
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;2 \cdot uy \leq 0.0006000000284984708:\\
\;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(\left(-ux\right) - -2\right)}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 6: 92.2% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(\left(-ux\right) - -2\right)} \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (* (sin (* PI (* 2.0 uy))) (sqrt (* ux (- (- ux) -2.0)))))
float code(float ux, float uy, float maxCos) {
	return sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * (-ux - -2.0f)));
}

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

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

\begin{array}{l}

\\
\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(\left(-ux\right) - -2\right)}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 7: 82.6% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 9.999999747378752e-5)
   (* (sin (* PI (* 2.0 uy))) (sqrt (* 2.0 ux)))
   (*
    2.0
    (*
     (* PI uy)
     (sqrt
      (+
       1.0
       (* (+ 1.0 (* ux (+ maxCos -1.0))) (+ ux (- -1.0 (* ux maxCos))))))))))
float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 9.999999747378752e-5f) {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((2.0f * ux));
	} else {
		tmp = 2.0f * ((((float) M_PI) * uy) * sqrtf((1.0f + ((1.0f + (ux * (maxCos + -1.0f))) * (ux + (-1.0f - (ux * maxCos)))))));
	}
	return tmp;
}

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(9.999999747378752e-5))
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(Float32(2.0) * ux)));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(Float32(pi) * uy) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(ux * Float32(maxCos + Float32(-1.0)))) * Float32(ux + Float32(Float32(-1.0) - Float32(ux * maxCos))))))));
	end
	return tmp
end

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(9.999999747378752e-5))
		tmp = sin((single(pi) * (single(2.0) * uy))) * sqrt((single(2.0) * ux));
	else
		tmp = single(2.0) * ((single(pi) * uy) * sqrt((single(1.0) + ((single(1.0) + (ux * (maxCos + single(-1.0)))) * (ux + (single(-1.0) - (ux * maxCos)))))));
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;ux \leq 9.999999747378752 \cdot 10^{-5}:\\
\;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{2 \cdot ux}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\right)\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 8: 76.9% accurate, 1.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00019999999494757503)
   (* 2.0 (* (* PI uy) (sqrt (- (* ux (- -2.0)) (* 2.0 (* ux maxCos))))))
   (*
    2.0
    (*
     (* PI uy)
     (sqrt
      (+
       1.0
       (* (+ 1.0 (* ux (+ maxCos -1.0))) (+ ux (- -1.0 (* ux maxCos))))))))))
float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00019999999494757503f) {
		tmp = 2.0f * ((((float) M_PI) * uy) * sqrtf(((ux * -(-2.0f)) - (2.0f * (ux * maxCos)))));
	} else {
		tmp = 2.0f * ((((float) M_PI) * uy) * sqrtf((1.0f + ((1.0f + (ux * (maxCos + -1.0f))) * (ux + (-1.0f - (ux * maxCos)))))));
	}
	return tmp;
}

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(2.0) * Float32(Float32(Float32(pi) * uy) * sqrt(Float32(Float32(ux * Float32(-Float32(-2.0))) - Float32(Float32(2.0) * Float32(ux * maxCos))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(Float32(pi) * uy) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(ux * Float32(maxCos + Float32(-1.0)))) * Float32(ux + Float32(Float32(-1.0) - Float32(ux * maxCos))))))));
	end
	return tmp
end

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = single(2.0) * ((single(pi) * uy) * sqrt(((ux * -single(-2.0)) - (single(2.0) * (ux * maxCos)))));
	else
		tmp = single(2.0) * ((single(pi) * uy) * sqrt((single(1.0) + ((single(1.0) + (ux * (maxCos + single(-1.0)))) * (ux + (single(-1.0) - (ux * maxCos)))))));
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;ux \leq 0.00019999999494757503:\\
\;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\right)\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 9: 75.5% accurate, 1.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00039999998989515007:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\\ \end{array} \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00039999998989515007)
   (* 2.0 (* (* PI uy) (sqrt (- (* ux (- -2.0)) (* 2.0 (* ux maxCos))))))
   (* 2.0 (* (* PI uy) (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0))))))))
float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00039999998989515007f) {
		tmp = 2.0f * ((((float) M_PI) * uy) * sqrtf(((ux * -(-2.0f)) - (2.0f * (ux * maxCos)))));
	} else {
		tmp = 2.0f * ((((float) M_PI) * uy) * sqrtf((1.0f + ((1.0f - ux) * (ux + -1.0f)))));
	}
	return tmp;
}

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

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00039999998989515007))
		tmp = single(2.0) * ((single(pi) * uy) * sqrt(((ux * -single(-2.0)) - (single(2.0) * (ux * maxCos)))));
	else
		tmp = single(2.0) * ((single(pi) * uy) * sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0))))));
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;ux \leq 0.00039999998989515007:\\
\;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\\


\end{array}
\end{array}
Derivation
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  1. Add Preprocessing

Alternative 10: 65.6% accurate, 1.5× speedup?

\[\begin{array}{l} \\ 2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\right) \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (* 2.0 (* (* PI uy) (sqrt (- (* ux (- -2.0)) (* 2.0 (* ux maxCos)))))))
float code(float ux, float uy, float maxCos) {
	return 2.0f * ((((float) M_PI) * uy) * sqrtf(((ux * -(-2.0f)) - (2.0f * (ux * maxCos)))));
}

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

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

\begin{array}{l}

\\
2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 11: 65.6% accurate, 1.5× speedup?

\[\begin{array}{l} \\ 2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (* 2.0 (* (* PI uy) (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))))
float code(float ux, float uy, float maxCos) {
	return 2.0f * ((((float) M_PI) * uy) * sqrtf((ux * (2.0f - (2.0f * maxCos)))));
}

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

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

\begin{array}{l}

\\
2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 12: 63.1% accurate, 1.5× speedup?

\[\begin{array}{l} \\ 2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(--2\right)}\right) \end{array} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (* 2.0 (* (* PI uy) (sqrt (* ux (- -2.0))))))
float code(float ux, float uy, float maxCos) {
	return 2.0f * ((((float) M_PI) * uy) * sqrtf((ux * -(-2.0f))));
}

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

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

\begin{array}{l}

\\
2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(--2\right)}\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2024003 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :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 (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))