UniformSampleCone, y

Percentage Accurate: 57.1% → 98.4%

Time: 24.9s

Alternatives: 26

Speedup: 2.0×

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

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* 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 sinf(((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(sin(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 = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.1% accurate, 1.0× speedup

Math

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

FPCore

(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))))))

C

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)));
}

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \sin \left(\sqrt[3]{{\left(uy \cdot 2\right)}^{3} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)\right)\right)}\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right)\right) + ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin
   (cbrt
    (*
     (pow (* uy 2.0) 3.0)
     (* (cbrt PI) (* (* PI PI) (* (cbrt PI) (cbrt PI)))))))
  (sqrt
   (+
    (* (pow ux 2.0) (* (- 1.0 maxCos) (+ -1.0 maxCos)))
    (* ux (+ (- 1.0 maxCos) (- 1.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	return sinf(cbrtf((powf((uy * 2.0f), 3.0f) * (cbrtf(((float) M_PI)) * ((((float) M_PI) * ((float) M_PI)) * (cbrtf(((float) M_PI)) * cbrtf(((float) M_PI)))))))) * sqrtf(((powf(ux, 2.0f) * ((1.0f - maxCos) * (-1.0f + maxCos))) + (ux * ((1.0f - maxCos) + (1.0f - maxCos)))));
}

Julia

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

Derivation

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Alternative 2: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right)\right) + ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right)} \cdot \sin \left(\sqrt[3]{{\left(uy \cdot 2\right)}^{3} \cdot {\pi}^{3}}\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (+
    (* (pow ux 2.0) (* (- 1.0 maxCos) (+ -1.0 maxCos)))
    (* ux (+ (- 1.0 maxCos) (- 1.0 maxCos)))))
  (sin (cbrt (* (pow (* uy 2.0) 3.0) (pow PI 3.0))))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(((powf(ux, 2.0f) * ((1.0f - maxCos) * (-1.0f + maxCos))) + (ux * ((1.0f - maxCos) + (1.0f - maxCos))))) * sinf(cbrtf((powf((uy * 2.0f), 3.0f) * powf(((float) M_PI), 3.0f))));
}

Julia

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

Derivation

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Alternative 3: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 4: 95.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(uy \cdot 2\right) \cdot \pi\\ \mathbf{if}\;maxCos \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{ux \cdot \left(\left(-ux\right) - -2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{-\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot -2, \left(-1 + maxCos\right) \cdot \left(\left(-1 + maxCos\right) \cdot {ux}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* uy 2.0) PI)))
   (if (<= maxCos 1.2000000424450263e-6)
     (* (sin t_0) (sqrt (* ux (- (- ux) -2.0))))
     (*
      t_0
      (sqrt
       (-
        (fma
         ux
         (* (- 1.0 maxCos) -2.0)
         (* (+ -1.0 maxCos) (* (+ -1.0 maxCos) (pow ux 2.0))))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (uy * 2.0f) * ((float) M_PI);
	float tmp;
	if (maxCos <= 1.2000000424450263e-6f) {
		tmp = sinf(t_0) * sqrtf((ux * (-ux - -2.0f)));
	} else {
		tmp = t_0 * sqrtf(-fmaf(ux, ((1.0f - maxCos) * -2.0f), ((-1.0f + maxCos) * ((-1.0f + maxCos) * powf(ux, 2.0f)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.2000000424450263e-6))
		tmp = Float32(sin(t_0) * sqrt(Float32(ux * Float32(Float32(-ux) - Float32(-2.0)))));
	else
		tmp = Float32(t_0 * sqrt(Float32(-fma(ux, Float32(Float32(Float32(1.0) - maxCos) * Float32(-2.0)), Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(Float32(-1.0) + maxCos) * (ux ^ Float32(2.0))))))));
	end
	return tmp
end

Derivation

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Alternative 5: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right)\right) + ux \cdot \left(\left(1 - \left(-1 + maxCos\right)\right) - maxCos\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 6: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right)\right) + ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 7: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(-ux\right) - -2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right)\right) + ux \cdot \left(\left(1 - \left(-1 + maxCos\right)\right) - maxCos\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.2000000424450263e-6)
   (* (sin (* (* uy 2.0) PI)) (sqrt (* ux (- (- ux) -2.0))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+
       (* (pow ux 2.0) (* (- 1.0 maxCos) (+ -1.0 maxCos)))
       (* ux (- (- 1.0 (+ -1.0 maxCos)) maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.2000000424450263e-6f) {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (-ux - -2.0f)));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf(((powf(ux, 2.0f) * ((1.0f - maxCos) * (-1.0f + maxCos))) + (ux * ((1.0f - (-1.0f + maxCos)) - maxCos)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.2000000424450263e-6))
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(-ux) - Float32(-2.0)))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(-1.0) + maxCos))) + Float32(ux * Float32(Float32(Float32(1.0) - Float32(Float32(-1.0) + maxCos)) - maxCos))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(1.2000000424450263e-6))
		tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((ux * (-ux - single(-2.0))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((((ux ^ single(2.0)) * ((single(1.0) - maxCos) * (single(-1.0) + maxCos))) + (ux * ((single(1.0) - (single(-1.0) + maxCos)) - maxCos)))));
	end
	tmp_2 = tmp;
end

Derivation

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Alternative 8: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(-ux\right) - -2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right)\right) + ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.2000000424450263e-6)
   (* (sin (* (* uy 2.0) PI)) (sqrt (* ux (- (- ux) -2.0))))
   (*
    2.0
    (*
     (sqrt
      (+
       (* (pow ux 2.0) (* (- 1.0 maxCos) (+ -1.0 maxCos)))
       (* ux (+ (- 1.0 maxCos) (- 1.0 maxCos)))))
     (* uy PI)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.2000000424450263e-6f) {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (-ux - -2.0f)));
	} else {
		tmp = 2.0f * (sqrtf(((powf(ux, 2.0f) * ((1.0f - maxCos) * (-1.0f + maxCos))) + (ux * ((1.0f - maxCos) + (1.0f - maxCos))))) * (uy * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.2000000424450263e-6))
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(-ux) - Float32(-2.0)))));
	else
		tmp = Float32(Float32(2.0) * Float32(sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(-1.0) + maxCos))) + Float32(ux * Float32(Float32(Float32(1.0) - maxCos) + Float32(Float32(1.0) - maxCos))))) * Float32(uy * Float32(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(1.2000000424450263e-6))
		tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((ux * (-ux - single(-2.0))));
	else
		tmp = single(2.0) * (sqrt((((ux ^ single(2.0)) * ((single(1.0) - maxCos) * (single(-1.0) + maxCos))) + (ux * ((single(1.0) - maxCos) + (single(1.0) - maxCos))))) * (uy * single(pi)));
	end
	tmp_2 = tmp;
end

Derivation

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Alternative 9: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(-ux\right) - -2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right) + {ux}^{2} \cdot \left(-1 - maxCos \cdot -2\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.2000000424450263e-6)
   (* (sin (* (* uy 2.0) PI)) (sqrt (* ux (- (- ux) -2.0))))
   (*
    2.0
    (*
     uy
     (*
      PI
      (sqrt
       (+
        (* ux (+ (- 1.0 maxCos) (- 1.0 maxCos)))
        (* (pow ux 2.0) (- -1.0 (* maxCos -2.0))))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.2000000424450263e-6f) {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (-ux - -2.0f)));
	} else {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((ux * ((1.0f - maxCos) + (1.0f - maxCos))) + (powf(ux, 2.0f) * (-1.0f - (maxCos * -2.0f)))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.2000000424450263e-6))
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(-ux) - Float32(-2.0)))));
	else
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(ux * Float32(Float32(Float32(1.0) - maxCos) + Float32(Float32(1.0) - maxCos))) + Float32((ux ^ Float32(2.0)) * Float32(Float32(-1.0) - Float32(maxCos * Float32(-2.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(1.2000000424450263e-6))
		tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((ux * (-ux - single(-2.0))));
	else
		tmp = single(2.0) * (uy * (single(pi) * sqrt(((ux * ((single(1.0) - maxCos) + (single(1.0) - maxCos))) + ((ux ^ single(2.0)) * (single(-1.0) - (maxCos * single(-2.0))))))));
	end
	tmp_2 = tmp;
end

Derivation

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Alternative 10: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;maxCos \leq 0.00044999999227002263:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(\left(-ux\right) - -2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* (* uy 2.0) PI))))
   (if (<= maxCos 0.00044999999227002263)
     (* t_0 (sqrt (* ux (- (- ux) -2.0))))
     (* t_0 (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf(((uy * 2.0f) * ((float) M_PI)));
	float tmp;
	if (maxCos <= 0.00044999999227002263f) {
		tmp = t_0 * sqrtf((ux * (-ux - -2.0f)));
	} else {
		tmp = t_0 * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	tmp = Float32(0.0)
	if (maxCos <= Float32(0.00044999999227002263))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(-ux) - Float32(-2.0)))));
	else
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = sin(((uy * single(2.0)) * single(pi)));
	tmp = single(0.0);
	if (maxCos <= single(0.00044999999227002263))
		tmp = t_0 * sqrt((ux * (-ux - single(-2.0))));
	else
		tmp = t_0 * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

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Alternative 11: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.000699999975040555:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(-1 + ux \cdot \left(1 - maxCos\right)\right) \cdot \left(\left(1 + ux \cdot maxCos\right) - ux\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.000699999975040555)
   (* (sin (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+
       1.0
       (* (+ -1.0 (* ux (- 1.0 maxCos))) (- (+ 1.0 (* ux maxCos)) ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.000699999975040555f) {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((-1.0f + (ux * (1.0f - maxCos))) * ((1.0f + (ux * maxCos)) - ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.000699999975040555))
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(-1.0) + Float32(ux * Float32(Float32(1.0) - maxCos))) * Float32(Float32(Float32(1.0) + Float32(ux * maxCos)) - ux))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.000699999975040555))
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(-1.0) + (ux * (single(1.0) - maxCos))) * ((single(1.0) + (ux * maxCos)) - ux)))));
	end
	tmp_2 = tmp;
end

Derivation

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Alternative 12: 92.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(-ux\right) - -2\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 13: 76.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0002099999983329326:\\ \;\;\;\;\left(uy \cdot \pi\right) \cdot \left(2 \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(-1 + ux \cdot \left(1 - maxCos\right)\right) \cdot \left(\left(1 + ux \cdot maxCos\right) - ux\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0002099999983329326)
   (* (* uy PI) (* 2.0 (sqrt (* ux (+ 2.0 (* maxCos -2.0))))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+
       1.0
       (* (+ -1.0 (* ux (- 1.0 maxCos))) (- (+ 1.0 (* ux maxCos)) ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0002099999983329326f) {
		tmp = (uy * ((float) M_PI)) * (2.0f * sqrtf((ux * (2.0f + (maxCos * -2.0f)))));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((-1.0f + (ux * (1.0f - maxCos))) * ((1.0f + (ux * maxCos)) - ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0002099999983329326))
		tmp = Float32(Float32(uy * Float32(pi)) * Float32(Float32(2.0) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0)))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(-1.0) + Float32(ux * Float32(Float32(1.0) - maxCos))) * Float32(Float32(Float32(1.0) + Float32(ux * maxCos)) - ux))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.0002099999983329326))
		tmp = (uy * single(pi)) * (single(2.0) * sqrt((ux * (single(2.0) + (maxCos * single(-2.0))))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(-1.0) + (ux * (single(1.0) - maxCos))) * ((single(1.0) + (ux * maxCos)) - ux)))));
	end
	tmp_2 = tmp;
end

Derivation

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Alternative 14: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0002099999983329326:\\ \;\;\;\;\left(uy \cdot \pi\right) \cdot \left(2 \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(-1 + ux\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0002099999983329326)
   (* (* uy PI) (* 2.0 (sqrt (* ux (+ 2.0 (* maxCos -2.0))))))
   (* 2.0 (* uy (* PI (sqrt (+ 1.0 (* (- 1.0 ux) (+ -1.0 ux)))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0002099999983329326f) {
		tmp = (uy * ((float) M_PI)) * (2.0f * sqrtf((ux * (2.0f + (maxCos * -2.0f)))));
	} else {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((1.0f + ((1.0f - ux) * (-1.0f + ux))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0002099999983329326))
		tmp = Float32(Float32(uy * Float32(pi)) * Float32(Float32(2.0) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0)))))));
	else
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(Float32(-1.0) + ux)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.0002099999983329326))
		tmp = (uy * single(pi)) * (single(2.0) * sqrt((ux * (single(2.0) + (maxCos * single(-2.0))))));
	else
		tmp = single(2.0) * (uy * (single(pi) * sqrt((single(1.0) + ((single(1.0) - ux) * (single(-1.0) + ux))))));
	end
	tmp_2 = tmp;
end

Derivation

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Alternative 15: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0002099999983329326:\\ \;\;\;\;\left(uy \cdot \pi\right) \cdot \left(2 \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\pi \cdot \left(uy \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(-1 + ux\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0002099999983329326)
   (* (* uy PI) (* 2.0 (sqrt (* ux (+ 2.0 (* maxCos -2.0))))))
   (* 2.0 (* PI (* uy (sqrt (+ 1.0 (* (- 1.0 ux) (+ -1.0 ux)))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0002099999983329326f) {
		tmp = (uy * ((float) M_PI)) * (2.0f * sqrtf((ux * (2.0f + (maxCos * -2.0f)))));
	} else {
		tmp = 2.0f * (((float) M_PI) * (uy * sqrtf((1.0f + ((1.0f - ux) * (-1.0f + ux))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0002099999983329326))
		tmp = Float32(Float32(uy * Float32(pi)) * Float32(Float32(2.0) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0)))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(pi) * Float32(uy * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(Float32(-1.0) + ux)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.0002099999983329326))
		tmp = (uy * single(pi)) * (single(2.0) * sqrt((ux * (single(2.0) + (maxCos * single(-2.0))))));
	else
		tmp = single(2.0) * (single(pi) * (uy * sqrt((single(1.0) + ((single(1.0) - ux) * (single(-1.0) + ux))))));
	end
	tmp_2 = tmp;
end

Derivation

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Alternative 16: 66.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 17: 63.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 18: 20.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{uy \cdot \left(\pi \cdot ux\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 19: 20.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{\pi \cdot \left(uy \cdot ux\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 20: 21.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(ux \cdot \sqrt{uy \cdot \pi}\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 21: 27.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(uy \cdot \sqrt[3]{\pi \cdot ux}\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (* 2.0 (* uy (cbrt (* PI ux)))))

C

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

Julia

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

Derivation

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Alternative 22: 20.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sin \left(uy \cdot \pi\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sin (* uy PI)))

C

float code(float ux, float uy, float maxCos) {
	return sinf((uy * ((float) M_PI)));
}

Julia

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

MATLAB

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

Derivation

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Alternative 23: 20.2% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(uy \cdot \left(\pi \cdot \left(ux + ux\right)\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (* 2.0 (* uy (* PI (+ ux ux)))))

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 24: 19.2% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(ux \cdot \left(uy \cdot \pi\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 25: 19.2% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(uy \cdot \left(\pi \cdot ux\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 26: 14.6% accurate, 223.0× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return 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 = 1.0e0
end function

Julia

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

MATLAB

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

Derivation

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Reproduce

herbie shell --seed 2024006 
(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)))))))