UniformSampleCone, x

Percentage Accurate: 57.7% → 99.0%

Time: 16.1s

Alternatives: 23

Speedup: 3.1×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* uy (* 2.0 PI)))
  (pow
   (fma
    ux
    (* 2.0 (- 1.0 maxCos))
    (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0))))
   0.5)))

C

float code(float ux, float uy, float maxCos) {
	return cosf((uy * (2.0f * ((float) M_PI)))) * powf(fmaf(ux, (2.0f * (1.0f - maxCos)), (powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f)))), 0.5f);
}

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * (fma(ux, Float32(Float32(2.0) * Float32(Float32(1.0) - maxCos)), Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))) ^ Float32(0.5)))
end

Derivation

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Alternative 2: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999997615814209:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* PI (* uy 2.0))) 0.9999997615814209)
   (* (cos (* uy (* 2.0 PI))) (sqrt (- (* 2.0 ux) (pow ux 2.0))))
   (pow
    (fma
     ux
     (* 2.0 (- 1.0 maxCos))
     (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0))))
    0.5)))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf((((float) M_PI) * (uy * 2.0f))) <= 0.9999997615814209f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf(((2.0f * ux) - powf(ux, 2.0f)));
	} else {
		tmp = powf(fmaf(ux, (2.0f * (1.0f - maxCos)), (powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f)))), 0.5f);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) <= Float32(0.9999997615814209))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(Float32(2.0) * ux) - (ux ^ Float32(2.0)))));
	else
		tmp = fma(ux, Float32(Float32(2.0) * Float32(Float32(1.0) - maxCos)), Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))) ^ Float32(0.5);
	end
	return tmp
end

Derivation

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Alternative 3: 90.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999920129776001:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* PI (* uy 2.0))) 0.9999920129776001)
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos))))
   (pow
    (fma
     ux
     (* 2.0 (- 1.0 maxCos))
     (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0))))
    0.5)))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf((((float) M_PI) * (uy * 2.0f))) <= 0.9999920129776001f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos)));
	} else {
		tmp = powf(fmaf(ux, (2.0f * (1.0f - maxCos)), (powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f)))), 0.5f);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) <= Float32(0.9999920129776001))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos))));
	else
		tmp = fma(ux, Float32(Float32(2.0) * Float32(Float32(1.0) - maxCos)), Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))) ^ Float32(0.5);
	end
	return tmp
end

Derivation

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Alternative 4: 90.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999920129776001:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + maxCos \cdot -2, {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* PI (* uy 2.0)))))
   (if (<= t_0 0.9999920129776001)
     (* t_0 (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))
     (sqrt
      (fma
       ux
       (+ 2.0 (* maxCos -2.0))
       (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf((((float) M_PI) * (uy * 2.0f)));
	float tmp;
	if (t_0 <= 0.9999920129776001f) {
		tmp = t_0 * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf(fmaf(ux, (2.0f + (maxCos * -2.0f)), (powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(pi) * Float32(uy * Float32(2.0))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999920129776001))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = sqrt(fma(ux, Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))), Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))));
	end
	return tmp
end

Derivation

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Alternative 5: 90.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999920129776001:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + maxCos \cdot -2, {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* PI (* uy 2.0))) 0.9999920129776001)
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos))))
   (sqrt
    (fma
     ux
     (+ 2.0 (* maxCos -2.0))
     (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf((((float) M_PI) * (uy * 2.0f))) <= 0.9999920129776001f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos)));
	} else {
		tmp = sqrtf(fmaf(ux, (2.0f + (maxCos * -2.0f)), (powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) <= Float32(0.9999920129776001))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos))));
	else
		tmp = sqrt(fma(ux, Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))), Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))));
	end
	return tmp
end

Derivation

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Alternative 6: 90.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ t_1 := ux \cdot \left(2 - 2 \cdot maxCos\right)\\ \mathbf{if}\;t_0 \leq 0.9999920129776001:\\ \;\;\;\;t_0 \cdot \sqrt{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* PI (* uy 2.0)))) (t_1 (* ux (- 2.0 (* 2.0 maxCos)))))
   (if (<= t_0 0.9999920129776001)
     (* t_0 (sqrt t_1))
     (sqrt (+ (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0))) t_1)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf((((float) M_PI) * (uy * 2.0f)));
	float t_1 = ux * (2.0f - (2.0f * maxCos));
	float tmp;
	if (t_0 <= 0.9999920129776001f) {
		tmp = t_0 * sqrtf(t_1);
	} else {
		tmp = sqrtf(((powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f))) + t_1));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(pi) * Float32(uy * Float32(2.0))))
	t_1 = Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999920129776001))
		tmp = Float32(t_0 * sqrt(t_1));
	else
		tmp = sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0)))) + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = cos((single(pi) * (uy * single(2.0))));
	t_1 = ux * (single(2.0) - (single(2.0) * maxCos));
	tmp = single(0.0);
	if (t_0 <= single(0.9999920129776001))
		tmp = t_0 * sqrt(t_1);
	else
		tmp = sqrt((((ux ^ single(2.0)) * ((single(1.0) - maxCos) * (maxCos + single(-1.0)))) + t_1));
	end
	tmp_2 = tmp;
end

Derivation

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Alternative 7: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 8: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 9: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 10: 97.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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Alternative 11: 91.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)\\ \mathbf{if}\;t_0 \leq 0.00022000000171829015:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (+
          1.0
          (* (+ (* ux maxCos) (- 1.0 ux)) (- (+ ux -1.0) (* ux maxCos))))))
   (if (<= t_0 0.00022000000171829015)
     (*
      (cos (* uy (* 2.0 PI)))
      (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos))))
     (* (cos (* PI (* uy 2.0))) (sqrt t_0)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = 1.0f + (((ux * maxCos) + (1.0f - ux)) * ((ux + -1.0f) - (ux * maxCos)));
	float tmp;
	if (t_0 <= 0.00022000000171829015f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos)));
	} else {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf(t_0);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(1.0) + Float32(Float32(Float32(ux * maxCos) + Float32(Float32(1.0) - ux)) * Float32(Float32(ux + Float32(-1.0)) - Float32(ux * maxCos))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00022000000171829015))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(1.0) + (((ux * maxCos) + (single(1.0) - ux)) * ((ux + single(-1.0)) - (ux * maxCos)));
	tmp = single(0.0);
	if (t_0 <= single(0.00022000000171829015))
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos)));
	else
		tmp = cos((single(pi) * (uy * single(2.0)))) * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Derivation

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1. Add Preprocessing

Alternative 12: 91.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right) \leq 0.00022000000171829015:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 + \left(1 - ux \cdot \left(1 - maxCos\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<=
      (+ 1.0 (* (+ (* ux maxCos) (- 1.0 ux)) (- (+ ux -1.0) (* ux maxCos))))
      0.00022000000171829015)
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos))))
   (*
    (cos (* 2.0 (* uy PI)))
    (sqrt
     (+
      1.0
      (* (- 1.0 (* ux (- 1.0 maxCos))) (+ ux (- -1.0 (* ux maxCos)))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((1.0f + (((ux * maxCos) + (1.0f - ux)) * ((ux + -1.0f) - (ux * maxCos)))) <= 0.00022000000171829015f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos)));
	} else {
		tmp = cosf((2.0f * (uy * ((float) M_PI)))) * sqrtf((1.0f + ((1.0f - (ux * (1.0f - maxCos))) * (ux + (-1.0f - (ux * maxCos))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + Float32(Float32(Float32(ux * maxCos) + Float32(Float32(1.0) - ux)) * Float32(Float32(ux + Float32(-1.0)) - Float32(ux * maxCos)))) <= Float32(0.00022000000171829015))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos))));
	else
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - Float32(ux * Float32(Float32(1.0) - maxCos))) * Float32(ux + Float32(Float32(-1.0) - Float32(ux * maxCos)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((single(1.0) + (((ux * maxCos) + (single(1.0) - ux)) * ((ux + single(-1.0)) - (ux * maxCos)))) <= single(0.00022000000171829015))
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos)));
	else
		tmp = cos((single(2.0) * (uy * single(pi)))) * sqrt((single(1.0) + ((single(1.0) - (ux * (single(1.0) - maxCos))) * (ux + (single(-1.0) - (ux * maxCos))))));
	end
	tmp_2 = tmp;
end

Derivation

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1. Add Preprocessing

Alternative 13: 89.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0006000000284984708)
   (sqrt
    (+
     (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0)))
     (* ux (- 2.0 (* 2.0 maxCos)))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0006000000284984708f) {
		tmp = sqrtf(((powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f))) + (ux * (2.0f - (2.0f * maxCos)))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0006000000284984708))
		tmp = 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(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

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

Derivation

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1. Add Preprocessing

Alternative 14: 80.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(ux, uy, maxCos)
	return 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

MATLAB

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

Derivation

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1. Add Preprocessing

Alternative 15: 79.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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1. Add Preprocessing

Alternative 16: 75.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(1 - maxCos\right)\\ \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - t_0\right) \cdot \left(-1 + t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- 1.0 maxCos))))
   (if (<= ux 0.00011000000085914508)
     (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos)))
     (sqrt (+ 1.0 (* (- 1.0 t_0) (+ -1.0 t_0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux * (1.0f - maxCos);
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos)));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - t_0) * (-1.0f + t_0))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: t_0
    real(4) :: tmp
    t_0 = ux * (1.0e0 - maxcos)
    if (ux <= 0.00011000000085914508e0) then
        tmp = sqrt((ux * ((1.0e0 + (1.0e0 - maxcos)) - maxcos)))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - t_0) * ((-1.0e0) + t_0))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(1.0) - maxCos))
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos)));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - t_0) * Float32(Float32(-1.0) + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = ux * (single(1.0) - maxCos);
	tmp = single(0.0);
	if (ux <= single(0.00011000000085914508))
		tmp = sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos)));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - t_0) * (single(-1.0) + t_0))));
	end
	tmp_2 = tmp;
end

Derivation

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1. Add Preprocessing

Alternative 17: 75.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux \cdot \left(1 - maxCos\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011000000085914508)
   (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos)))
   (sqrt
    (+ 1.0 (* (- 1.0 (* ux (- 1.0 maxCos))) (+ ux (- -1.0 (* ux maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos)));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - (ux * (1.0f - maxCos))) * (ux + (-1.0f - (ux * maxCos))))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00011000000085914508e0) then
        tmp = sqrt((ux * ((1.0e0 + (1.0e0 - maxcos)) - maxcos)))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - (ux * (1.0e0 - maxcos))) * (ux + ((-1.0e0) - (ux * maxcos))))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos)));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - Float32(ux * Float32(Float32(1.0) - maxCos))) * Float32(ux + Float32(Float32(-1.0) - Float32(ux * maxCos))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00011000000085914508))
		tmp = sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos)));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - (ux * (single(1.0) - maxCos))) * (ux + (single(-1.0) - (ux * maxCos))))));
	end
	tmp_2 = tmp;
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 18: 74.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011000000085914508)
   (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos)))
   (sqrt (+ 1.0 (* (- 1.0 ux) (+ -1.0 (* ux (- 1.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos)));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - ux) * (-1.0f + (ux * (1.0f - maxCos))))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00011000000085914508e0) then
        tmp = sqrt((ux * ((1.0e0 + (1.0e0 - maxcos)) - maxcos)))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - ux) * ((-1.0e0) + (ux * (1.0e0 - maxcos))))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos)));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(Float32(-1.0) + Float32(ux * Float32(Float32(1.0) - maxCos))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00011000000085914508))
		tmp = sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos)));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - ux) * (single(-1.0) + (ux * (single(1.0) - maxCos))))));
	end
	tmp_2 = tmp;
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 19: 73.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011000000085914508)
   (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
   (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - ux) * (ux + -1.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00011000000085914508e0) then
        tmp = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - ux) * (ux + (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00011000000085914508))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 20: 73.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011000000085914508)
   (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos)))
   (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos)));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - ux) * (ux + -1.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00011000000085914508e0) then
        tmp = sqrt((ux * ((1.0e0 + (1.0e0 - maxcos)) - maxcos)))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - ux) * (ux + (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos)));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00011000000085914508))
		tmp = sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos)));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

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1. Add Preprocessing

Alternative 21: 64.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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1. Add Preprocessing

Alternative 22: 61.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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1. Add Preprocessing

Alternative 23: 6.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{0} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt 0.0))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(0.0f);
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt(0.0e0)
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(0.0))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt(single(0.0));
end

Derivation

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1. Add Preprocessing

Reproduce

herbie shell --seed 2023350 
(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)))))))