Beckmann Sample, near normal, slope_x

Percentage Accurate: 58.6% → 99.0%

Time: 11.9s

Alternatives: 7

Speedup: 4.0×

Specification

\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

Initial Program: 58.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (cos (* 2.0 (* PI u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * cosf((2.0f * (((float) M_PI) * u2)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(Float32(Float32(2.0) * Float32(Float32(pi) * u2))))
end

Derivation

1. Initial program 56.5%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 56.5%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 98.9%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. associate-*l* 98.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

4. Final simplification 98.9%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

Alternative 2: 92.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9804999828338623:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9804999828338623)
   (sqrt (- (log (- 1.0 u1))))
   (* (cos (* u2 (* 2.0 PI))) (sqrt (- u1 (* u1 (* u1 -0.5)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9804999828338623f) {
		tmp = sqrtf(-logf((1.0f - u1)));
	} else {
		tmp = cosf((u2 * (2.0f * ((float) M_PI)))) * sqrtf((u1 - (u1 * (u1 * -0.5f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9804999828338623))
		tmp = sqrt(Float32(-log(Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(u1 - Float32(u1 * Float32(u1 * Float32(-0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.9804999828338623))
		tmp = sqrt(-log((single(1.0) - u1)));
	else
		tmp = cos((u2 * (single(2.0) * single(pi)))) * sqrt((u1 - (u1 * (u1 * single(-0.5)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.980499983

   1. Initial program 96.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 84.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]

   if 0.980499983 < (-.f32 1 u1)

   1. Initial program 46.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0 97.4%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. +-commutative 76.1%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot 1 \]

      2. mul-1-neg 76.1%

         \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot 1 \]

      3. unsub-neg 76.1%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot 1 \]

      4. unpow2 76.1%

         \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot 1 \]

      5. associate-*r* 76.1%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot 1 \]

   4. Simplified 97.4%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9804999828338623:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \end{array} \]

Alternative 3: 84.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9992420077323914:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9992420077323914)
   (sqrt (- (log (- 1.0 u1))))
   (* (cos (* PI (* 2.0 u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9992420077323914f) {
		tmp = sqrtf(-logf((1.0f - u1)));
	} else {
		tmp = cosf((((float) M_PI) * (2.0f * u2))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9992420077323914))
		tmp = sqrt(Float32(-log(Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.9992420077323914))
		tmp = sqrt(-log((single(1.0) - u1)));
	else
		tmp = cos((single(pi) * (single(2.0) * u2))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.999242008

   1. Initial program 93.5%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 78.6%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]

   if 0.999242008 < (-.f32 1 u1)

   1. Initial program 42.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 38.5%

         \[\leadsto \color{blue}{\sqrt{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \sqrt{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}} \]

      2. pow2 38.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)}^{2}} \]

   3. Applied egg-rr 82.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{2}} \]

   4. Taylor expanded in u1 around 0 90.1%

      \[\leadsto \color{blue}{\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1}} \]
   5. Step-by-step derivation

      1. associate-*r* 90.1%

         \[\leadsto \cos \color{blue}{\left(\left(2 \cdot u2\right) \cdot \pi\right)} \cdot \sqrt{u1} \]

      2. *-commutative 90.1%

         \[\leadsto \cos \left(\color{blue}{\left(u2 \cdot 2\right)} \cdot \pi\right) \cdot \sqrt{u1} \]

      3. *-commutative 90.1%

         \[\leadsto \cos \color{blue}{\left(\pi \cdot \left(u2 \cdot 2\right)\right)} \cdot \sqrt{u1} \]

   6. Simplified 90.1%

      \[\leadsto \color{blue}{\cos \left(\pi \cdot \left(u2 \cdot 2\right)\right) \cdot \sqrt{u1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9992420077323914:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 79.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{u1 - \left(u1 \cdot \left(u1 \cdot -0.5\right) + -0.3333333333333333 \cdot {u1}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.019999999552965164)
   (sqrt (- u1 (+ (* u1 (* u1 -0.5)) (* -0.3333333333333333 (pow u1 3.0)))))
   (sqrt (- (log (- 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.019999999552965164f) {
		tmp = sqrtf((u1 - ((u1 * (u1 * -0.5f)) + (-0.3333333333333333f * powf(u1, 3.0f)))));
	} else {
		tmp = sqrtf(-logf((1.0f - u1)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if (u1 <= 0.019999999552965164e0) then
        tmp = sqrt((u1 - ((u1 * (u1 * (-0.5e0))) + ((-0.3333333333333333e0) * (u1 ** 3.0e0)))))
    else
        tmp = sqrt(-log((1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.019999999552965164))
		tmp = sqrt(Float32(u1 - Float32(Float32(u1 * Float32(u1 * Float32(-0.5))) + Float32(Float32(-0.3333333333333333) * (u1 ^ Float32(3.0))))));
	else
		tmp = sqrt(Float32(-log(Float32(Float32(1.0) - u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.019999999552965164))
		tmp = sqrt((u1 - ((u1 * (u1 * single(-0.5))) + (single(-0.3333333333333333) * (u1 ^ single(3.0))))));
	else
		tmp = sqrt(-log((single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0199999996

   1. Initial program 46.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 39.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]

   3. Taylor expanded in u1 around 0 76.8%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)}} \cdot 1 \]
   4. Step-by-step derivation

      1. +-commutative 76.8%

         \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right) + -1 \cdot u1\right)}} \cdot 1 \]

      2. mul-1-neg 76.8%

         \[\leadsto \sqrt{-\left(\left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right) + \color{blue}{\left(-u1\right)}\right)} \cdot 1 \]

      3. unsub-neg 76.8%

         \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right) - u1\right)}} \cdot 1 \]

      4. fma-def 76.8%

         \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot {u1}^{2}\right)} - u1\right)} \cdot 1 \]

      5. unpow2 76.8%

         \[\leadsto \sqrt{-\left(\mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)}\right) - u1\right)} \cdot 1 \]

      6. associate-*r* 76.8%

         \[\leadsto \sqrt{-\left(\mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, \color{blue}{\left(-0.5 \cdot u1\right) \cdot u1}\right) - u1\right)} \cdot 1 \]

   5. Simplified 76.8%

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, \left(-0.5 \cdot u1\right) \cdot u1\right) - u1\right)}} \cdot 1 \]
   6. Step-by-step derivation

      1. fma-udef 76.8%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.3333333333333333 \cdot {u1}^{3} + \left(-0.5 \cdot u1\right) \cdot u1\right)} - u1\right)} \cdot 1 \]

      2. *-commutative 76.8%

         \[\leadsto \sqrt{-\left(\left(-0.3333333333333333 \cdot {u1}^{3} + \color{blue}{u1 \cdot \left(-0.5 \cdot u1\right)}\right) - u1\right)} \cdot 1 \]

      3. *-commutative 76.8%

         \[\leadsto \sqrt{-\left(\left(-0.3333333333333333 \cdot {u1}^{3} + u1 \cdot \color{blue}{\left(u1 \cdot -0.5\right)}\right) - u1\right)} \cdot 1 \]

   7. Applied egg-rr 76.8%

      \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.3333333333333333 \cdot {u1}^{3} + u1 \cdot \left(u1 \cdot -0.5\right)\right)} - u1\right)} \cdot 1 \]

   if 0.0199999996 < u1

   1. Initial program 96.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 84.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{u1 - \left(u1 \cdot \left(u1 \cdot -0.5\right) + -0.3333333333333333 \cdot {u1}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]

Alternative 5: 78.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.006000000052154064)
   (sqrt (- u1 (* u1 (* u1 -0.5))))
   (sqrt (- (log (- 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.006000000052154064f) {
		tmp = sqrtf((u1 - (u1 * (u1 * -0.5f))));
	} else {
		tmp = sqrtf(-logf((1.0f - u1)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if (u1 <= 0.006000000052154064e0) then
        tmp = sqrt((u1 - (u1 * (u1 * (-0.5e0)))))
    else
        tmp = sqrt(-log((1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.006000000052154064))
		tmp = sqrt(Float32(u1 - Float32(u1 * Float32(u1 * Float32(-0.5)))));
	else
		tmp = sqrt(Float32(-log(Float32(Float32(1.0) - u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.006000000052154064))
		tmp = sqrt((u1 - (u1 * (u1 * single(-0.5)))));
	else
		tmp = sqrt(-log((single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00600000005

   1. Initial program 45.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 38.6%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]

   3. Taylor expanded in u1 around 0 76.6%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot 1 \]
   4. Step-by-step derivation

      1. +-commutative 76.6%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot 1 \]

      2. mul-1-neg 76.6%

         \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot 1 \]

      3. unsub-neg 76.6%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot 1 \]

      4. unpow2 76.6%

         \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot 1 \]

      5. associate-*r* 76.6%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot 1 \]

   5. Simplified 76.6%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot 1 \]
   6. Step-by-step derivation

      1. expm1-log1p-u 76.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}\right)\right)} \cdot 1 \]

      2. expm1-udef 61.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{-\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}\right)} - 1\right)} \cdot 1 \]

      3. *-commutative 61.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{-\left(\color{blue}{\left(u1 \cdot -0.5\right)} \cdot u1 - u1\right)}\right)} - 1\right) \cdot 1 \]

      4. *-commutative 61.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{-\left(\color{blue}{u1 \cdot \left(u1 \cdot -0.5\right)} - u1\right)}\right)} - 1\right) \cdot 1 \]

   7. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{-\left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}\right)} - 1\right)} \cdot 1 \]
   8. Step-by-step derivation

      1. expm1-def 76.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-\left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}\right)\right)} \cdot 1 \]

      2. expm1-log1p 76.6%

         \[\leadsto \color{blue}{\sqrt{-\left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}} \cdot 1 \]

      3. neg-sub0 76.6%

         \[\leadsto \sqrt{\color{blue}{0 - \left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}} \cdot 1 \]

      4. metadata-eval 76.6%

         \[\leadsto \sqrt{\color{blue}{\log 1} - \left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)} \cdot 1 \]

      5. sub-neg 76.6%

         \[\leadsto \sqrt{\log 1 - \color{blue}{\left(u1 \cdot \left(u1 \cdot -0.5\right) + \left(-u1\right)\right)}} \cdot 1 \]

      6. +-commutative 76.6%

         \[\leadsto \sqrt{\log 1 - \color{blue}{\left(\left(-u1\right) + u1 \cdot \left(u1 \cdot -0.5\right)\right)}} \cdot 1 \]

      7. neg-mul-1 76.6%

         \[\leadsto \sqrt{\log 1 - \left(\color{blue}{-1 \cdot u1} + u1 \cdot \left(u1 \cdot -0.5\right)\right)} \cdot 1 \]

      8. *-commutative 76.6%

         \[\leadsto \sqrt{\log 1 - \left(\color{blue}{u1 \cdot -1} + u1 \cdot \left(u1 \cdot -0.5\right)\right)} \cdot 1 \]

      9. metadata-eval 76.6%

         \[\leadsto \sqrt{\log 1 - \left(u1 \cdot \color{blue}{\left(-1\right)} + u1 \cdot \left(u1 \cdot -0.5\right)\right)} \cdot 1 \]

      10. distribute-rgt-neg-in 76.6%

          \[\leadsto \sqrt{\log 1 - \left(\color{blue}{\left(-u1 \cdot 1\right)} + u1 \cdot \left(u1 \cdot -0.5\right)\right)} \cdot 1 \]

      11. associate--r+ 76.6%

          \[\leadsto \sqrt{\color{blue}{\left(\log 1 - \left(-u1 \cdot 1\right)\right) - u1 \cdot \left(u1 \cdot -0.5\right)}} \cdot 1 \]

      12. metadata-eval 76.6%

          \[\leadsto \sqrt{\left(\color{blue}{0} - \left(-u1 \cdot 1\right)\right) - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot 1 \]

      13. neg-sub0 76.6%

          \[\leadsto \sqrt{\color{blue}{\left(-\left(-u1 \cdot 1\right)\right)} - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot 1 \]

      14. *-rgt-identity 76.6%

          \[\leadsto \sqrt{\left(-\left(-\color{blue}{u1}\right)\right) - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot 1 \]

      15. remove-double-neg 76.6%

          \[\leadsto \sqrt{\color{blue}{u1} - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot 1 \]

   9. Simplified 76.6%

      \[\leadsto \color{blue}{\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}} \cdot 1 \]

   if 0.00600000005 < u1

   1. Initial program 96.3%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 82.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]

Alternative 6: 72.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (- u1 (* u1 (* u1 -0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 - (u1 * (u1 * -0.5f))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 - (u1 * (u1 * (-0.5e0)))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 - Float32(u1 * Float32(u1 * Float32(-0.5)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 - (u1 * (u1 * single(-0.5)))));
end

Derivation

1. Initial program 56.5%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Taylor expanded in u2 around 0 48.0%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]

3. Taylor expanded in u1 around 0 71.1%

   \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot 1 \]
4. Step-by-step derivation

   1. +-commutative 71.1%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot 1 \]

   2. mul-1-neg 71.1%

      \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot 1 \]

   3. unsub-neg 71.1%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot 1 \]

   4. unpow2 71.1%

      \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot 1 \]

   5. associate-*r* 71.1%

      \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot 1 \]

5. Simplified 71.1%

   \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot 1 \]
6. Step-by-step derivation

   1. expm1-log1p-u 71.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}\right)\right)} \cdot 1 \]

   2. expm1-udef 59.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{-\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}\right)} - 1\right)} \cdot 1 \]

   3. *-commutative 59.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{-\left(\color{blue}{\left(u1 \cdot -0.5\right)} \cdot u1 - u1\right)}\right)} - 1\right) \cdot 1 \]

   4. *-commutative 59.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{-\left(\color{blue}{u1 \cdot \left(u1 \cdot -0.5\right)} - u1\right)}\right)} - 1\right) \cdot 1 \]

7. Applied egg-rr 59.0%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{-\left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}\right)} - 1\right)} \cdot 1 \]
8. Step-by-step derivation

   1. expm1-def 71.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-\left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}\right)\right)} \cdot 1 \]

   2. expm1-log1p 71.1%

      \[\leadsto \color{blue}{\sqrt{-\left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}} \cdot 1 \]

   3. neg-sub0 71.1%

      \[\leadsto \sqrt{\color{blue}{0 - \left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}} \cdot 1 \]

   4. metadata-eval 71.1%

      \[\leadsto \sqrt{\color{blue}{\log 1} - \left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)} \cdot 1 \]

   5. sub-neg 71.1%

      \[\leadsto \sqrt{\log 1 - \color{blue}{\left(u1 \cdot \left(u1 \cdot -0.5\right) + \left(-u1\right)\right)}} \cdot 1 \]

   6. +-commutative 71.1%

      \[\leadsto \sqrt{\log 1 - \color{blue}{\left(\left(-u1\right) + u1 \cdot \left(u1 \cdot -0.5\right)\right)}} \cdot 1 \]

   7. neg-mul-1 71.1%

      \[\leadsto \sqrt{\log 1 - \left(\color{blue}{-1 \cdot u1} + u1 \cdot \left(u1 \cdot -0.5\right)\right)} \cdot 1 \]

   8. *-commutative 71.1%

      \[\leadsto \sqrt{\log 1 - \left(\color{blue}{u1 \cdot -1} + u1 \cdot \left(u1 \cdot -0.5\right)\right)} \cdot 1 \]

   9. metadata-eval 71.1%

      \[\leadsto \sqrt{\log 1 - \left(u1 \cdot \color{blue}{\left(-1\right)} + u1 \cdot \left(u1 \cdot -0.5\right)\right)} \cdot 1 \]

   10. distribute-rgt-neg-in 71.1%

       \[\leadsto \sqrt{\log 1 - \left(\color{blue}{\left(-u1 \cdot 1\right)} + u1 \cdot \left(u1 \cdot -0.5\right)\right)} \cdot 1 \]

   11. associate--r+ 71.1%

       \[\leadsto \sqrt{\color{blue}{\left(\log 1 - \left(-u1 \cdot 1\right)\right) - u1 \cdot \left(u1 \cdot -0.5\right)}} \cdot 1 \]

   12. metadata-eval 71.1%

       \[\leadsto \sqrt{\left(\color{blue}{0} - \left(-u1 \cdot 1\right)\right) - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot 1 \]

   13. neg-sub0 71.1%

       \[\leadsto \sqrt{\color{blue}{\left(-\left(-u1 \cdot 1\right)\right)} - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot 1 \]

   14. *-rgt-identity 71.1%

       \[\leadsto \sqrt{\left(-\left(-\color{blue}{u1}\right)\right) - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot 1 \]

   15. remove-double-neg 71.1%

       \[\leadsto \sqrt{\color{blue}{u1} - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot 1 \]

9. Simplified 71.1%

   \[\leadsto \color{blue}{\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}} \cdot 1 \]

10. Final simplification 71.1%

    \[\leadsto \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \]

Alternative 7: 64.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

Derivation

1. Initial program 56.5%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 56.5%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 98.9%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. associate-*l* 98.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]
4. Step-by-step derivation

   1. neg-mul-1 98.9%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

   2. log1p-udef 56.5%

      \[\leadsto \sqrt{-1 \cdot \color{blue}{\log \left(1 + \left(-u1\right)\right)}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

   3. sub-neg 56.5%

      \[\leadsto \sqrt{-1 \cdot \log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

   4. neg-mul-1 56.5%

      \[\leadsto \sqrt{\color{blue}{-\log \left(1 - u1\right)}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

   5. add-cube-cbrt 56.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)}} \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)}}\right) \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)}}\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

   6. pow3 56.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)}}\right)}^{3}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

5. Applied egg-rr 75.5%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{log1p}\left(u1\right)}}\right)}^{3}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

6. Taylor expanded in u2 around 0 62.4%

   \[\leadsto {\left(\sqrt[3]{\sqrt{\mathsf{log1p}\left(u1\right)}}\right)}^{3} \cdot \color{blue}{1} \]

7. Taylor expanded in u1 around 0 64.1%

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot 1 \]

8. Final simplification 64.1%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023238 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))