Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.4% → 99.1%

Time: 10.6s

Alternatives: 6

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: 57.4% 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: 99.1% 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 57.0%

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

   1. sub-neg 57.0%

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

   2. log1p-def 99.1%

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

   3. associate-*l* 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 2: 93.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9865000247955322:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9865000247955322)
   (sqrt (- (log (- 1.0 u1))))
   (* (cos (* PI (* 2.0 u2))) (sqrt (+ u1 (* 0.5 (* u1 u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9865000247955322f) {
		tmp = sqrtf(-logf((1.0f - u1)));
	} else {
		tmp = cosf((((float) M_PI) * (2.0f * u2))) * sqrtf((u1 + (0.5f * (u1 * u1))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9865000247955322))
		tmp = sqrt(Float32(-log(Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) * sqrt(Float32(u1 + Float32(Float32(0.5) * Float32(u1 * u1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.9865000247955322))
		tmp = sqrt(-log((single(1.0) - u1)));
	else
		tmp = cos((single(pi) * (single(2.0) * u2))) * sqrt((u1 + (single(0.5) * (u1 * u1))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.986500025

   1. Initial program 96.4%

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

   2. Taylor expanded in u2 around 0 77.8%

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

   if 0.986500025 < (-.f32 1 u1)

   1. Initial program 48.2%

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

      1. associate-*r* 48.2%

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

      2. add-log-exp 48.2%

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

   3. Applied egg-rr 48.2%

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

   4. Taylor expanded in u1 around 0 95.4%

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

      1. +-commutative 95.4%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

      2. mul-1-neg 95.4%

         \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

      3. unsub-neg 95.4%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

      4. unpow2 95.4%

         \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

      5. associate-*r* 95.4%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

   6. Simplified 95.4%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

   7. Taylor expanded in u2 around inf 96.8%

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

      1. *-commutative 96.8%

         \[\leadsto \color{blue}{\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1 - -0.5 \cdot {u1}^{2}}} \]

      2. *-commutative 96.8%

         \[\leadsto \cos \color{blue}{\left(\left(u2 \cdot \pi\right) \cdot 2\right)} \cdot \sqrt{u1 - -0.5 \cdot {u1}^{2}} \]

      3. *-commutative 96.8%

         \[\leadsto \cos \left(\color{blue}{\left(\pi \cdot u2\right)} \cdot 2\right) \cdot \sqrt{u1 - -0.5 \cdot {u1}^{2}} \]

      4. associate-*l* 96.8%

         \[\leadsto \cos \color{blue}{\left(\pi \cdot \left(u2 \cdot 2\right)\right)} \cdot \sqrt{u1 - -0.5 \cdot {u1}^{2}} \]

      5. cancel-sign-sub-inv 96.8%

         \[\leadsto \cos \left(\pi \cdot \left(u2 \cdot 2\right)\right) \cdot \sqrt{\color{blue}{u1 + \left(--0.5\right) \cdot {u1}^{2}}} \]

      6. metadata-eval 96.8%

         \[\leadsto \cos \left(\pi \cdot \left(u2 \cdot 2\right)\right) \cdot \sqrt{u1 + \color{blue}{0.5} \cdot {u1}^{2}} \]

      7. unpow2 96.8%

         \[\leadsto \cos \left(\pi \cdot \left(u2 \cdot 2\right)\right) \cdot \sqrt{u1 + 0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)}} \]

   9. Simplified 96.8%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9865000247955322:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\\ \end{array} \]

Alternative 3: 85.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998019933700562:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9998019933700562)
   (sqrt (- (log (- 1.0 u1))))
   (* (cos (* 2.0 (* PI u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9998019933700562f) {
		tmp = sqrtf(-logf((1.0f - u1)));
	} else {
		tmp = cosf((2.0f * (((float) M_PI) * u2))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9998019933700562))
		tmp = sqrt(Float32(-log(Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(Float32(pi) * 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(0.9998019933700562))
		tmp = sqrt(-log((single(1.0) - u1)));
	else
		tmp = cos((single(2.0) * (single(pi) * u2))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.999801993

   1. Initial program 90.6%

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

   2. Taylor expanded in u2 around 0 74.8%

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

   if 0.999801993 < (-.f32 1 u1)

   1. Initial program 38.2%

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

      1. sub-neg 38.2%

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

      2. log1p-def 99.0%

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

      3. associate-*l* 99.0%

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

   3. Simplified 99.0%

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

      1. add-cbrt-cube 99.0%

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

      2. pow1/3 94.1%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{0.3333333333333333}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

   5. Applied egg-rr 87.0%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

   6. Taylor expanded in u1 around 0 91.6%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998019933700562:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 79.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9965000152587891:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9965000152587891)
   (sqrt (- (log (- 1.0 u1))))
   (sqrt (+ u1 (* 0.5 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9965000152587891f) {
		tmp = sqrtf(-logf((1.0f - u1)));
	} else {
		tmp = sqrtf((u1 + (0.5f * (u1 * 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 ((1.0e0 - u1) <= 0.9965000152587891e0) then
        tmp = sqrt(-log((1.0e0 - u1)))
    else
        tmp = sqrt((u1 + (0.5e0 * (u1 * u1))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9965000152587891))
		tmp = sqrt(Float32(-log(Float32(Float32(1.0) - u1))));
	else
		tmp = sqrt(Float32(u1 + Float32(Float32(0.5) * Float32(u1 * u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.9965000152587891))
		tmp = sqrt(-log((single(1.0) - u1)));
	else
		tmp = sqrt((u1 + (single(0.5) * (u1 * u1))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.99650002

   1. Initial program 94.8%

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

   2. Taylor expanded in u2 around 0 75.6%

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

   if 0.99650002 < (-.f32 1 u1)

   1. Initial program 44.5%

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

      1. associate-*r* 44.5%

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

      2. add-log-exp 44.5%

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

   3. Applied egg-rr 44.5%

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

   4. Taylor expanded in u1 around 0 96.7%

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

      1. +-commutative 96.7%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

      2. mul-1-neg 96.7%

         \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

      3. unsub-neg 96.7%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

      4. unpow2 96.7%

         \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

      5. associate-*r* 96.7%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

   6. Simplified 96.7%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

   7. Taylor expanded in u2 around 0 77.1%

      \[\leadsto \color{blue}{\sqrt{u1 - -0.5 \cdot {u1}^{2}}} \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 77.1%

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

      2. metadata-eval 77.1%

         \[\leadsto \sqrt{u1 + \color{blue}{0.5} \cdot {u1}^{2}} \]

      3. unpow2 77.1%

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

   9. Simplified 77.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9965000152587891:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\\ \end{array} \]

Alternative 5: 73.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 + (0.5f * (u1 * 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 + (0.5e0 * (u1 * u1))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

   1. associate-*r* 57.0%

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

   2. add-log-exp 57.0%

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

3. Applied egg-rr 57.0%

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

4. Taylor expanded in u1 around 0 87.8%

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

   1. +-commutative 87.8%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

   2. mul-1-neg 87.8%

      \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

   3. unsub-neg 87.8%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

   4. unpow2 87.8%

      \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

   5. associate-*r* 87.8%

      \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

6. Simplified 87.8%

   \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot \log \left(e^{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

7. Taylor expanded in u2 around 0 71.4%

   \[\leadsto \color{blue}{\sqrt{u1 - -0.5 \cdot {u1}^{2}}} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 71.4%

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

   2. metadata-eval 71.4%

      \[\leadsto \sqrt{u1 + \color{blue}{0.5} \cdot {u1}^{2}} \]

   3. unpow2 71.4%

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

9. Simplified 71.4%

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

10. Final simplification 71.4%

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

Alternative 6: 65.1% 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 57.0%

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

   1. sub-neg 57.0%

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

   2. log1p-def 99.1%

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

   3. associate-*l* 99.1%

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

3. Simplified 99.1%

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

   1. add-cbrt-cube 99.0%

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

   2. pow1/3 95.4%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{0.3333333333333333}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

5. Applied egg-rr 73.1%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

6. Taylor expanded in u1 around 0 77.1%

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

7. Taylor expanded in u2 around 0 63.7%

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

8. Final simplification 63.7%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023199 
(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))))