Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.1% → 99.0%

Time: 27.7s

Alternatives: 11

Speedup: 2.4×

Specification

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* (* (- alpha) alpha) (log (- 1.0 u0))))

C

float code(float alpha, float u0) {
	return (-alpha * alpha) * logf((1.0f - u0));
}

Fortran

real(4) function code(alpha, u0)
use fmin_fmax_functions
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (-alpha * alpha) * log((1.0e0 - u0))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (-alpha * alpha) * log((single(1.0) - u0));
end

Initial Program: 56.1% accurate, 1.0× speedup

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* (* (- alpha) alpha) (log (- 1.0 u0))))

C

float code(float alpha, float u0) {
	return (-alpha * alpha) * logf((1.0f - u0));
}

Fortran

real(4) function code(alpha, u0)
use fmin_fmax_functions
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (-alpha * alpha) * log((1.0e0 - u0))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (-alpha * alpha) * log((single(1.0) - u0));
end

Alternative 1: 99.0% accurate, 0.9× speedup

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\alpha \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-\alpha\right)\right) \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* alpha (* (log1p (- u0)) (- alpha))))

C

float code(float alpha, float u0) {
	return alpha * (log1pf(-u0) * -alpha);
}

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(log1p(Float32(-u0)) * Float32(-alpha)))
end

Derivation

1. Initial program 56.1%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

3. Applied rewrites 56.1%

   \[\leadsto \alpha \cdot \left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \]

5. Applied rewrites 99.0%

   \[\leadsto \alpha \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-\alpha\right)\right) \]
6. Add Preprocessing

Alternative 2: 96.8% accurate, 0.7× speedup

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9980000257492065:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \mathsf{fma}\left(\alpha, u0, \left(u0 \cdot 0.5\right) \cdot \left(\alpha \cdot u0\right)\right)\\ \end{array} \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (if (<= (- 1.0 u0) 0.9980000257492065)
  (* (* (- alpha) alpha) (log (- 1.0 u0)))
  (* alpha (fma alpha u0 (* (* u0 0.5) (* alpha u0))))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9980000257492065f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = alpha * fmaf(alpha, u0, ((u0 * 0.5f) * (alpha * u0)));
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9980000257492065))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(alpha * fma(alpha, u0, Float32(Float32(u0 * Float32(0.5)) * Float32(alpha * u0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.998000026

   1. Initial program 56.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   if 0.998000026 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 56.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   3. Applied rewrites 56.1%

      \[\leadsto \alpha \cdot \left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \]

   4. Taylor expanded in u0 around 0

      \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + \frac{1}{2} \cdot \left(\alpha \cdot u0\right)\right)\right) \]

   6. Applied rewrites 87.0%

      \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + 0.5 \cdot \left(\alpha \cdot u0\right)\right)\right) \]

   8. Applied rewrites 87.1%

      \[\leadsto \alpha \cdot \mathsf{fma}\left(\alpha, u0, \left(u0 \cdot 0.5\right) \cdot \left(\alpha \cdot u0\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.7% accurate, 0.7× speedup

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9980000257492065:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(\alpha \cdot u0\right) \cdot \left(\alpha \cdot 0.5\right)\right)\\ \end{array} \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (if (<= (- 1.0 u0) 0.9980000257492065)
  (* (* (- alpha) alpha) (log (- 1.0 u0)))
  (* u0 (fma alpha alpha (* (* alpha u0) (* alpha 0.5))))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9980000257492065f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = u0 * fmaf(alpha, alpha, ((alpha * u0) * (alpha * 0.5f)));
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9980000257492065))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(u0 * fma(alpha, alpha, Float32(Float32(alpha * u0) * Float32(alpha * Float32(0.5)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.998000026

   1. Initial program 56.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   if 0.998000026 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 56.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   2. Taylor expanded in u0 around 0

      \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right) \]

   4. Applied rewrites 87.0%

      \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, {\alpha}^{2} \cdot u0, {\alpha}^{2}\right) \]

   6. Applied rewrites 87.1%

      \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(\alpha \cdot u0\right) \cdot \left(\alpha \cdot 0.5\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.6% accurate, 0.7× speedup

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9980000257492065:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot u0, 0.5, u0\right)\\ \end{array} \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (if (<= (- 1.0 u0) 0.9980000257492065)
  (* (* (- alpha) alpha) (log (- 1.0 u0)))
  (* (* alpha alpha) (fma (* u0 u0) 0.5 u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9980000257492065f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = (alpha * alpha) * fmaf((u0 * u0), 0.5f, u0);
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9980000257492065))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(alpha * alpha) * fma(Float32(u0 * u0), Float32(0.5), u0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.998000026

   1. Initial program 56.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   if 0.998000026 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 56.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   2. Taylor expanded in u0 around 0

      \[\leadsto u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{2} \cdot {\alpha}^{2}\right) + {\alpha}^{2}\right) \]

   4. Applied rewrites 91.2%

      \[\leadsto u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, {\alpha}^{2} \cdot u0, 0.5 \cdot {\alpha}^{2}\right), {\alpha}^{2}\right) \]

   6. Applied rewrites 91.3%

      \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)\right) \cdot u0\right) \]

   8. Applied rewrites 91.2%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right) \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \frac{1}{2}, u0\right) \]

   11. Applied rewrites 87.0%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot u0, 0.5, u0\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.6% accurate, 0.7× speedup

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9980000257492065:\\ \;\;\;\;\alpha \cdot \left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot u0, 0.5, u0\right)\\ \end{array} \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (if (<= (- 1.0 u0) 0.9980000257492065)
  (* alpha (* (log (- 1.0 u0)) (- alpha)))
  (* (* alpha alpha) (fma (* u0 u0) 0.5 u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9980000257492065f) {
		tmp = alpha * (logf((1.0f - u0)) * -alpha);
	} else {
		tmp = (alpha * alpha) * fmaf((u0 * u0), 0.5f, u0);
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9980000257492065))
		tmp = Float32(alpha * Float32(log(Float32(Float32(1.0) - u0)) * Float32(-alpha)));
	else
		tmp = Float32(Float32(alpha * alpha) * fma(Float32(u0 * u0), Float32(0.5), u0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.998000026

   1. Initial program 56.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   3. Applied rewrites 56.1%

      \[\leadsto \alpha \cdot \left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \]

   if 0.998000026 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 56.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   2. Taylor expanded in u0 around 0

      \[\leadsto u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{2} \cdot {\alpha}^{2}\right) + {\alpha}^{2}\right) \]

   4. Applied rewrites 91.2%

      \[\leadsto u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, {\alpha}^{2} \cdot u0, 0.5 \cdot {\alpha}^{2}\right), {\alpha}^{2}\right) \]

   6. Applied rewrites 91.3%

      \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)\right) \cdot u0\right) \]

   8. Applied rewrites 91.2%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right) \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \frac{1}{2}, u0\right) \]

   11. Applied rewrites 87.0%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot u0, 0.5, u0\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 87.0% accurate, 1.1× speedup

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot u0, 0.5, u0\right) \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* (* alpha alpha) (fma (* u0 u0) 0.5 u0)))

C

float code(float alpha, float u0) {
	return (alpha * alpha) * fmaf((u0 * u0), 0.5f, u0);
}

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * alpha) * fma(Float32(u0 * u0), Float32(0.5), u0))
end

Derivation

1. Initial program 56.1%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

2. Taylor expanded in u0 around 0

   \[\leadsto u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{2} \cdot {\alpha}^{2}\right) + {\alpha}^{2}\right) \]

4. Applied rewrites 91.2%

   \[\leadsto u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(0.3333333333333333, {\alpha}^{2} \cdot u0, 0.5 \cdot {\alpha}^{2}\right), {\alpha}^{2}\right) \]

6. Applied rewrites 91.3%

   \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)\right) \cdot u0\right) \]

8. Applied rewrites 91.2%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right) \]

9. Taylor expanded in u0 around 0

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot u0, \frac{1}{2}, u0\right) \]

11. Applied rewrites 87.0%

    \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot u0, 0.5, u0\right) \]
12. Add Preprocessing

Alternative 7: 87.0% accurate, 1.1× speedup

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[u0 \cdot \left(\alpha \cdot \mathsf{fma}\left(0.5, \alpha \cdot u0, \alpha\right)\right) \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* u0 (* alpha (fma 0.5 (* alpha u0) alpha))))

C

float code(float alpha, float u0) {
	return u0 * (alpha * fmaf(0.5f, (alpha * u0), alpha));
}

Julia

function code(alpha, u0)
	return Float32(u0 * Float32(alpha * fma(Float32(0.5), Float32(alpha * u0), alpha)))
end

Derivation

1. Initial program 56.1%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

2. Taylor expanded in u0 around 0

   \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right) \]

4. Applied rewrites 87.0%

   \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, {\alpha}^{2} \cdot u0, {\alpha}^{2}\right) \]

6. Applied rewrites 87.0%

   \[\leadsto u0 \cdot \mathsf{fma}\left(u0, 0.5 \cdot \left(\alpha \cdot \alpha\right), \alpha \cdot \alpha\right) \]

8. Applied rewrites 86.8%

   \[\leadsto u0 \cdot \left(\alpha \cdot \left(\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \alpha\right)\right) \]

10. Applied rewrites 87.0%

    \[\leadsto u0 \cdot \left(\alpha \cdot \mathsf{fma}\left(0.5, \alpha \cdot u0, \alpha\right)\right) \]
11. Add Preprocessing

Alternative 8: 86.8% accurate, 1.1× speedup

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[u0 \cdot \left(\alpha \cdot \left(\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \alpha\right)\right) \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* u0 (* alpha (* (fma u0 0.5 1.0) alpha))))

C

float code(float alpha, float u0) {
	return u0 * (alpha * (fmaf(u0, 0.5f, 1.0f) * alpha));
}

Julia

function code(alpha, u0)
	return Float32(u0 * Float32(alpha * Float32(fma(u0, Float32(0.5), Float32(1.0)) * alpha)))
end

Derivation

1. Initial program 56.1%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

2. Taylor expanded in u0 around 0

   \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right) \]

4. Applied rewrites 87.0%

   \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, {\alpha}^{2} \cdot u0, {\alpha}^{2}\right) \]

6. Applied rewrites 87.0%

   \[\leadsto u0 \cdot \mathsf{fma}\left(u0, 0.5 \cdot \left(\alpha \cdot \alpha\right), \alpha \cdot \alpha\right) \]

8. Applied rewrites 86.8%

   \[\leadsto u0 \cdot \left(\alpha \cdot \left(\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \alpha\right)\right) \]
9. Add Preprocessing

Alternative 9: 86.8% accurate, 1.1× speedup

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\alpha \cdot \left(\left(\alpha \cdot u0\right) \cdot \mathsf{fma}\left(u0, 0.5, 1\right)\right) \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* alpha (* (* alpha u0) (fma u0 0.5 1.0))))

C

float code(float alpha, float u0) {
	return alpha * ((alpha * u0) * fmaf(u0, 0.5f, 1.0f));
}

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(Float32(alpha * u0) * fma(u0, Float32(0.5), Float32(1.0))))
end

Derivation

1. Initial program 56.1%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

2. Taylor expanded in u0 around 0

   \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right) \]

4. Applied rewrites 87.0%

   \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, {\alpha}^{2} \cdot u0, {\alpha}^{2}\right) \]

6. Applied rewrites 87.1%

   \[\leadsto \mathsf{fma}\left(\alpha, \alpha \cdot u0, \left(\left(\alpha \cdot u0\right) \cdot \left(\alpha \cdot 0.5\right)\right) \cdot u0\right) \]

8. Applied rewrites 86.8%

   \[\leadsto \alpha \cdot \left(\left(\alpha \cdot u0\right) \cdot \mathsf{fma}\left(u0, 0.5, 1\right)\right) \]
9. Add Preprocessing

Alternative 10: 74.3% accurate, 1.8× speedup

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(-u0\right) \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* (* (- alpha) alpha) (- u0)))

C

float code(float alpha, float u0) {
	return (-alpha * alpha) * -u0;
}

Fortran

real(4) function code(alpha, u0)
use fmin_fmax_functions
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (-alpha * alpha) * -u0
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * Float32(-u0))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (-alpha * alpha) * -u0;
end

Derivation

1. Initial program 56.1%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

2. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(-1 \cdot u0\right) \]

4. Applied rewrites 74.3%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(-1 \cdot u0\right) \]

6. Applied rewrites 74.3%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(-u0\right) \]
7. Add Preprocessing

Alternative 11: 74.3% accurate, 2.4× speedup

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\alpha \cdot \left(\alpha \cdot u0\right) \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* alpha (* alpha u0)))

C

float code(float alpha, float u0) {
	return alpha * (alpha * u0);
}

Fortran

real(4) function code(alpha, u0)
use fmin_fmax_functions
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (alpha * u0)
end function

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(alpha * u0))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * u0);
end

Derivation

1. Initial program 56.1%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

3. Applied rewrites 56.1%

   \[\leadsto \alpha \cdot \left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \]

4. Taylor expanded in u0 around 0

   \[\leadsto \alpha \cdot \left(\alpha \cdot u0\right) \]

6. Applied rewrites 74.3%

   \[\leadsto \alpha \cdot \left(\alpha \cdot u0\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(FPCore (alpha u0)
  :name "Beckmann Distribution sample, tan2theta, alphax == alphay"
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0)) (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* (* (- alpha) alpha) (log (- 1.0 u0))))