Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.2% → 99.0%

Time: 3.7s

Alternatives: 13

Speedup: 1.8×

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: 56.2% 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

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

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) (log1p (- u0))))

C

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

Julia

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

Derivation

1. Initial program 56.2%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Step-by-step derivation

3. Applied rewrites 99.0%

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

Alternative 2: 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.2%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Step-by-step derivation

3. Applied rewrites 56.2%

   \[\leadsto -\left(\log \left(1 - u0\right) \cdot \alpha\right) \cdot \alpha \]
4. Step-by-step derivation

5. Applied rewrites 99.0%

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

Alternative 3: 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

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.996999979019165:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\alpha, u0, \left(0.5 \cdot u0\right) \cdot \left(u0 \cdot \alpha\right)\right) \cdot \alpha\\ \end{array} \]

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) 0.996999979019165)
  (* (* (- alpha) alpha) (log (- 1.0 u0)))
  (* (fma alpha u0 (* (* 0.5 u0) (* u0 alpha))) alpha)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.996999979019165f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = fmaf(alpha, u0, ((0.5f * u0) * (u0 * alpha))) * alpha;
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.996999979019165))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(fma(alpha, u0, Float32(Float32(Float32(0.5) * u0) * Float32(u0 * alpha))) * alpha);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996999979

   1. Initial program 56.2%

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

   if 0.996999979 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 56.2%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 87.1%

      \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, {\alpha}^{2} \cdot u0, {\alpha}^{2}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 87.3%

      \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(0.5 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 87.0%

      \[\leadsto \left(\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot u0\right) \cdot \alpha\right) \cdot \alpha \]
   9. Step-by-step derivation

   10. Applied rewrites 87.3%

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

Alternative 4: 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

\[\begin{array}{l} t_0 := \left(-\alpha\right) \cdot \alpha\\ \mathbf{if}\;1 - u0 \leq 0.996999979019165:\\ \;\;\;\;t\_0 \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(u0 \cdot \left(-0.5 \cdot u0\right) - u0\right)\\ \end{array} \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (let* ((t_0 (* (- alpha) alpha)))
  (if (<= (- 1.0 u0) 0.996999979019165)
    (* t_0 (log (- 1.0 u0)))
    (* t_0 (- (* u0 (* -0.5 u0)) u0)))))

C

float code(float alpha, float u0) {
	float t_0 = -alpha * alpha;
	float tmp;
	if ((1.0f - u0) <= 0.996999979019165f) {
		tmp = t_0 * logf((1.0f - u0));
	} else {
		tmp = t_0 * ((u0 * (-0.5f * u0)) - u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
use fmin_fmax_functions
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -alpha * alpha
    if ((1.0e0 - u0) <= 0.996999979019165e0) then
        tmp = t_0 * log((1.0e0 - u0))
    else
        tmp = t_0 * ((u0 * ((-0.5e0) * u0)) - u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(-alpha) * alpha)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.996999979019165))
		tmp = Float32(t_0 * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(t_0 * Float32(Float32(u0 * Float32(Float32(-0.5) * u0)) - u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = -alpha * alpha;
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.996999979019165))
		tmp = t_0 * log((single(1.0) - u0));
	else
		tmp = t_0 * ((u0 * (single(-0.5) * u0)) - u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996999979

   1. Initial program 56.2%

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

   if 0.996999979 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 56.2%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 87.0%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 87.1%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0\right) - u0\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.7% accurate, 0.8× 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

\[\begin{array}{l} \mathbf{if}\;u0 \leq 0.002285516122356057:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0\right) - u0\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\log \left(1 - u0\right) \cdot \alpha\right) \cdot \alpha\\ \end{array} \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (if (<= u0 0.002285516122356057)
  (* (* (- alpha) alpha) (- (* u0 (* -0.5 u0)) u0))
  (- (* (* (log (- 1.0 u0)) alpha) alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.002285516122356057f) {
		tmp = (-alpha * alpha) * ((u0 * (-0.5f * u0)) - u0);
	} else {
		tmp = -((logf((1.0f - u0)) * alpha) * alpha);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
use fmin_fmax_functions
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if (u0 <= 0.002285516122356057e0) then
        tmp = (-alpha * alpha) * ((u0 * ((-0.5e0) * u0)) - u0)
    else
        tmp = -((log((1.0e0 - u0)) * alpha) * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.002285516122356057))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * Float32(Float32(u0 * Float32(Float32(-0.5) * u0)) - u0));
	else
		tmp = Float32(-Float32(Float32(log(Float32(Float32(1.0) - u0)) * alpha) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.002285516122356057))
		tmp = (-alpha * alpha) * ((u0 * (single(-0.5) * u0)) - u0);
	else
		tmp = -((log((single(1.0) - u0)) * alpha) * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00228551612

   1. Initial program 56.2%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 87.0%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-0.5 \cdot u0 - 1\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 87.1%

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

   if 0.00228551612 < u0

   1. Initial program 56.2%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Step-by-step derivation

   3. Applied rewrites 56.2%

      \[\leadsto -\left(\log \left(1 - u0\right) \cdot \alpha\right) \cdot \alpha \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.6% accurate, 0.8× 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

\[\begin{array}{l} \mathbf{if}\;u0 \leq 0.002285516122356057:\\ \;\;\;\;\left(u0 \cdot \left(\alpha + 0.5 \cdot \left(\alpha \cdot u0\right)\right)\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;-\left(\log \left(1 - u0\right) \cdot \alpha\right) \cdot \alpha\\ \end{array} \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (if (<= u0 0.002285516122356057)
  (* (* u0 (+ alpha (* 0.5 (* alpha u0)))) alpha)
  (- (* (* (log (- 1.0 u0)) alpha) alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.002285516122356057f) {
		tmp = (u0 * (alpha + (0.5f * (alpha * u0)))) * alpha;
	} else {
		tmp = -((logf((1.0f - u0)) * alpha) * alpha);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
use fmin_fmax_functions
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if (u0 <= 0.002285516122356057e0) then
        tmp = (u0 * (alpha + (0.5e0 * (alpha * u0)))) * alpha
    else
        tmp = -((log((1.0e0 - u0)) * alpha) * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.002285516122356057))
		tmp = Float32(Float32(u0 * Float32(alpha + Float32(Float32(0.5) * Float32(alpha * u0)))) * alpha);
	else
		tmp = Float32(-Float32(Float32(log(Float32(Float32(1.0) - u0)) * alpha) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.002285516122356057))
		tmp = (u0 * (alpha + (single(0.5) * (alpha * u0)))) * alpha;
	else
		tmp = -((log((single(1.0) - u0)) * alpha) * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00228551612

   1. Initial program 56.2%

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

   2. Taylor expanded in u0 around 0

      \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 87.1%

      \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, {\alpha}^{2} \cdot u0, {\alpha}^{2}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 87.3%

      \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(0.5 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 87.0%

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

   9. Taylor expanded in u0 around 0

      \[\leadsto \left(u0 \cdot \left(\alpha + \frac{1}{2} \cdot \left(\alpha \cdot u0\right)\right)\right) \cdot \alpha \]
   10. Step-by-step derivation

   11. Applied rewrites 87.2%

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

   if 0.00228551612 < u0

   1. Initial program 56.2%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Step-by-step derivation

   3. Applied rewrites 56.2%

      \[\leadsto -\left(\log \left(1 - u0\right) \cdot \alpha\right) \cdot \alpha \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 87.2% 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

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

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 (* 0.5 (* alpha u0)))) alpha))

C

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

Fortran

real(4) function code(alpha, u0)
use fmin_fmax_functions
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (u0 * (alpha + (0.5e0 * (alpha * u0)))) * alpha
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = (u0 * (alpha + (single(0.5) * (alpha * u0)))) * alpha;
end

Derivation

1. Initial program 56.2%

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

2. Taylor expanded in u0 around 0

   \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right) \]
3. Step-by-step derivation

4. Applied rewrites 87.1%

   \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, {\alpha}^{2} \cdot u0, {\alpha}^{2}\right) \]
5. Step-by-step derivation

6. Applied rewrites 87.3%

   \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(0.5 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\right) \]
7. Step-by-step derivation

8. Applied rewrites 87.0%

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

9. Taylor expanded in u0 around 0

   \[\leadsto \left(u0 \cdot \left(\alpha + \frac{1}{2} \cdot \left(\alpha \cdot u0\right)\right)\right) \cdot \alpha \]
10. Step-by-step derivation

11. Applied rewrites 87.2%

    \[\leadsto \left(u0 \cdot \left(\alpha + 0.5 \cdot \left(\alpha \cdot u0\right)\right)\right) \cdot \alpha \]
12. Add Preprocessing

Alternative 8: 87.2% 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

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

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 (+ alpha (* 0.5 (* alpha u0))))))

C

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

Fortran

real(4) function code(alpha, u0)
use fmin_fmax_functions
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = u0 * (alpha * (alpha + (0.5e0 * (alpha * u0))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = u0 * (alpha * (alpha + (single(0.5) * (alpha * u0))));
end

Derivation

1. Initial program 56.2%

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

2. Taylor expanded in u0 around 0

   \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right) \]
3. Step-by-step derivation

4. Applied rewrites 87.1%

   \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, {\alpha}^{2} \cdot u0, {\alpha}^{2}\right) \]
5. Step-by-step derivation

6. Applied rewrites 87.0%

   \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{1}{{\alpha}^{-2}} \cdot u0, \frac{1}{{\alpha}^{-2}}\right) \]
7. Step-by-step derivation

8. Applied rewrites 87.0%

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

9. Taylor expanded in u0 around 0

   \[\leadsto u0 \cdot \left(\alpha \cdot \left(\alpha + \frac{1}{2} \cdot \left(\alpha \cdot u0\right)\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 87.2%

    \[\leadsto u0 \cdot \left(\alpha \cdot \left(\alpha + 0.5 \cdot \left(\alpha \cdot u0\right)\right)\right) \]
12. Add Preprocessing

Alternative 9: 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

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

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) (* (fma 0.5 u0 1.0) u0)))

C

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

Julia

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

Derivation

1. Initial program 56.2%

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

2. Taylor expanded in u0 around 0

   \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right) \]
3. Step-by-step derivation

4. Applied rewrites 87.1%

   \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, {\alpha}^{2} \cdot u0, {\alpha}^{2}\right) \]
5. Step-by-step derivation

6. Applied rewrites 87.3%

   \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(0.5 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\right) \]
7. Step-by-step derivation

8. Applied rewrites 87.0%

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

Alternative 10: 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

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

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 0.5 u0 1.0) alpha))))

C

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

Julia

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

Derivation

1. Initial program 56.2%

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

2. Taylor expanded in u0 around 0

   \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right) \]
3. Step-by-step derivation

4. Applied rewrites 87.1%

   \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, {\alpha}^{2} \cdot u0, {\alpha}^{2}\right) \]
5. Step-by-step derivation

6. Applied rewrites 87.0%

   \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, \frac{1}{{\alpha}^{-2}} \cdot u0, \frac{1}{{\alpha}^{-2}}\right) \]
7. Step-by-step derivation

8. Applied rewrites 87.0%

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

Alternative 11: 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.2%

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

2. Taylor expanded in u0 around 0

   \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right) \]
3. Step-by-step derivation

4. Applied rewrites 87.1%

   \[\leadsto u0 \cdot \mathsf{fma}\left(0.5, {\alpha}^{2} \cdot u0, {\alpha}^{2}\right) \]
5. Step-by-step derivation

6. Applied rewrites 87.3%

   \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(0.5 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\right) \]
7. Step-by-step derivation

8. Applied rewrites 87.0%

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

Alternative 12: 74.5% 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 56.2%

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

2. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(-1 \cdot u0\right) \]
3. Step-by-step derivation

4. Applied rewrites 74.4%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(-1 \cdot u0\right) \]
5. Step-by-step derivation

6. Applied rewrites 74.4%

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

Alternative 13: 74.4% 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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.2%

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

2. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(-1 \cdot u0\right) \]
3. Step-by-step derivation

4. Applied rewrites 74.4%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(-1 \cdot u0\right) \]
5. Step-by-step derivation

6. Applied rewrites 74.5%

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

Reproduce

herbie shell --seed 2026084 
(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))))