Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.2% → 99.0%

Time: 3.3s

Alternatives: 9

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: 56.2% 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

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

FPCore

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

C

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

Julia

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

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

5. Applied rewrites 56.2%

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

7. Applied rewrites 99.0%

   \[\leadsto \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha \]
8. 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

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9959999918937683:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u0 \cdot 0.5, u0 \cdot \alpha, u0 \cdot \alpha\right) \cdot \alpha\\ \end{array} \]

FPCore

(FPCore (alpha u0)
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0))
     (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (if (<= (- 1.0 u0) 0.9959999918937683)
  (* (* (- alpha) alpha) (log (- 1.0 u0)))
  (* (fma (* u0 0.5) (* u0 alpha) (* u0 alpha)) alpha)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9959999918937683f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = fmaf((u0 * 0.5f), (u0 * alpha), (u0 * alpha)) * alpha;
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9959999918937683))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(fma(Float32(u0 * Float32(0.5)), Float32(u0 * alpha), Float32(u0 * alpha)) * alpha);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.995999992

   1. Initial program 56.2%

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

   if 0.995999992 < (-.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. Step-by-step derivation

   3. Applied rewrites 56.2%

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 87.1%

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

   10. Applied rewrites 87.1%

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

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9959999918937683:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \left(\alpha + 0.5 \cdot \left(\alpha \cdot u0\right)\right)\right) \cdot \alpha\\ \end{array} \]

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) 0.9959999918937683)
  (* (* (- alpha) alpha) (log (- 1.0 u0)))
  (* (* u0 (+ alpha (* 0.5 (* alpha u0)))) alpha)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9959999918937683f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = (u0 * (alpha + (0.5f * (alpha * u0)))) * 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 ((1.0e0 - u0) <= 0.9959999918937683e0) then
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    else
        tmp = (u0 * (alpha + (0.5e0 * (alpha * u0)))) * alpha
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9959999918937683))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(u0 * Float32(alpha + Float32(Float32(0.5) * Float32(alpha * u0)))) * alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9959999918937683))
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	else
		tmp = (u0 * (alpha + (single(0.5) * (alpha * u0)))) * alpha;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.995999992

   1. Initial program 56.2%

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

   if 0.995999992 < (-.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. Step-by-step derivation

   3. Applied rewrites 56.2%

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 87.1%

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

Alternative 5: 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

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9959999918937683:\\ \;\;\;\;\left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \left(\alpha + 0.5 \cdot \left(\alpha \cdot u0\right)\right)\right) \cdot \alpha\\ \end{array} \]

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) 0.9959999918937683)
  (* (* (log (- 1.0 u0)) (- alpha)) alpha)
  (* (* u0 (+ alpha (* 0.5 (* alpha u0)))) alpha)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9959999918937683f) {
		tmp = (logf((1.0f - u0)) * -alpha) * alpha;
	} else {
		tmp = (u0 * (alpha + (0.5f * (alpha * u0)))) * 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 ((1.0e0 - u0) <= 0.9959999918937683e0) then
        tmp = (log((1.0e0 - u0)) * -alpha) * alpha
    else
        tmp = (u0 * (alpha + (0.5e0 * (alpha * u0)))) * alpha
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9959999918937683))
		tmp = Float32(Float32(log(Float32(Float32(1.0) - u0)) * Float32(-alpha)) * alpha);
	else
		tmp = Float32(Float32(u0 * Float32(alpha + Float32(Float32(0.5) * Float32(alpha * u0)))) * alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9959999918937683))
		tmp = (log((single(1.0) - u0)) * -alpha) * alpha;
	else
		tmp = (u0 * (alpha + (single(0.5) * (alpha * u0)))) * alpha;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.995999992

   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%

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

   5. Applied rewrites 56.2%

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

   if 0.995999992 < (-.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. Step-by-step derivation

   3. Applied rewrites 56.2%

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in u0 around 0

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

   8. Applied rewrites 87.1%

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

Alternative 6: 87.1% 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. Step-by-step derivation

3. Applied rewrites 56.2%

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

5. Applied rewrites 56.2%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 87.1%

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

Alternative 7: 87.1% 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

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

FPCore

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

C

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

Julia

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

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

5. Applied rewrites 56.2%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 87.1%

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

10. Applied rewrites 87.1%

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

Alternative 8: 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

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

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)) alpha))

C

float code(float alpha, float u0) {
	return (-u0 * -alpha) * 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) * alpha
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = (-u0 * -alpha) * 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 74.3%

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

6. Applied rewrites 74.3%

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

Alternative 9: 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

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

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(alpha * alpha) * 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

   \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right) \]
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 87.2%

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

7. Taylor expanded in alpha around inf

   \[\leadsto u0 \cdot \left({\alpha}^{4} \cdot {\left(\left|\frac{1}{\alpha}\right|\right)}^{2}\right) \]
8. Step-by-step derivation

9. Applied rewrites 74.0%

   \[\leadsto u0 \cdot \left({\alpha}^{4} \cdot {\left(\left|\frac{1}{\alpha}\right|\right)}^{2}\right) \]

11. Applied rewrites 74.3%

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

Reproduce

herbie shell --seed 2026070 
(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))))