Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 7.0s

Alternatives: 12

Speedup: 1.0×

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

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2));
}

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 / (1.0e0 - u1))) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin((single(6.28318530718) * u2));
end

Initial Program: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2));
}

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 / (1.0e0 - u1))) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin((single(6.28318530718) * u2));
end

Alternative 1: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{3}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (cbrt (* (pow (/ u1 (- 1.0 u1)) 1.5) (pow (sin (* 6.28318530718 u2)) 3.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	return cbrtf((powf((u1 / (1.0f - u1)), 1.5f) * powf(sinf((6.28318530718f * u2)), 3.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return cbrt(Float32((Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(1.5)) * (sin(Float32(Float32(6.28318530718) * u2)) ^ Float32(3.0))))
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 98.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. add-cbrt-cube 98.4%

      \[\leadsto \sqrt[3]{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \cdot \color{blue}{\sqrt[3]{\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)}} \]

   3. cbrt-unprod 98.2%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)}} \]

   4. add-sqr-sqrt 98.5%

      \[\leadsto \sqrt[3]{\left(\color{blue}{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   5. pow1 98.5%

      \[\leadsto \sqrt[3]{\left(\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   6. pow1/2 98.5%

      \[\leadsto \sqrt[3]{\left({\left(\frac{u1}{1 - u1}\right)}^{1} \cdot \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}\right) \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   7. pow-prod-up 98.4%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{\left(1 + 0.5\right)}} \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   8. metadata-eval 98.4%

      \[\leadsto \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{\color{blue}{1.5}} \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   9. pow3 98.4%

      \[\leadsto \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot \color{blue}{{\sin \left(6.28318530718 \cdot u2\right)}^{3}}} \]

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{3}}} \]
5. Add Preprocessing

Alternative 2: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.11999999731779099:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.11999999731779099)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (* u2 (+ 6.28318530718 (* -41.341702240407926 (* u2 u2)))))
   (* (sqrt (* u1 (+ 1.0 u1))) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.11999999731779099f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
	} else {
		tmp = sqrtf((u1 * (1.0f + u1))) * sinf((6.28318530718f * u2));
	}
	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 ((6.28318530718e0 * u2) <= 0.11999999731779099e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (u2 * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
    else
        tmp = sqrt((u1 * (1.0e0 + u1))) * sin((6.28318530718e0 * u2))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.11999999731779099))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + u1))) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.11999999731779099))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
	else
		tmp = sqrt((u1 * (single(1.0) + u1))) * sin((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.119999997

   1. Initial program 98.6%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0 98.0%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot {u2}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 98.0%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]

   5. Applied egg-rr 98.0%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]

   if 0.119999997 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0 87.7%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.1599999964237213:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.1599999964237213)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (* u2 (+ 6.28318530718 (* -41.341702240407926 (* u2 u2)))))
   (* (sqrt u1) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.1599999964237213f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
	} else {
		tmp = sqrtf(u1) * sinf((6.28318530718f * u2));
	}
	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 ((6.28318530718e0 * u2) <= 0.1599999964237213e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (u2 * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
    else
        tmp = sqrt(u1) * sin((6.28318530718e0 * u2))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.1599999964237213))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.1599999964237213))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
	else
		tmp = sqrt(u1) * sin((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.159999996

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0 97.6%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot {u2}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 97.6%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]

   5. Applied egg-rr 97.6%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]

   if 0.159999996 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0 75.0%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2));
}

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 / (1.0e0 - u1))) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin((single(6.28318530718) * u2));
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 5: 89.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (* u2 (+ 6.28318530718 (* -41.341702240407926 (* u2 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
}

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 / (1.0e0 - u1))) * (u2 * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 89.4%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot {u2}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 89.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]

5. Applied egg-rr 89.4%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]
6. Add Preprocessing

Alternative 6: 81.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (* 6.28318530718 u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (6.28318530718f * u2);
}

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 / (1.0e0 - u1))) * (6.28318530718e0 * u2)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(6.28318530718) * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (single(6.28318530718) * u2);
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 79.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(6.28318530718 \cdot u2\right)} \]
4. Add Preprocessing

Alternative 7: 81.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* (sqrt (/ u1 (- 1.0 u1))) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (sqrtf((u1 / (1.0f - u1))) * u2);
}

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 = 6.28318530718e0 * (sqrt((u1 / (1.0e0 - u1))) * u2)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (sqrt((u1 / (single(1.0) - u1))) * u2);
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 79.3%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
4. Add Preprocessing

Alternative 8: 72.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(\sqrt{u1 \cdot \left(1 + u1\right)} \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* (sqrt (* u1 (+ 1.0 u1))) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (sqrtf((u1 * (1.0f + u1))) * u2);
}

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 = 6.28318530718e0 * (sqrt((u1 * (1.0e0 + u1))) * u2)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + u1))) * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (sqrt((u1 * (single(1.0) + u1))) * u2);
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 79.3%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]

4. Taylor expanded in u1 around 0 70.5%

   \[\leadsto 6.28318530718 \cdot \left(\sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot u2\right) \]
5. Add Preprocessing

Alternative 9: 64.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \cdot \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt u1) (* 6.28318530718 u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1) * (6.28318530718f * u2);
}

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) * (6.28318530718e0 * u2)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(u1) * Float32(Float32(6.28318530718) * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1) * (single(6.28318530718) * u2);
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 73.0%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)} \]

4. Taylor expanded in u2 around 0 62.9%

   \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(6.28318530718 \cdot u2\right)} \]
5. Add Preprocessing

Alternative 10: 64.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(\sqrt{u1} \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* (sqrt u1) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (sqrtf(u1) * u2);
}

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 = 6.28318530718e0 * (sqrt(u1) * u2)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(sqrt(u1) * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (sqrt(u1) * u2);
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 79.3%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]

4. Taylor expanded in u1 around 0 62.9%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{u1} \cdot u2\right)} \]
5. Add Preprocessing

Alternative 11: 20.6% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(u1 \cdot \left(u2 + 0.5 \cdot \frac{u2}{u1}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u1 (+ u2 (* 0.5 (/ u2 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u1 * (u2 + (0.5f * (u2 / 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 = 6.28318530718e0 * (u1 * (u2 + (0.5e0 * (u2 / u1))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u1 * Float32(u2 + Float32(Float32(0.5) * Float32(u2 / u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u1 * (u2 + (single(0.5) * (u2 / u1))));
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 79.3%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]

4. Taylor expanded in u1 around 0 70.5%

   \[\leadsto 6.28318530718 \cdot \left(\sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot u2\right) \]

5. Taylor expanded in u1 around inf 20.8%

   \[\leadsto 6.28318530718 \cdot \color{blue}{\left(u1 \cdot \left(u2 + 0.5 \cdot \frac{u2}{u1}\right)\right)} \]
6. Add Preprocessing

Alternative 12: 19.4% accurate, 41.8× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(u1 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* 6.28318530718 (* u1 u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u1 * u2);
}

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 = 6.28318530718e0 * (u1 * u2)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u1 * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u1 * u2);
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 79.3%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]

4. Taylor expanded in u1 around 0 70.5%

   \[\leadsto 6.28318530718 \cdot \left(\sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot u2\right) \]

5. Taylor expanded in u1 around inf 19.7%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(u1 \cdot u2\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024116 -o generate:simplify
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :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 (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))