Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 9.9s

Alternatives: 13

Speedup: 1.0×

• Rival profile vector overflowed, profile may not be complete

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 98.5%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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 98.4%

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

   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 98.6%

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

   8. metadata-eval 98.6%

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

   9. pow3 98.6%

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

4. Applied egg-rr 98.6%

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

Alternative 2: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0020000000949949026:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{\frac{4 + \left(u1 + 1\right) \cdot \left(-1 - u1\right)}{\left(u1 + 1\right) + 2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0020000000949949026)
   (*
    (* 6.28318530718 u2)
    (sqrt (/ u1 (/ (+ 4.0 (* (+ u1 1.0) (- -1.0 u1))) (+ (+ u1 1.0) 2.0)))))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0020000000949949026f) {
		tmp = (6.28318530718f * u2) * sqrtf((u1 / ((4.0f + ((u1 + 1.0f) * (-1.0f - u1))) / ((u1 + 1.0f) + 2.0f))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	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) <= 0.0020000000949949026e0) then
        tmp = (6.28318530718e0 * u2) * sqrt((u1 / ((4.0e0 + ((u1 + 1.0e0) * ((-1.0e0) - u1))) / ((u1 + 1.0e0) + 2.0e0))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(Float32(u1 / Float32(Float32(Float32(4.0) + Float32(Float32(u1 + Float32(1.0)) * Float32(Float32(-1.0) - u1))) / Float32(Float32(u1 + Float32(1.0)) + Float32(2.0))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.0020000000949949026))
		tmp = (single(6.28318530718) * u2) * sqrt((u1 / ((single(4.0) + ((u1 + single(1.0)) * (single(-1.0) - u1))) / ((u1 + single(1.0)) + single(2.0)))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00200000009

   1. Initial program 98.6%

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

      1. expm1-log1p-u 98.5%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. Applied egg-rr 98.5%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   5. Step-by-step derivation

      1. expm1-undefine 98.5%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} - 1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      2. sub-neg 98.5%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} + \left(-1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      3. log1p-undefine 98.6%

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

      4. rem-exp-log 98.6%

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

      5. associate-+r- 98.6%

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

      6. metadata-eval 98.6%

         \[\leadsto \sqrt{\frac{u1}{\left(\color{blue}{2} - u1\right) + \left(-1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      7. metadata-eval 98.6%

         \[\leadsto \sqrt{\frac{u1}{\left(2 - u1\right) + \color{blue}{-1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. Simplified 98.6%

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

      1. associate-+l- 98.6%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{2 - \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      2. flip-- 98.7%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{2 \cdot 2 - \left(u1 - -1\right) \cdot \left(u1 - -1\right)}{2 + \left(u1 - -1\right)}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      3. metadata-eval 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{4} - \left(u1 - -1\right) \cdot \left(u1 - -1\right)}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      4. sub-neg 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{4 - \color{blue}{\left(u1 + \left(--1\right)\right)} \cdot \left(u1 - -1\right)}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      5. metadata-eval 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + \color{blue}{1}\right) \cdot \left(u1 - -1\right)}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      6. sub-neg 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \color{blue}{\left(u1 + \left(--1\right)\right)}}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      7. metadata-eval 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + \color{blue}{1}\right)}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      8. sub-neg 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + 1\right)}{2 + \color{blue}{\left(u1 + \left(--1\right)\right)}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      9. metadata-eval 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + 1\right)}{2 + \left(u1 + \color{blue}{1}\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   8. Applied egg-rr 98.7%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + 1\right)}{2 + \left(u1 + 1\right)}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   9. Taylor expanded in u2 around 0 98.3%

      \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + 1\right)}{2 + \left(u1 + 1\right)}}} \cdot \color{blue}{\left(6.28318530718 \cdot u2\right)} \]

   if 0.00200000009 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.1%

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

   3. Taylor expanded in u1 around 0 88.0%

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

      1. +-commutative 88.0%

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

   5. Simplified 88.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0020000000949949026:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{\frac{4 + \left(u1 + 1\right) \cdot \left(-1 - u1\right)}{\left(u1 + 1\right) + 2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0024999999441206455:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{\frac{4 + \left(u1 + 1\right) \cdot \left(-1 - u1\right)}{\left(u1 + 1\right) + 2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0024999999441206455)
   (*
    (* 6.28318530718 u2)
    (sqrt (/ u1 (/ (+ 4.0 (* (+ u1 1.0) (- -1.0 u1))) (+ (+ u1 1.0) 2.0)))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0024999999441206455f) {
		tmp = (6.28318530718f * u2) * sqrtf((u1 / ((4.0f + ((u1 + 1.0f) * (-1.0f - u1))) / ((u1 + 1.0f) + 2.0f))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.0024999999441206455e0) then
        tmp = (6.28318530718e0 * u2) * sqrt((u1 / ((4.0e0 + ((u1 + 1.0e0) * ((-1.0e0) - u1))) / ((u1 + 1.0e0) + 2.0e0))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.0024999999441206455))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(Float32(u1 / Float32(Float32(Float32(4.0) + Float32(Float32(u1 + Float32(1.0)) * Float32(Float32(-1.0) - u1))) / Float32(Float32(u1 + Float32(1.0)) + Float32(2.0))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.0024999999441206455))
		tmp = (single(6.28318530718) * u2) * sqrt((u1 / ((single(4.0) + ((u1 + single(1.0)) * (single(-1.0) - u1))) / ((u1 + single(1.0)) + single(2.0)))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00249999994

   1. Initial program 98.6%

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

      1. expm1-log1p-u 98.5%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. Applied egg-rr 98.5%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   5. Step-by-step derivation

      1. expm1-undefine 98.5%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} - 1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      2. sub-neg 98.5%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} + \left(-1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      3. log1p-undefine 98.6%

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

      4. rem-exp-log 98.6%

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

      5. associate-+r- 98.6%

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

      6. metadata-eval 98.6%

         \[\leadsto \sqrt{\frac{u1}{\left(\color{blue}{2} - u1\right) + \left(-1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      7. metadata-eval 98.6%

         \[\leadsto \sqrt{\frac{u1}{\left(2 - u1\right) + \color{blue}{-1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. Simplified 98.6%

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

      1. associate-+l- 98.6%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{2 - \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      2. flip-- 98.7%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{2 \cdot 2 - \left(u1 - -1\right) \cdot \left(u1 - -1\right)}{2 + \left(u1 - -1\right)}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      3. metadata-eval 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{4} - \left(u1 - -1\right) \cdot \left(u1 - -1\right)}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      4. sub-neg 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{4 - \color{blue}{\left(u1 + \left(--1\right)\right)} \cdot \left(u1 - -1\right)}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      5. metadata-eval 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + \color{blue}{1}\right) \cdot \left(u1 - -1\right)}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      6. sub-neg 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \color{blue}{\left(u1 + \left(--1\right)\right)}}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      7. metadata-eval 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + \color{blue}{1}\right)}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      8. sub-neg 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + 1\right)}{2 + \color{blue}{\left(u1 + \left(--1\right)\right)}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      9. metadata-eval 98.7%

         \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + 1\right)}{2 + \left(u1 + \color{blue}{1}\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   8. Applied egg-rr 98.7%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + 1\right)}{2 + \left(u1 + 1\right)}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   9. Taylor expanded in u2 around 0 98.2%

      \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + 1\right)}{2 + \left(u1 + 1\right)}}} \cdot \color{blue}{\left(6.28318530718 \cdot u2\right)} \]

   if 0.00249999994 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.1%

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

   3. Taylor expanded in u1 around 0 76.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0024999999441206455:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{\frac{4 + \left(u1 + 1\right) \cdot \left(-1 - u1\right)}{\left(u1 + 1\right) + 2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((6.28318530718f * u2)) * sqrtf((u1 / (1.0f - 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 = sin((6.28318530718e0 * u2)) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Final simplification 98.5%

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

Alternative 5: 81.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{\frac{4 + \left(u1 + 1\right) \cdot \left(-1 - u1\right)}{\left(u1 + 1\right) + 2}}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (* 6.28318530718 u2)
  (sqrt (/ u1 (/ (+ 4.0 (* (+ u1 1.0) (- -1.0 u1))) (+ (+ u1 1.0) 2.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.28318530718f * u2) * sqrtf((u1 / ((4.0f + ((u1 + 1.0f) * (-1.0f - u1))) / ((u1 + 1.0f) + 2.0f))));
}

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 * u2) * sqrt((u1 / ((4.0e0 + ((u1 + 1.0e0) * ((-1.0e0) - u1))) / ((u1 + 1.0e0) + 2.0e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(6.28318530718) * u2) * sqrt(Float32(u1 / Float32(Float32(Float32(4.0) + Float32(Float32(u1 + Float32(1.0)) * Float32(Float32(-1.0) - u1))) / Float32(Float32(u1 + Float32(1.0)) + Float32(2.0))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(6.28318530718) * u2) * sqrt((u1 / ((single(4.0) + ((u1 + single(1.0)) * (single(-1.0) - u1))) / ((u1 + single(1.0)) + single(2.0)))));
end

Derivation

1. Initial program 98.5%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

4. Applied egg-rr 98.4%

   \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
5. Step-by-step derivation

   1. expm1-undefine 98.3%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} - 1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. sub-neg 98.3%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} + \left(-1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. log1p-undefine 98.4%

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

   4. rem-exp-log 98.4%

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

   5. associate-+r- 98.4%

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

   6. metadata-eval 98.4%

      \[\leadsto \sqrt{\frac{u1}{\left(\color{blue}{2} - u1\right) + \left(-1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   7. metadata-eval 98.4%

      \[\leadsto \sqrt{\frac{u1}{\left(2 - u1\right) + \color{blue}{-1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

6. Simplified 98.4%

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

   1. associate-+l- 98.4%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{2 - \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. flip-- 98.5%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{2 \cdot 2 - \left(u1 - -1\right) \cdot \left(u1 - -1\right)}{2 + \left(u1 - -1\right)}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. metadata-eval 98.5%

      \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{4} - \left(u1 - -1\right) \cdot \left(u1 - -1\right)}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. sub-neg 98.5%

      \[\leadsto \sqrt{\frac{u1}{\frac{4 - \color{blue}{\left(u1 + \left(--1\right)\right)} \cdot \left(u1 - -1\right)}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. metadata-eval 98.5%

      \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + \color{blue}{1}\right) \cdot \left(u1 - -1\right)}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. sub-neg 98.5%

      \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \color{blue}{\left(u1 + \left(--1\right)\right)}}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   7. metadata-eval 98.5%

      \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + \color{blue}{1}\right)}{2 + \left(u1 - -1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   8. sub-neg 98.5%

      \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + 1\right)}{2 + \color{blue}{\left(u1 + \left(--1\right)\right)}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   9. metadata-eval 98.5%

      \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + 1\right)}{2 + \left(u1 + \color{blue}{1}\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

8. Applied egg-rr 98.5%

   \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + 1\right)}{2 + \left(u1 + 1\right)}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

9. Taylor expanded in u2 around 0 85.0%

   \[\leadsto \sqrt{\frac{u1}{\frac{4 - \left(u1 + 1\right) \cdot \left(u1 + 1\right)}{2 + \left(u1 + 1\right)}}} \cdot \color{blue}{\left(6.28318530718 \cdot u2\right)} \]

10. Final simplification 85.0%

    \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{\frac{4 + \left(u1 + 1\right) \cdot \left(-1 - u1\right)}{\left(u1 + 1\right) + 2}}} \]
11. Add Preprocessing

Alternative 6: 81.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(2 - u1\right) + -1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* 6.28318530718 u2) (sqrt (/ u1 (+ (- 2.0 u1) -1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.28318530718f * u2) * sqrtf((u1 / ((2.0f - u1) + -1.0f)));
}

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 * u2) * sqrt((u1 / ((2.0e0 - u1) + (-1.0e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(6.28318530718) * u2) * sqrt(Float32(u1 / Float32(Float32(Float32(2.0) - u1) + Float32(-1.0)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(6.28318530718) * u2) * sqrt((u1 / ((single(2.0) - u1) + single(-1.0))));
end

Derivation

1. Initial program 98.5%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

4. Applied egg-rr 98.4%

   \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
5. Step-by-step derivation

   1. expm1-undefine 98.3%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} - 1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. sub-neg 98.3%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} + \left(-1\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. log1p-undefine 98.4%

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

   4. rem-exp-log 98.4%

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

   5. associate-+r- 98.4%

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

   6. metadata-eval 98.4%

      \[\leadsto \sqrt{\frac{u1}{\left(\color{blue}{2} - u1\right) + \left(-1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   7. metadata-eval 98.4%

      \[\leadsto \sqrt{\frac{u1}{\left(2 - u1\right) + \color{blue}{-1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

6. Simplified 98.4%

   \[\leadsto \sqrt{\frac{u1}{\color{blue}{\left(2 - u1\right) + -1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

7. Taylor expanded in u2 around 0 84.9%

   \[\leadsto \sqrt{\frac{u1}{\left(2 - u1\right) + -1}} \cdot \color{blue}{\left(6.28318530718 \cdot u2\right)} \]

8. Final simplification 84.9%

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

Alternative 7: 81.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.28318530718f * u2) * sqrtf((u1 / (1.0f - 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 * u2) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

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 84.8%

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

   1. *-commutative 84.8%

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

   2. associate-*r* 84.9%

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

5. Simplified 84.9%

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

Alternative 8: 81.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * sqrtf((u1 / (1.0f - 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 * (u2 * sqrt((u1 / (1.0e0 - u1))))
end function

Julia

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

MATLAB

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

Derivation

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 84.8%

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

4. Final simplification 84.8%

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

Alternative 9: 73.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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

Julia

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

MATLAB

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

Derivation

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 84.8%

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

4. Taylor expanded in u1 around 0 77.2%

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

   1. +-commutative 88.0%

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

6. Simplified 77.2%

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

7. Final simplification 77.2%

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

Alternative 10: 65.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (6.28318530718e0 * u2) * sqrt(u1)
end function

Julia

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

MATLAB

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

Derivation

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 84.8%

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

4. Taylor expanded in u1 around 0 69.1%

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

5. Taylor expanded in u1 around -inf -0.0%

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

   1. *-commutative -0.0%

      \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot \left(u2 \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot -6.28318530718} \]

   2. associate-*r* -0.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{u1} \cdot u2\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \cdot -6.28318530718 \]

   3. unpow2 -0.0%

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

   4. rem-square-sqrt 69.1%

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

   5. associate-*l* 69.1%

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

   6. metadata-eval 69.1%

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

   7. associate-*l* 69.2%

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

7. Simplified 69.2%

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

8. Final simplification 69.2%

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

Alternative 11: 65.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (u2 * sqrt(u1))
end function

Julia

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

MATLAB

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

Derivation

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 84.8%

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

4. Taylor expanded in u1 around 0 69.1%

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

5. Final simplification 69.1%

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

Alternative 12: 4.8% accurate, 41.8× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(u1 \cdot -3.14159265359\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* u2 (* u1 -3.14159265359)))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (u1 * -3.14159265359f);
}

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 = u2 * (u1 * (-3.14159265359e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(u1 * Float32(-3.14159265359)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (u1 * single(-3.14159265359));
end

Derivation

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 84.8%

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

4. Taylor expanded in u1 around 0 82.0%

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

5. Taylor expanded in u1 around -inf -0.0%

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({u1}^{2} \cdot \left({\left(\sqrt{-1}\right)}^{2} - 0.5 \cdot \frac{1}{u1}\right)\right)} \cdot u2\right) \]
6. Step-by-step derivation

   1. unpow2 -0.0%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} - 0.5 \cdot \frac{1}{u1}\right)\right) \cdot u2\right) \]

   2. rem-square-sqrt 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(\color{blue}{-1} - 0.5 \cdot \frac{1}{u1}\right)\right) \cdot u2\right) \]

   3. associate-*r/ 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(-1 - \color{blue}{\frac{0.5 \cdot 1}{u1}}\right)\right) \cdot u2\right) \]

   4. metadata-eval 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(-1 - \frac{\color{blue}{0.5}}{u1}\right)\right) \cdot u2\right) \]

7. Simplified 4.6%

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({u1}^{2} \cdot \left(-1 - \frac{0.5}{u1}\right)\right)} \cdot u2\right) \]

8. Taylor expanded in u1 around 0 4.6%

   \[\leadsto \color{blue}{-3.14159265359 \cdot \left(u1 \cdot u2\right)} \]
9. Step-by-step derivation

   1. associate-*r* 4.6%

      \[\leadsto \color{blue}{\left(-3.14159265359 \cdot u1\right) \cdot u2} \]

10. Simplified 4.6%

    \[\leadsto \color{blue}{\left(-3.14159265359 \cdot u1\right) \cdot u2} \]

11. Final simplification 4.6%

    \[\leadsto u2 \cdot \left(u1 \cdot -3.14159265359\right) \]
12. Add Preprocessing

Alternative 13: 4.8% accurate, 41.8× speedup

Math

\[\begin{array}{l} \\ -3.14159265359 \cdot \left(u1 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* -3.14159265359 (* u1 u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return -3.14159265359f * (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 = (-3.14159265359e0) * (u1 * u2)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(-3.14159265359) * Float32(u1 * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(-3.14159265359) * (u1 * u2);
end

Derivation

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 84.8%

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

4. Taylor expanded in u1 around 0 82.0%

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

5. Taylor expanded in u1 around -inf -0.0%

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({u1}^{2} \cdot \left({\left(\sqrt{-1}\right)}^{2} - 0.5 \cdot \frac{1}{u1}\right)\right)} \cdot u2\right) \]
6. Step-by-step derivation

   1. unpow2 -0.0%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} - 0.5 \cdot \frac{1}{u1}\right)\right) \cdot u2\right) \]

   2. rem-square-sqrt 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(\color{blue}{-1} - 0.5 \cdot \frac{1}{u1}\right)\right) \cdot u2\right) \]

   3. associate-*r/ 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(-1 - \color{blue}{\frac{0.5 \cdot 1}{u1}}\right)\right) \cdot u2\right) \]

   4. metadata-eval 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(-1 - \frac{\color{blue}{0.5}}{u1}\right)\right) \cdot u2\right) \]

7. Simplified 4.6%

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({u1}^{2} \cdot \left(-1 - \frac{0.5}{u1}\right)\right)} \cdot u2\right) \]

8. Taylor expanded in u1 around 0 4.6%

   \[\leadsto \color{blue}{-3.14159265359 \cdot \left(u1 \cdot u2\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(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))))