Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.5%

Time: 8.8s

Alternatives: 10

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: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (sqrt (* (pow u2 2.0) 39.47841760436263)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf(sqrtf((powf(u2, 2.0f) * 39.47841760436263f)));
}

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(sqrt(((u2 ** 2.0e0) * 39.47841760436263e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(sqrt(Float32((u2 ^ Float32(2.0)) * Float32(39.47841760436263)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263))));
end

Derivation

1. Initial program 98.3%

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

   1. add-sqr-sqrt 97.7%

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

   2. sqrt-unprod 98.3%

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

   3. *-commutative 98.3%

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

   4. *-commutative 98.3%

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

   5. swap-sqr 98.2%

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

   6. pow2 98.2%

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

   7. metadata-eval 98.5%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot \color{blue}{39.47841760436263}}\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)} \]
5. Add Preprocessing

Alternative 2: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0015899999998509884:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{\left(2 - u1\right) + -1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0015899999998509884)
   (* 6.28318530718 (* u2 (sqrt (/ u1 (+ (- 2.0 u1) -1.0)))))
   (* (sin (* u2 6.28318530718)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.0015899999998509884f) {
		tmp = 6.28318530718f * (u2 * sqrtf((u1 / ((2.0f - u1) + -1.0f))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * 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 ((u2 * 6.28318530718e0) <= 0.0015899999998509884e0) then
        tmp = 6.28318530718e0 * (u2 * sqrt((u1 / ((2.0e0 - u1) + (-1.0e0)))))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.0015899999998509884))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 / Float32(Float32(Float32(2.0) - u1) + Float32(-1.0))))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * 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 ((u2 * single(6.28318530718)) <= single(0.0015899999998509884))
		tmp = single(6.28318530718) * (u2 * sqrt((u1 / ((single(2.0) - u1) + single(-1.0)))));
	else
		tmp = sin((u2 * single(6.28318530718))) * 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) < 0.00159

   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 98.4%

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

      1. expm1-log1p-u 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. expm1-undefine 98.2%

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

      2. sub-neg 98.2%

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

      3. log1p-undefine 98.4%

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

      4. rem-exp-log 98.4%

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

      5. associate-+r- 98.4%

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

      6. metadata-eval 98.4%

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

      7. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

   if 0.00159 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0 89.8%

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

      1. +-commutative 89.8%

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

   5. Simplified 89.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0015899999998509884:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{\left(2 - u1\right) + -1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004900000058114529:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{\left(2 - u1\right) + -1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.004900000058114529)
   (* 6.28318530718 (* u2 (sqrt (/ u1 (+ (- 2.0 u1) -1.0)))))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.004900000058114529f) {
		tmp = 6.28318530718f * (u2 * sqrtf((u1 / ((2.0f - u1) + -1.0f))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * 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 ((u2 * 6.28318530718e0) <= 0.004900000058114529e0) then
        tmp = 6.28318530718e0 * (u2 * sqrt((u1 / ((2.0e0 - u1) + (-1.0e0)))))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.004900000058114529))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 / Float32(Float32(Float32(2.0) - u1) + Float32(-1.0))))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.004900000058114529))
		tmp = single(6.28318530718) * (u2 * sqrt((u1 / ((single(2.0) - u1) + single(-1.0)))));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00490000006

   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 97.7%

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

      1. expm1-log1p-u 97.5%

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

   5. Applied egg-rr 97.5%

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

      1. expm1-undefine 97.5%

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

      2. sub-neg 97.5%

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

      3. log1p-undefine 97.7%

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

      4. rem-exp-log 97.7%

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

      5. associate-+r- 97.7%

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

      6. metadata-eval 97.7%

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

      7. metadata-eval 97.7%

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

   7. Simplified 97.7%

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

   if 0.00490000006 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.8%

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

   3. Taylor expanded in u1 around 0 77.3%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004900000058114529:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{\left(2 - u1\right) + -1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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((u2 * 6.28318530718e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Final simplification 98.3%

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

Alternative 5: 81.6% accurate, 1.9× speedup

Math

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

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(6.28318530718) * Float32(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 98.3%

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

3. Taylor expanded in u2 around 0 82.3%

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

   1. expm1-log1p-u 82.2%

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

5. Applied egg-rr 82.2%

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

   1. expm1-undefine 82.2%

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

   2. sub-neg 82.2%

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

   3. log1p-undefine 82.3%

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

   4. rem-exp-log 82.3%

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

   5. associate-+r- 82.3%

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

   6. metadata-eval 82.3%

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

   7. metadata-eval 82.3%

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

7. Simplified 82.3%

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

8. Final simplification 82.3%

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

Alternative 6: 81.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0 82.3%

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

   1. *-commutative 82.3%

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

   2. associate-*r* 82.3%

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

5. Simplified 82.3%

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

6. Final simplification 82.3%

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

Alternative 7: 81.6% 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 98.3%

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

3. Taylor expanded in u2 around 0 82.3%

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

Alternative 8: 73.1% 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 98.3%

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

3. Taylor expanded in u2 around 0 82.3%

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

4. Taylor expanded in u1 around 0 73.7%

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

   1. +-commutative 87.2%

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

6. Simplified 73.7%

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

7. Final simplification 73.7%

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

Alternative 9: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * powf(u1, 0.5f));
}

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 * (u1 ** 0.5e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0 82.3%

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

4. Taylor expanded in u1 around 0 65.3%

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

   1. pow1/2 65.3%

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

6. Applied egg-rr 65.3%

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

7. Final simplification 65.3%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot {u1}^{0.5}\right) \]
8. Add Preprocessing

Alternative 10: 64.7% 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 98.3%

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

3. Taylor expanded in u2 around 0 82.3%

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

4. Taylor expanded in u1 around 0 65.3%

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

5. Final simplification 65.3%

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

Reproduce

herbie shell --seed 2024185 
(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))))