Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 12.2s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Final simplification 99.1%

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

Alternative 2: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ t_1 := \frac{1}{u1} + -1\\ \mathbf{if}\;t\_0 \leq 0.999019980430603:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-39.47841760436263 \cdot \left(u2 \cdot \frac{u2}{t\_1}\right) + \frac{1}{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))) (t_1 (+ (/ 1.0 u1) -1.0)))
   (if (<= t_0 0.999019980430603)
     (* t_0 (sqrt (* u1 (+ u1 1.0))))
     (sqrt (+ (* -39.47841760436263 (* u2 (/ u2 t_1))) (/ 1.0 t_1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float t_1 = (1.0f / u1) + -1.0f;
	float tmp;
	if (t_0 <= 0.999019980430603f) {
		tmp = t_0 * sqrtf((u1 * (u1 + 1.0f)));
	} else {
		tmp = sqrtf(((-39.47841760436263f * (u2 * (u2 / t_1))) + (1.0f / t_1)));
	}
	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) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    t_1 = (1.0e0 / u1) + (-1.0e0)
    if (t_0 <= 0.999019980430603e0) then
        tmp = t_0 * sqrt((u1 * (u1 + 1.0e0)))
    else
        tmp = sqrt((((-39.47841760436263e0) * (u2 * (u2 / t_1))) + (1.0e0 / t_1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	t_1 = Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.999019980430603))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	else
		tmp = sqrt(Float32(Float32(Float32(-39.47841760436263) * Float32(u2 * Float32(u2 / t_1))) + Float32(Float32(1.0) / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	t_1 = (single(1.0) / u1) + single(-1.0);
	tmp = single(0.0);
	if (t_0 <= single(0.999019980430603))
		tmp = t_0 * sqrt((u1 * (u1 + single(1.0))));
	else
		tmp = sqrt(((single(-39.47841760436263) * (u2 * (u2 / t_1))) + (single(1.0) / t_1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.99901998

   1. Initial program 98.2%

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

   3. Taylor expanded in u1 around 0 86.9%

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

      1. +-commutative 34.4%

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

   5. Simplified 86.9%

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

   if 0.99901998 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 98.2%

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

      2. sqrt-unprod 99.5%

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

      3. swap-sqr 99.4%

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

      4. add-sqr-sqrt 99.6%

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

      5. pow2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. *-lft-identity 99.6%

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

      2. associate-*l/ 99.3%

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

      3. associate-/r/ 99.3%

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

      4. associate-*l/ 99.3%

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

      5. *-lft-identity 99.3%

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

      6. div-sub 99.1%

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

      7. sub-neg 99.1%

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

      8. *-inverses 99.1%

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

      9. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in u2 around 0 98.8%

      \[\leadsto \sqrt{\color{blue}{-39.47841760436263 \cdot \frac{{u2}^{2}}{\frac{1}{u1} - 1} + \frac{1}{\frac{1}{u1} - 1}}} \]
   8. Step-by-step derivation

      1. unpow2 98.8%

         \[\leadsto \sqrt{-39.47841760436263 \cdot \frac{\color{blue}{u2 \cdot u2}}{\frac{1}{u1} - 1} + \frac{1}{\frac{1}{u1} - 1}} \]

      2. associate-/l* 98.8%

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

      3. sub-neg 98.8%

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

      4. metadata-eval 98.8%

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

   9. Applied egg-rr 98.8%

      \[\leadsto \sqrt{-39.47841760436263 \cdot \color{blue}{\left(u2 \cdot \frac{u2}{\frac{1}{u1} + -1}\right)} + \frac{1}{\frac{1}{u1} - 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.999019980430603:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-39.47841760436263 \cdot \left(u2 \cdot \frac{u2}{\frac{1}{u1} + -1}\right) + \frac{1}{\frac{1}{u1} + -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{u1} + -1\\ t_1 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_1 \leq 0.9869999885559082:\\ \;\;\;\;t\_1 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-39.47841760436263 \cdot \left(u2 \cdot \frac{u2}{t\_0}\right) + \frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (+ (/ 1.0 u1) -1.0)) (t_1 (cos (* 6.28318530718 u2))))
   (if (<= t_1 0.9869999885559082)
     (* t_1 (sqrt u1))
     (sqrt (+ (* -39.47841760436263 (* u2 (/ u2 t_0))) (/ 1.0 t_0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (1.0f / u1) + -1.0f;
	float t_1 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_1 <= 0.9869999885559082f) {
		tmp = t_1 * sqrtf(u1);
	} else {
		tmp = sqrtf(((-39.47841760436263f * (u2 * (u2 / t_0))) + (1.0f / t_0)));
	}
	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) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = (1.0e0 / u1) + (-1.0e0)
    t_1 = cos((6.28318530718e0 * u2))
    if (t_1 <= 0.9869999885559082e0) then
        tmp = t_1 * sqrt(u1)
    else
        tmp = sqrt((((-39.47841760436263e0) * (u2 * (u2 / t_0))) + (1.0e0 / t_0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))
	t_1 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.9869999885559082))
		tmp = Float32(t_1 * sqrt(u1));
	else
		tmp = sqrt(Float32(Float32(Float32(-39.47841760436263) * Float32(u2 * Float32(u2 / t_0))) + Float32(Float32(1.0) / t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = (single(1.0) / u1) + single(-1.0);
	t_1 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_1 <= single(0.9869999885559082))
		tmp = t_1 * sqrt(u1);
	else
		tmp = sqrt(((single(-39.47841760436263) * (u2 * (u2 / t_0))) + (single(1.0) / t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.986999989

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0 76.8%

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

   if 0.986999989 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.4%

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

      1. add-sqr-sqrt 98.2%

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

      2. sqrt-unprod 99.4%

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

      3. swap-sqr 99.4%

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

      4. add-sqr-sqrt 99.6%

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

      5. pow2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. *-lft-identity 99.6%

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

      2. associate-*l/ 99.3%

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

      3. associate-/r/ 99.2%

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

      4. associate-*l/ 99.2%

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

      5. *-lft-identity 99.2%

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

      6. div-sub 99.1%

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

      7. sub-neg 99.1%

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

      8. *-inverses 99.1%

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

      9. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in u2 around 0 97.4%

      \[\leadsto \sqrt{\color{blue}{-39.47841760436263 \cdot \frac{{u2}^{2}}{\frac{1}{u1} - 1} + \frac{1}{\frac{1}{u1} - 1}}} \]
   8. Step-by-step derivation

      1. unpow2 97.4%

         \[\leadsto \sqrt{-39.47841760436263 \cdot \frac{\color{blue}{u2 \cdot u2}}{\frac{1}{u1} - 1} + \frac{1}{\frac{1}{u1} - 1}} \]

      2. associate-/l* 97.4%

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

      3. sub-neg 97.4%

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

      4. metadata-eval 97.4%

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

   9. Applied egg-rr 97.4%

      \[\leadsto \sqrt{-39.47841760436263 \cdot \color{blue}{\left(u2 \cdot \frac{u2}{\frac{1}{u1} + -1}\right)} + \frac{1}{\frac{1}{u1} - 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9869999885559082:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-39.47841760436263 \cdot \left(u2 \cdot \frac{u2}{\frac{1}{u1} + -1}\right) + \frac{1}{\frac{1}{u1} + -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{u1} + -1\\ \sqrt{-39.47841760436263 \cdot \left(u2 \cdot \frac{u2}{t\_0}\right) + \frac{1}{t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (+ (/ 1.0 u1) -1.0)))
   (sqrt (+ (* -39.47841760436263 (* u2 (/ u2 t_0))) (/ 1.0 t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (1.0f / u1) + -1.0f;
	return sqrtf(((-39.47841760436263f * (u2 * (u2 / t_0))) + (1.0f / t_0)));
}

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) :: t_0
    t_0 = (1.0e0 / u1) + (-1.0e0)
    code = sqrt((((-39.47841760436263e0) * (u2 * (u2 / t_0))) + (1.0e0 / t_0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))
	return sqrt(Float32(Float32(Float32(-39.47841760436263) * Float32(u2 * Float32(u2 / t_0))) + Float32(Float32(1.0) / t_0)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = (single(1.0) / u1) + single(-1.0);
	tmp = sqrt(((single(-39.47841760436263) * (u2 * (u2 / t_0))) + (single(1.0) / t_0)));
end

Derivation

1. Initial program 99.1%

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

   1. add-sqr-sqrt 93.9%

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

   2. sqrt-unprod 95.2%

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

   3. swap-sqr 95.1%

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

   4. add-sqr-sqrt 95.2%

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

   5. pow2 95.2%

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

4. Applied egg-rr 95.2%

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

   1. *-lft-identity 95.2%

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

   2. associate-*l/ 95.0%

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

   3. associate-/r/ 95.0%

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

   4. associate-*l/ 95.0%

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

   5. *-lft-identity 95.0%

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

   6. div-sub 94.9%

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

   7. sub-neg 94.9%

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

   8. *-inverses 94.9%

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

   9. metadata-eval 94.9%

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

6. Simplified 94.9%

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

7. Taylor expanded in u2 around 0 85.0%

   \[\leadsto \sqrt{\color{blue}{-39.47841760436263 \cdot \frac{{u2}^{2}}{\frac{1}{u1} - 1} + \frac{1}{\frac{1}{u1} - 1}}} \]
8. Step-by-step derivation

   1. unpow2 85.0%

      \[\leadsto \sqrt{-39.47841760436263 \cdot \frac{\color{blue}{u2 \cdot u2}}{\frac{1}{u1} - 1} + \frac{1}{\frac{1}{u1} - 1}} \]

   2. associate-/l* 85.0%

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

   3. sub-neg 85.0%

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

   4. metadata-eval 85.0%

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

9. Applied egg-rr 85.0%

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

10. Final simplification 85.0%

    \[\leadsto \sqrt{-39.47841760436263 \cdot \left(u2 \cdot \frac{u2}{\frac{1}{u1} + -1}\right) + \frac{1}{\frac{1}{u1} + -1}} \]
11. Add Preprocessing

Alternative 5: 71.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = sqrt((u1 * (u1 + 1.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0 77.9%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

4. Taylor expanded in u1 around 0 68.8%

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

   1. +-commutative 68.8%

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

6. Simplified 68.8%

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

7. Final simplification 68.8%

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

Alternative 6: 80.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0 77.9%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

4. Final simplification 77.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0 77.9%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

4. Taylor expanded in u1 around 0 61.4%

   \[\leadsto \color{blue}{\sqrt{u1}} \]

5. Final simplification 61.4%

   \[\leadsto \sqrt{u1} \]
6. Add Preprocessing

Alternative 8: 20.2% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ u1 + 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (+ u1 0.5))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0 77.9%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

4. Taylor expanded in u1 around 0 68.8%

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

   1. +-commutative 68.8%

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

6. Simplified 68.8%

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

7. Taylor expanded in u1 around inf 20.5%

   \[\leadsto \color{blue}{u1 \cdot \left(1 + 0.5 \cdot \frac{1}{u1}\right)} \]
8. Step-by-step derivation

   1. distribute-lft-in 20.5%

      \[\leadsto \color{blue}{u1 \cdot 1 + u1 \cdot \left(0.5 \cdot \frac{1}{u1}\right)} \]

   2. *-rgt-identity 20.5%

      \[\leadsto \color{blue}{u1} + u1 \cdot \left(0.5 \cdot \frac{1}{u1}\right) \]

   3. *-commutative 20.5%

      \[\leadsto u1 + u1 \cdot \color{blue}{\left(\frac{1}{u1} \cdot 0.5\right)} \]

   4. associate-*r* 20.5%

      \[\leadsto u1 + \color{blue}{\left(u1 \cdot \frac{1}{u1}\right) \cdot 0.5} \]

   5. *-inverses 20.5%

      \[\leadsto u1 + \left(u1 \cdot \frac{\color{blue}{\frac{u1}{u1}}}{u1}\right) \cdot 0.5 \]

   6. associate-/r* 20.5%

      \[\leadsto u1 + \left(u1 \cdot \color{blue}{\frac{u1}{u1 \cdot u1}}\right) \cdot 0.5 \]

   7. unpow2 20.5%

      \[\leadsto u1 + \left(u1 \cdot \frac{u1}{\color{blue}{{u1}^{2}}}\right) \cdot 0.5 \]

   8. associate-/l* 20.5%

      \[\leadsto u1 + \color{blue}{\frac{u1 \cdot u1}{{u1}^{2}}} \cdot 0.5 \]

   9. unpow2 20.5%

      \[\leadsto u1 + \frac{\color{blue}{{u1}^{2}}}{{u1}^{2}} \cdot 0.5 \]

   10. *-inverses 20.5%

       \[\leadsto u1 + \color{blue}{1} \cdot 0.5 \]

   11. metadata-eval 20.5%

       \[\leadsto u1 + \color{blue}{0.5} \]

9. Simplified 20.5%

   \[\leadsto \color{blue}{u1 + 0.5} \]

10. Final simplification 20.5%

    \[\leadsto u1 + 0.5 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :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))) (cos (* 6.28318530718 u2))))