Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 11.8s

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt[3]{\left(6.28318530718 \cdot u2\right) \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (cos (cbrt (* (* 6.28318530718 u2) (* 39.47841760436263 (* u2 u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf(cbrtf(((6.28318530718f * u2) * (39.47841760436263f * (u2 * u2)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(cbrt(Float32(Float32(Float32(6.28318530718) * u2) * Float32(Float32(39.47841760436263) * Float32(u2 * u2))))))
end

Derivation

1. Initial program 99.2%

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

   1. add-cbrt-cube_binary32 99.2%

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

3. Applied rewrite-once 99.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt[3]{\left(\left(6.28318530718 \cdot u2\right) \cdot \left(6.28318530718 \cdot u2\right)\right) \cdot \left(6.28318530718 \cdot u2\right)}\right)} \]
4. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. swap-sqr 99.2%

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

   3. metadata-eval 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt[3]{\left(6.28318530718 \cdot u2\right) \cdot \left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}\right) \]

5. Simplified 99.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt[3]{\left(6.28318530718 \cdot u2\right) \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}\right)} \]

6. Final simplification 99.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt[3]{\left(6.28318530718 \cdot u2\right) \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}\right) \]

Alternative 2: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.11999999731779099:\\ \;\;\;\;\left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \cdot \sqrt{\frac{1}{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.11999999731779099)
   (*
    (+ 1.0 (* (* u2 u2) -19.739208802181317))
    (sqrt (/ 1.0 (+ -1.0 (/ 1.0 u1)))))
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.11999999731779099f) {
		tmp = (1.0f + ((u2 * u2) * -19.739208802181317f)) * sqrtf((1.0f / (-1.0f + (1.0f / u1))));
	} else {
		tmp = cosf((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) <= 0.11999999731779099e0) then
        tmp = (1.0e0 + ((u2 * u2) * (-19.739208802181317e0))) * sqrt((1.0e0 / ((-1.0e0) + (1.0e0 / u1))))
    else
        tmp = cos((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(0.11999999731779099))
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))) * sqrt(Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)))));
	else
		tmp = Float32(cos(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(0.11999999731779099))
		tmp = (single(1.0) + ((u2 * u2) * single(-19.739208802181317))) * sqrt((single(1.0) / (single(-1.0) + (single(1.0) / u1))));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.119999997

   1. Initial program 99.3%

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

      1. add-exp-log_binary32 96.6%

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

   3. Applied rewrite-once 96.6%

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

      1. rem-exp-log 99.3%

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

      2. clear-num 99.1%

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

      3. inv-pow 99.1%

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

      4. metadata-eval 99.1%

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

      5. sqrt-pow1 99.2%

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

      6. div-sub 99.2%

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

      7. sub-neg 99.2%

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

      8. *-inverses 99.2%

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

      9. metadata-eval 99.2%

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

      10. metadata-eval 99.2%

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

      11. metadata-eval 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in u2 around 0 98.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} - 1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 98.7%

         \[\leadsto \sqrt{\frac{1}{\frac{1}{u1} - 1}} + \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      2. distribute-rgt1-in 98.6%

         \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      3. unpow2 98.6%

         \[\leadsto \left(-19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      4. sub-neg 98.6%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \]

      5. metadata-eval 98.6%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \]

      6. +-commutative 98.6%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{-1 + \frac{1}{u1}}}} \]

   8. Simplified 98.6%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{-1 + \frac{1}{u1}}}} \]

   if 0.119999997 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.6%

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

   2. Taylor expanded in u1 around 0 79.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.11999999731779099:\\ \;\;\;\;\left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \cdot \sqrt{\frac{1}{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3: 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 99.2%

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

2. Final simplification 99.2%

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

Alternative 4: 88.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \cdot \sqrt{\frac{1}{-1 + \frac{1}{u1}}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (+ 1.0 (* (* u2 u2) -19.739208802181317))
  (sqrt (/ 1.0 (+ -1.0 (/ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (1.0f + ((u2 * u2) * -19.739208802181317f)) * sqrtf((1.0f / (-1.0f + (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 = (1.0e0 + ((u2 * u2) * (-19.739208802181317e0))) * sqrt((1.0e0 / ((-1.0e0) + (1.0e0 / u1))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))) * sqrt(Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(1.0) + ((u2 * u2) * single(-19.739208802181317))) * sqrt((single(1.0) / (single(-1.0) + (single(1.0) / u1))));
end

Derivation

1. Initial program 99.2%

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

   1. add-exp-log_binary32 96.6%

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

3. Applied rewrite-once 96.6%

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

   1. rem-exp-log 99.2%

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

   2. clear-num 99.0%

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

   3. inv-pow 99.0%

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

   4. metadata-eval 99.0%

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

   5. sqrt-pow1 99.0%

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

   6. div-sub 99.1%

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

   7. sub-neg 99.1%

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

   8. *-inverses 99.1%

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

   9. metadata-eval 99.1%

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

   10. metadata-eval 99.1%

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

   11. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in u2 around 0 88.2%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} - 1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
7. Step-by-step derivation

   1. associate-*r* 88.2%

      \[\leadsto \sqrt{\frac{1}{\frac{1}{u1} - 1}} + \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

   2. distribute-rgt1-in 88.2%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

   3. unpow2 88.2%

      \[\leadsto \left(-19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

   4. sub-neg 88.2%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \]

   5. metadata-eval 88.2%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \]

   6. +-commutative 88.2%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{-1 + \frac{1}{u1}}}} \]

8. Simplified 88.2%

   \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{-1 + \frac{1}{u1}}}} \]

9. Final simplification 88.2%

   \[\leadsto \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \cdot \sqrt{\frac{1}{-1 + \frac{1}{u1}}} \]

Alternative 5: 79.9% 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 99.2%

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

2. Taylor expanded in u2 around 0 78.6%

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

3. Final simplification 78.6%

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

Alternative 6: 63.2% 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 99.2%

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

2. Taylor expanded in u2 around 0 78.6%

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

3. Taylor expanded in u1 around 0 62.5%

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

4. Final simplification 62.5%

   \[\leadsto \sqrt{u1} \]

Alternative 7: 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 99.2%

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

2. Taylor expanded in u2 around 0 78.6%

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

3. Taylor expanded in u1 around 0 69.6%

   \[\leadsto \sqrt{\color{blue}{u1 + {u1}^{2}}} \]
4. Step-by-step derivation

   1. +-commutative 69.6%

      \[\leadsto \sqrt{\color{blue}{{u1}^{2} + u1}} \]

   2. unpow2 69.6%

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

   3. fma-udef 69.6%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

5. Simplified 69.6%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

6. Taylor expanded in u1 around inf 20.4%

   \[\leadsto \color{blue}{0.5 + u1} \]
7. Step-by-step derivation

   1. +-commutative 20.4%

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

8. Simplified 20.4%

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

9. Final simplification 20.4%

   \[\leadsto u1 + 0.5 \]

Alternative 8: 19.2% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 1.0)

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in u2 around 0 78.6%

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

4. Applied egg-rr 60.9%

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

5. Taylor expanded in u1 around inf 19.4%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 19.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023297 
(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))))