Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 12.0s

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

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

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

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

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 97.6%

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

   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 98.6%

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0013800000306218863:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0013800000306218863)
   (* 6.28318530718 (* (sqrt (/ u1 (- 1.0 u1))) u2))
   (* (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) <= 0.0013800000306218863f) {
		tmp = 6.28318530718f * (sqrtf((u1 / (1.0f - u1))) * u2);
	} 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) <= 0.0013800000306218863e0) then
        tmp = 6.28318530718e0 * (sqrt((u1 / (1.0e0 - u1))) * u2)
    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(0.0013800000306218863))
		tmp = Float32(Float32(6.28318530718) * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * u2));
	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(0.0013800000306218863))
		tmp = single(6.28318530718) * (sqrt((u1 / (single(1.0) - u1))) * u2);
	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 314159265359/50000000000 u2) < 0.00138000003

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0 98.3%

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

   if 0.00138000003 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.1%

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

      1. clear-num 98.0%

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

      2. associate-/r/ 97.9%

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

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in u1 around 0 86.4%

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

      1. +-commutative 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0013800000306218863:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.009999999776482582)
   (* 6.28318530718 (* (sqrt (/ u1 (- 1.0 u1))) u2))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.009999999776482582f) {
		tmp = 6.28318530718f * (sqrtf((u1 / (1.0f - u1))) * u2);
	} 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) <= 0.009999999776482582e0) then
        tmp = 6.28318530718e0 * (sqrt((u1 / (1.0e0 - u1))) * u2)
    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(0.009999999776482582))
		tmp = Float32(Float32(6.28318530718) * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * u2));
	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(0.009999999776482582))
		tmp = single(6.28318530718) * (sqrt((u1 / (single(1.0) - u1))) * u2);
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00999999978

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0 96.5%

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

   if 0.00999999978 < (*.f32 314159265359/50000000000 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 74.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot \sin \left(u2 \cdot 6.28318530718\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (+ -1.0 (/ 1.0 u1)) -0.5) (sin (* u2 6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf((-1.0f + (1.0f / u1)), -0.5f) * 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 = (((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0)) * sin((u2 * 6.28318530718e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5)) * sin(Float32(u2 * Float32(6.28318530718))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5)) * 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. Step-by-step derivation

   1. add-sqr-sqrt 97.6%

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

   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 98.6%

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

4. Applied egg-rr 98.6%

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

   1. add-sqr-sqrt 95.7%

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

   2. clear-num 95.7%

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

   3. unpow-1 95.7%

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

   4. add-sqr-sqrt 98.6%

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

   5. *-commutative 98.6%

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

   6. sqrt-prod 98.3%

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

   7. metadata-eval 98.2%

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

   8. unpow2 98.2%

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

   9. sqrt-prod 97.7%

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

   10. add-sqr-sqrt 98.2%

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

   11. expm1-log1p-u 98.2%

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

   12. expm1-udef 37.3%

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

6. Applied egg-rr 37.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \sin \left(u2 \cdot 6.28318530718\right)\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 98.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \sin \left(u2 \cdot 6.28318530718\right)\right)\right)} \]

   2. expm1-log1p 98.3%

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

   3. *-commutative 98.3%

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

   4. +-commutative 98.3%

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

8. Simplified 98.3%

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

9. Final simplification 98.3%

   \[\leadsto {\left(-1 + \frac{1}{u1}\right)}^{-0.5} \cdot \sin \left(u2 \cdot 6.28318530718\right) \]
10. Add Preprocessing

Alternative 5: 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 6: 81.7% 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 83.1%

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

4. Final simplification 83.1%

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

Alternative 7: 64.9% 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 83.1%

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

4. Taylor expanded in u1 around 0 65.6%

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

5. Final simplification 65.6%

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

Alternative 8: 19.3% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u1 around 0 85.8%

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

   1. +-commutative 85.8%

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

   2. unpow2 85.8%

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

   3. fma-def 85.8%

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

5. Simplified 85.8%

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

6. Taylor expanded in u2 around 0 73.8%

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

   1. associate-*r* 73.7%

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

   2. *-commutative 73.7%

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

   3. +-commutative 73.7%

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

   4. unpow2 73.7%

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

   5. rem-square-sqrt 73.7%

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

   6. hypot-def 73.7%

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

8. Simplified 73.7%

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

9. Taylor expanded in u1 around inf 19.7%

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

    1. *-commutative 19.7%

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

11. Simplified 19.7%

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

12. Final simplification 19.7%

    \[\leadsto 6.28318530718 \cdot \left(u1 \cdot u2\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(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))))