Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 10.0s

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

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

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. Final simplification 99.1%

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

Alternative 2: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9999995231628418:\\ \;\;\;\;t_0 \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999995231628418)
     (* t_0 (sqrt (* u1 (+ u1 1.0))))
     (sqrt (/ u1 (- 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9999995231628418f) {
		tmp = t_0 * sqrtf((u1 * (u1 + 1.0f)));
	} else {
		tmp = sqrtf((u1 / (1.0f - 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) :: t_0
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    if (t_0 <= 0.9999995231628418e0) then
        tmp = t_0 * sqrt((u1 * (u1 + 1.0e0)))
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999995231628418))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	else
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_0 <= single(0.9999995231628418))
		tmp = t_0 * sqrt((u1 * (u1 + single(1.0))));
	else
		tmp = sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.999999523

   1. Initial program 98.5%

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

      1. clear-num 98.4%

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

      2. associate-/r/ 98.5%

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

   3. Applied egg-rr 98.5%

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

   4. Taylor expanded in u1 around 0 86.8%

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

      1. +-commutative 86.8%

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

   6. Simplified 86.8%

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

   if 0.999999523 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.4%

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

   2. Taylor expanded in u2 around 0 99.1%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999995231628418:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]

Alternative 3: 90.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9999949932098389:\\ \;\;\;\;t_0 \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999949932098389)
     (* t_0 (pow (/ 1.0 u1) -0.5))
     (sqrt (/ u1 (- 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9999949932098389f) {
		tmp = t_0 * powf((1.0f / u1), -0.5f);
	} else {
		tmp = sqrtf((u1 / (1.0f - 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) :: t_0
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    if (t_0 <= 0.9999949932098389e0) then
        tmp = t_0 * ((1.0e0 / u1) ** (-0.5e0))
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999949932098389))
		tmp = Float32(t_0 * (Float32(Float32(1.0) / u1) ^ Float32(-0.5)));
	else
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_0 <= single(0.9999949932098389))
		tmp = t_0 * ((single(1.0) / u1) ^ single(-0.5));
	else
		tmp = sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.999994993

   1. Initial program 98.4%

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

      1. clear-num 98.3%

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

      2. associate-/r/ 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. *-commutative 98.4%

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

      2. sqrt-prod 97.8%

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

      3. associate-*r* 97.9%

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

      4. sqrt-div 97.6%

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

      5. metadata-eval 97.6%

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

      6. div-inv 98.2%

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

      7. associate-/r/ 98.3%

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

      8. expm1-log1p-u 97.0%

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

      9. expm1-udef 78.8%

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

      10. sqrt-undiv 78.8%

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

   5. Applied egg-rr 78.8%

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

      1. expm1-def 97.1%

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

      2. expm1-log1p 98.3%

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

   7. Simplified 98.3%

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

      1. clear-num 98.1%

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

      2. associate-/r/ 98.0%

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

      3. pow1/2 98.0%

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

      4. pow-flip 98.3%

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

      5. div-sub 98.3%

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

      6. *-inverses 98.3%

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

      7. sub-neg 98.3%

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

      8. metadata-eval 98.3%

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

      9. metadata-eval 98.3%

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

   9. Applied egg-rr 98.3%

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

   10. Taylor expanded in u1 around 0 74.9%

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

   if 0.999994993 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.4%

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

   2. Taylor expanded in u2 around 0 98.4%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999949932098389:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]

Alternative 4: 90.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9999949932098389:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999949932098389)
     (* t_0 (sqrt u1))
     (sqrt (/ u1 (- 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9999949932098389f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf((u1 / (1.0f - 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) :: t_0
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    if (t_0 <= 0.9999949932098389e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999949932098389))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_0 <= single(0.9999949932098389))
		tmp = t_0 * sqrt(u1);
	else
		tmp = sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.999994993

   1. Initial program 98.4%

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

   2. Taylor expanded in u1 around 0 74.9%

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

   if 0.999994993 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.4%

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

   2. Taylor expanded in u2 around 0 98.4%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999949932098389:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]

Alternative 5: 71.3% 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. Taylor expanded in u2 around 0 80.0%

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

3. Taylor expanded in u1 around 0 74.4%

   \[\leadsto \sqrt{\color{blue}{u1 + \left({u1}^{2} + {u1}^{3}\right)}} \]
4. Step-by-step derivation

   1. +-commutative 74.4%

      \[\leadsto \sqrt{\color{blue}{\left({u1}^{2} + {u1}^{3}\right) + u1}} \]

   2. unpow2 74.4%

      \[\leadsto \sqrt{\left(\color{blue}{u1 \cdot u1} + {u1}^{3}\right) + u1} \]

   3. cube-mult 74.4%

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

   4. unpow2 74.4%

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

   5. distribute-lft-out 74.4%

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

   6. unpow2 74.4%

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

   7. distribute-rgt1-in 74.4%

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

   8. *-commutative 74.4%

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

   9. *-rgt-identity 74.4%

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

   10. distribute-lft-in 74.2%

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

   11. fma-udef 74.2%

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

5. Simplified 74.2%

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

6. Taylor expanded in u1 around 0 71.8%

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

   1. +-commutative 71.8%

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

8. Simplified 71.8%

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

9. Final simplification 71.8%

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

Alternative 6: 80.0% 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. Taylor expanded in u2 around 0 80.0%

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

3. Final simplification 80.0%

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

Alternative 7: 63.1% 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. Taylor expanded in u2 around 0 80.0%

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

3. Taylor expanded in u1 around 0 64.2%

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

4. Final simplification 64.2%

   \[\leadsto \sqrt{u1} \]

Alternative 8: 20.3% 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. Taylor expanded in u2 around 0 80.0%

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

3. Taylor expanded in u1 around 0 74.4%

   \[\leadsto \sqrt{\color{blue}{u1 + \left({u1}^{2} + {u1}^{3}\right)}} \]
4. Step-by-step derivation

   1. +-commutative 74.4%

      \[\leadsto \sqrt{\color{blue}{\left({u1}^{2} + {u1}^{3}\right) + u1}} \]

   2. unpow2 74.4%

      \[\leadsto \sqrt{\left(\color{blue}{u1 \cdot u1} + {u1}^{3}\right) + u1} \]

   3. cube-mult 74.4%

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

   4. unpow2 74.4%

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

   5. distribute-lft-out 74.4%

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

   6. unpow2 74.4%

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

   7. distribute-rgt1-in 74.4%

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

   8. *-commutative 74.4%

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

   9. *-rgt-identity 74.4%

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

   10. distribute-lft-in 74.2%

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

   11. fma-udef 74.2%

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

5. Simplified 74.2%

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

6. Taylor expanded in u1 around 0 71.8%

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

   1. +-commutative 71.8%

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

8. Simplified 71.8%

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

9. Taylor expanded in u1 around inf 20.4%

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

    1. +-commutative 20.4%

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

11. Simplified 20.4%

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

12. Final simplification 20.4%

    \[\leadsto u1 + 0.5 \]

Reproduce

herbie shell --seed 2023318 
(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))))