Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 10.3s

Alternatives: 10

Speedup: 1.0×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

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

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, 0.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf(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))) * cos(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))) * cos(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))) * cos(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263))));
end

Derivation

1. Initial program 99.0%

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

   1. add-sqr-sqrt 98.8%

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

   2. sqrt-unprod 99.0%

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

   3. *-commutative 99.0%

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

   4. *-commutative 99.0%

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

   5. swap-sqr 98.9%

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

   6. pow2 98.9%

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

   7. metadata-eval 99.0%

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

4. Applied egg-rr 99.0%

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

Alternative 2: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \leq 0.9999995231628418:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9999995231628418)
     (* t_0 (sqrt (* u1 (+ u1 1.0))))
     (pow (+ (/ 1.0 u1) -1.0) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9999995231628418f) {
		tmp = t_0 * sqrtf((u1 * (u1 + 1.0f)));
	} else {
		tmp = powf(((1.0f / u1) + -1.0f), -0.5f);
	}
	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((u2 * 6.28318530718e0))
    if (t_0 <= 0.9999995231628418e0) then
        tmp = t_0 * sqrt((u1 * (u1 + 1.0e0)))
    else
        tmp = ((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(6.28318530718)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999995231628418))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	else
		tmp = Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((u2 * single(6.28318530718)));
	tmp = single(0.0);
	if (t_0 <= single(0.9999995231628418))
		tmp = t_0 * sqrt((u1 * (u1 + single(1.0))));
	else
		tmp = ((single(1.0) / u1) + single(-1.0)) ^ single(-0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999999523

   1. Initial program 98.3%

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

   3. Taylor expanded in u1 around 0 87.9%

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

      1. +-commutative 87.9%

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

   5. Simplified 87.9%

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

   if 0.999999523 < (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. clear-num 99.3%

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

      2. sqrt-div 99.0%

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

      3. metadata-eval 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. *-un-lft-identity 99.0%

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

      2. inv-pow 99.0%

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

      3. sqrt-pow2 99.6%

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

      4. div-sub 99.5%

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

      5. *-inverses 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

      8. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. *-lft-identity 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in u2 around 0 99.2%

      \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999995231628418:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \leq 0.999983012676239:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.999983012676239)
     (* t_0 (sqrt u1))
     (pow (+ (/ 1.0 u1) -1.0) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.999983012676239f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = powf(((1.0f / u1) + -1.0f), -0.5f);
	}
	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((u2 * 6.28318530718e0))
    if (t_0 <= 0.999983012676239e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = ((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(6.28318530718)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.999983012676239))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((u2 * single(6.28318530718)));
	tmp = single(0.0);
	if (t_0 <= single(0.999983012676239))
		tmp = t_0 * sqrt(u1);
	else
		tmp = ((single(1.0) / u1) + single(-1.0)) ^ single(-0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999983013

   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 76.2%

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

   if 0.999983013 < (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. clear-num 99.3%

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

      2. sqrt-div 98.9%

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

      3. metadata-eval 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. inv-pow 98.9%

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

      3. sqrt-pow2 99.5%

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

      4. div-sub 99.4%

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

      5. *-inverses 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

      8. metadata-eval 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. *-lft-identity 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in u2 around 0 97.7%

      \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.999983012676239:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Final simplification 99.0%

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

Alternative 5: 79.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((1.0e0 / ((1.0e0 / u1) + (-1.0e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. clear-num 98.9%

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

   2. sqrt-div 98.6%

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

   3. metadata-eval 98.6%

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

4. Applied egg-rr 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. inv-pow 98.6%

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

   3. sqrt-pow2 99.0%

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

   4. div-sub 98.9%

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

   5. *-inverses 98.9%

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

   6. sub-neg 98.9%

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

   7. metadata-eval 98.9%

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

   8. metadata-eval 98.9%

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

6. Applied egg-rr 98.9%

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

   1. *-lft-identity 98.9%

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

8. Simplified 98.9%

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

9. Taylor expanded in u2 around 0 77.4%

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

10. Final simplification 77.4%

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

Alternative 6: 79.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((single(1.0) / u1) + single(-1.0)) ^ single(-0.5);
end

Derivation

1. Initial program 99.0%

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

   1. clear-num 98.9%

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

   2. sqrt-div 98.6%

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

   3. metadata-eval 98.6%

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

4. Applied egg-rr 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. inv-pow 98.6%

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

   3. sqrt-pow2 99.0%

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

   4. div-sub 98.9%

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

   5. *-inverses 98.9%

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

   6. sub-neg 98.9%

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

   7. metadata-eval 98.9%

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

   8. metadata-eval 98.9%

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

6. Applied egg-rr 98.9%

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

   1. *-lft-identity 98.9%

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

8. Simplified 98.9%

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

9. Taylor expanded in u2 around 0 77.5%

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

10. Final simplification 77.5%

    \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 7: 79.7% 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 99.0%

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

3. Taylor expanded in u2 around 0 77.4%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Add Preprocessing

Alternative 8: 71.2% 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 99.0%

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

3. Taylor expanded in u2 around 0 77.4%

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

4. Taylor expanded in u1 around 0 70.5%

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

   1. +-commutative 87.6%

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

6. Simplified 70.5%

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

Alternative 9: 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 99.0%

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

3. Taylor expanded in u2 around 0 77.4%

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

4. Taylor expanded in u1 around 0 64.5%

   \[\leadsto \sqrt{\color{blue}{u1}} \]
5. Add Preprocessing

Alternative 10: 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 99.0%

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

3. Taylor expanded in u2 around 0 77.4%

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

4. Taylor expanded in u1 around 0 70.5%

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

   1. +-commutative 87.6%

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

6. Simplified 70.5%

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

7. Taylor expanded in u1 around inf 19.9%

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

   1. distribute-rgt-in 19.9%

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

   2. *-lft-identity 19.9%

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

   3. associate-*l* 19.9%

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

   4. lft-mult-inverse 19.9%

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

   5. metadata-eval 19.9%

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

9. Simplified 19.9%

   \[\leadsto \color{blue}{u1 + 0.5} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024114 
(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))))