Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 15.6s

Alternatives: 12

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. pow2 99.0%

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

   4. *-commutative 99.0%

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

   5. unpow-prod-down 99.0%

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

   6. metadata-eval 99.1%

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

4. Applied egg-rr 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \leq 0.9999300241470337:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 - u1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9999300241470337)
     (* t_0 (sqrt u1))
     (pow (/ (- 1.0 u1) u1) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9999300241470337f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = powf(((1.0f - u1) / u1), -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 <= 0.9999300241470337e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = ((1.0e0 - u1) / u1) ** (-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(0.9999300241470337))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = Float32(Float32(Float32(1.0) - u1) / u1) ^ 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(0.9999300241470337))
		tmp = t_0 * sqrt(u1);
	else
		tmp = ((single(1.0) - u1) / u1) ^ 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)) < 0.999930024

   1. Initial program 98.0%

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

   3. Taylor expanded in u1 around 0 75.2%

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

   if 0.999930024 < (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. Taylor expanded in u2 around 0 96.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{1} \]
   4. Step-by-step derivation

      1. pow1/2 96.8%

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

      2. clear-num 96.6%

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

      3. inv-pow 96.6%

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

      4. pow-pow 96.8%

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

      5. metadata-eval 96.8%

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

   5. Applied egg-rr 96.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999300241470337:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 - u1}{u1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.012000000104308128:\\ \;\;\;\;{\left(\frac{1 - u1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.012000000104308128)
   (pow (/ (- 1.0 u1) u1) -0.5)
   (* (cos (* u2 6.28318530718)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.012000000104308128f) {
		tmp = powf(((1.0f - u1) / u1), -0.5f);
	} else {
		tmp = cosf((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) <= 0.012000000104308128e0) then
        tmp = ((1.0e0 - u1) / u1) ** (-0.5e0)
    else
        tmp = cos((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(0.012000000104308128))
		tmp = Float32(Float32(Float32(1.0) - u1) / u1) ^ Float32(-0.5);
	else
		tmp = Float32(cos(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(0.012000000104308128))
		tmp = ((single(1.0) - u1) / u1) ^ single(-0.5);
	else
		tmp = cos((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 #s(literal 314159265359/50000000000 binary32) u2) < 0.0120000001

   1. Initial program 99.5%

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

   3. Taylor expanded in u2 around 0 96.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{1} \]
   4. Step-by-step derivation

      1. pow1/2 96.8%

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

      2. clear-num 96.6%

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

      3. inv-pow 96.6%

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

      4. pow-pow 96.8%

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

      5. metadata-eval 96.8%

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

   5. Applied egg-rr 96.8%

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

   if 0.0120000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.0%

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

   3. Taylor expanded in u1 around 0 86.7%

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

      1. +-commutative 40.9%

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

   5. Simplified 86.7%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.012000000104308128:\\ \;\;\;\;{\left(\frac{1 - u1}{u1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = cos((u2 * single(6.28318530718))) * (((single(1.0) - u1) / 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. Step-by-step derivation

   1. pow1/2 80.8%

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

   2. clear-num 80.7%

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

   3. inv-pow 80.7%

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

   4. pow-pow 80.9%

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

   5. metadata-eval 80.9%

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

4. Applied egg-rr 99.1%

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

5. Final simplification 99.1%

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

Alternative 5: 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 6: 80.4% 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. Taylor expanded in u2 around 0 80.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{1} \]
4. Step-by-step derivation

   1. pow1/2 80.8%

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

   2. clear-num 80.7%

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

   3. inv-pow 80.7%

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

   4. pow-pow 80.9%

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

   5. metadata-eval 80.9%

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

5. Applied egg-rr 80.9%

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

   1. div-sub 80.9%

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

   2. *-inverses 80.9%

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

   3. sub-neg 80.9%

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

   4. metadata-eval 80.9%

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

7. Simplified 80.9%

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

8. Final simplification 80.9%

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

Alternative 7: 80.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((single(1.0) - u1) / 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 80.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{1} \]
4. Step-by-step derivation

   1. pow1/2 80.8%

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

   2. clear-num 80.7%

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

   3. inv-pow 80.7%

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

   4. pow-pow 80.9%

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

   5. metadata-eval 80.9%

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

5. Applied egg-rr 80.9%

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

6. Final simplification 80.9%

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

Alternative 8: 72.0% 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 80.8%

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

4. Taylor expanded in u1 around 0 70.9%

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

   1. +-commutative 70.9%

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

6. Simplified 70.9%

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

7. Final simplification 70.9%

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

Alternative 9: 80.4% 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 80.8%

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

4. Final simplification 80.8%

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

Alternative 10: 63.6% 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 80.8%

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

4. Taylor expanded in u1 around 0 62.2%

   \[\leadsto \sqrt{\color{blue}{u1}} \cdot 1 \]

5. Final simplification 62.2%

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

Alternative 11: 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 99.0%

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

3. Taylor expanded in u1 around 0 84.8%

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

   1. distribute-lft-in 85.0%

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

   2. *-rgt-identity 85.0%

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

   3. unpow2 85.0%

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

5. Simplified 85.0%

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

6. Taylor expanded in u2 around 0 71.0%

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

   1. +-commutative 71.0%

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

   2. unpow2 71.0%

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

   3. fma-undefine 71.0%

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

   4. rem-square-sqrt 70.8%

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

   5. fma-define 70.8%

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

   6. hypot-undefine 70.9%

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

8. Simplified 70.9%

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

9. Taylor expanded in u1 around inf 20.4%

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

    1. distribute-rgt-in 20.4%

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

    2. *-lft-identity 20.4%

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

    3. associate-*l* 20.4%

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

    4. lft-mult-inverse 20.4%

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

    5. metadata-eval 20.4%

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

11. Simplified 20.4%

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

12. Final simplification 20.4%

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

Alternative 12: 19.2% accurate, 209.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = 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 u1 around 0 84.8%

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

   1. distribute-lft-in 85.0%

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

   2. *-rgt-identity 85.0%

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

   3. unpow2 85.0%

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

5. Simplified 85.0%

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

6. Taylor expanded in u2 around 0 71.0%

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

   1. +-commutative 71.0%

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

   2. unpow2 71.0%

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

   3. fma-undefine 71.0%

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

   4. rem-square-sqrt 70.8%

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

   5. fma-define 70.8%

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

   6. hypot-undefine 70.9%

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

8. Simplified 70.9%

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

9. Taylor expanded in u1 around inf 19.4%

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

10. Final simplification 19.4%

    \[\leadsto u1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024069 
(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))))