Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 12.8s

Alternatives: 9

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. 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. swap-sqr 98.1%

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

   4. *-commutative 98.1%

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

   5. pow2 98.1%

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

   6. metadata-eval 98.5%

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

3. Applied egg-rr 98.5%

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

4. Final simplification 98.5%

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

Alternative 2: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.029999999329447746:\\ \;\;\;\;u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.029999999329447746)
   (* u2 (/ 6.28318530718 (sqrt (+ -1.0 (/ 1.0 u1)))))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.029999999329447746f) {
		tmp = u2 * (6.28318530718f / sqrtf((-1.0f + (1.0f / u1))));
	} 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) <= 0.029999999329447746e0) then
        tmp = u2 * (6.28318530718e0 / sqrt(((-1.0e0) + (1.0e0 / u1))))
    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(0.029999999329447746))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) / sqrt(Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)))));
	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(0.029999999329447746))
		tmp = u2 * (single(6.28318530718) / sqrt((single(-1.0) + (single(1.0) / u1))));
	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) < 0.0299999993

   1. Initial program 98.4%

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

   2. Taylor expanded in u2 around 0 94.8%

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

      1. expm1-log1p-u 94.8%

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

      2. expm1-udef 28.0%

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

      3. associate-*r* 28.0%

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

      4. *-commutative 28.0%

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

      5. sqrt-div 28.0%

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

      6. associate-*r/ 28.0%

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

      7. associate-/l* 28.0%

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

      8. sqrt-div 28.0%

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

      9. div-sub 28.0%

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

      10. *-inverses 28.0%

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

      11. sub-neg 28.0%

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

      12. metadata-eval 28.0%

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

      13. +-commutative 28.0%

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

   4. Applied egg-rr 28.0%

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

      1. expm1-def 95.1%

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

      2. expm1-log1p 95.1%

         \[\leadsto \color{blue}{u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}}} \]

   6. Simplified 95.1%

      \[\leadsto \color{blue}{u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}}} \]

   if 0.0299999993 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.8%

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

   2. Taylor expanded in u1 around 0 73.2%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.029999999329447746:\\ \;\;\;\;u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3: 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. Final simplification 98.3%

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

Alternative 4: 81.5% 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. Taylor expanded in u2 around 0 80.7%

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

3. Final simplification 80.7%

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

Alternative 5: 81.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \frac{u2}{\sqrt{-1 + \frac{1}{u1}}} \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (u2 / sqrt(((-1.0e0) + (1.0e0 / u1))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in u2 around 0 80.7%

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

   1. expm1-log1p-u 80.7%

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

   2. expm1-udef 26.9%

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

   3. *-commutative 26.9%

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

   4. sqrt-div 26.9%

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

   5. associate-*r/ 26.9%

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

   6. associate-/l* 26.9%

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

   7. sqrt-div 26.9%

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

   8. div-sub 26.9%

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

   9. *-inverses 26.9%

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

   10. sub-neg 26.9%

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

   11. metadata-eval 26.9%

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

   12. +-commutative 26.9%

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

4. Applied egg-rr 26.9%

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

   1. expm1-def 80.7%

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

   2. expm1-log1p 80.7%

      \[\leadsto 6.28318530718 \cdot \color{blue}{\frac{u2}{\sqrt{-1 + \frac{1}{u1}}}} \]

6. Simplified 80.7%

   \[\leadsto 6.28318530718 \cdot \color{blue}{\frac{u2}{\sqrt{-1 + \frac{1}{u1}}}} \]

7. Final simplification 80.7%

   \[\leadsto 6.28318530718 \cdot \frac{u2}{\sqrt{-1 + \frac{1}{u1}}} \]

Alternative 6: 81.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}} \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = u2 * (6.28318530718e0 / sqrt(((-1.0e0) + (1.0e0 / u1))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in u2 around 0 80.7%

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

   1. expm1-log1p-u 80.7%

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

   2. expm1-udef 31.1%

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

   3. associate-*r* 31.1%

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

   4. *-commutative 31.1%

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

   5. sqrt-div 31.1%

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

   6. associate-*r/ 31.1%

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

   7. associate-/l* 31.1%

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

   8. sqrt-div 31.1%

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

   9. div-sub 31.1%

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

   10. *-inverses 31.1%

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

   11. sub-neg 31.1%

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

   12. metadata-eval 31.1%

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

   13. +-commutative 31.1%

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

4. Applied egg-rr 31.1%

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

   1. expm1-def 80.9%

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

   2. expm1-log1p 80.9%

      \[\leadsto \color{blue}{u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}}} \]

6. Simplified 80.9%

   \[\leadsto \color{blue}{u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}}} \]

7. Final simplification 80.9%

   \[\leadsto u2 \cdot \frac{6.28318530718}{\sqrt{-1 + \frac{1}{u1}}} \]

Alternative 7: 4.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * sqrtf(u1)) * -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 = (u2 * sqrt(u1)) * (-6.28318530718e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in u2 around 0 80.7%

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

3. Taylor expanded in u1 around 0 64.2%

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

   1. add-sqr-sqrt 64.1%

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

   2. sqrt-unprod 64.2%

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

   3. *-commutative 64.2%

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

   4. *-commutative 64.2%

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

   5. swap-sqr 64.2%

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

   6. swap-sqr 64.2%

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

   7. add-sqr-sqrt 64.3%

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

   8. pow2 64.3%

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

   9. metadata-eval 64.3%

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

5. Applied egg-rr 64.3%

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

6. Taylor expanded in u2 around -inf 4.4%

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

7. Final simplification 4.4%

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

Alternative 8: 64.4% 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. Taylor expanded in u2 around 0 80.7%

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

3. Taylor expanded in u1 around 0 64.2%

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

4. Final simplification 64.2%

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

Alternative 9: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * 6.28318530718f) * 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 = (u2 * 6.28318530718e0) * sqrt(u1)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. add-exp-log 96.3%

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

3. Applied egg-rr 96.3%

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

4. Taylor expanded in u1 around 0 72.5%

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

5. Taylor expanded in u2 around 0 64.2%

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

   1. *-commutative 64.2%

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

   2. associate-*r* 64.3%

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

   3. *-commutative 64.3%

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

7. Simplified 64.3%

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

8. Final simplification 64.3%

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

Reproduce

herbie shell --seed 2023301 
(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))))