Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 97.2%

Time: 15.7s

Alternatives: 11

Speedup: 1.0×

Specification

\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* (- cosTheta) cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((1.0f / sqrtf(((float) M_PI))) * (sqrtf(((1.0f - cosTheta) - cosTheta)) / cosTheta)) * expf((-cosTheta * cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp(Float32(Float32(-cosTheta) * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) / sqrt(single(pi))) * (sqrt(((single(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp((-cosTheta * cosTheta))));
end

Initial Program: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* (- cosTheta) cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((1.0f / sqrtf(((float) M_PI))) * (sqrtf(((1.0f - cosTheta) - cosTheta)) / cosTheta)) * expf((-cosTheta * cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp(Float32(Float32(-cosTheta) * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) / sqrt(single(pi))) * (sqrt(((single(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp((-cosTheta * cosTheta))));
end

Alternative 1: 97.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := t_0 - c\\ t_2 := 1 - t_1\\ t_3 := \sqrt{{\pi}^{3}}\\ t_4 := t_0 \cdot -1.5\\ t_5 := \mathsf{fma}\left(\pi, t_4, {t_2}^{2} \cdot \left(-t_3\right)\right)\\ \left(\left(cosTheta \cdot \sqrt{\pi} + {cosTheta}^{4} \cdot \left(t_5 \cdot \left(\sqrt{\pi} \cdot t_2\right) + \left(t_2 \cdot \left(t_4 \cdot t_3\right) - 0.5 \cdot \left(\pi \cdot t_0\right)\right)\right)\right) - t_5 \cdot {cosTheta}^{3}\right) + \pi \cdot \left({cosTheta}^{2} \cdot \left(-1 + t_1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI)))
        (t_1 (- t_0 c))
        (t_2 (- 1.0 t_1))
        (t_3 (sqrt (pow PI 3.0)))
        (t_4 (* t_0 -1.5))
        (t_5 (fma PI t_4 (* (pow t_2 2.0) (- t_3)))))
   (+
    (-
     (+
      (* cosTheta (sqrt PI))
      (*
       (pow cosTheta 4.0)
       (+
        (* t_5 (* (sqrt PI) t_2))
        (- (* t_2 (* t_4 t_3)) (* 0.5 (* PI t_0))))))
     (* t_5 (pow cosTheta 3.0)))
    (* PI (* (pow cosTheta 2.0) (+ -1.0 t_1))))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	float t_1 = t_0 - c;
	float t_2 = 1.0f - t_1;
	float t_3 = sqrtf(powf(((float) M_PI), 3.0f));
	float t_4 = t_0 * -1.5f;
	float t_5 = fmaf(((float) M_PI), t_4, (powf(t_2, 2.0f) * -t_3));
	return (((cosTheta * sqrtf(((float) M_PI))) + (powf(cosTheta, 4.0f) * ((t_5 * (sqrtf(((float) M_PI)) * t_2)) + ((t_2 * (t_4 * t_3)) - (0.5f * (((float) M_PI) * t_0)))))) - (t_5 * powf(cosTheta, 3.0f))) + (((float) M_PI) * (powf(cosTheta, 2.0f) * (-1.0f + t_1)));
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	t_1 = Float32(t_0 - c)
	t_2 = Float32(Float32(1.0) - t_1)
	t_3 = sqrt((Float32(pi) ^ Float32(3.0)))
	t_4 = Float32(t_0 * Float32(-1.5))
	t_5 = fma(Float32(pi), t_4, Float32((t_2 ^ Float32(2.0)) * Float32(-t_3)))
	return Float32(Float32(Float32(Float32(cosTheta * sqrt(Float32(pi))) + Float32((cosTheta ^ Float32(4.0)) * Float32(Float32(t_5 * Float32(sqrt(Float32(pi)) * t_2)) + Float32(Float32(t_2 * Float32(t_4 * t_3)) - Float32(Float32(0.5) * Float32(Float32(pi) * t_0)))))) - Float32(t_5 * (cosTheta ^ Float32(3.0)))) + Float32(Float32(pi) * Float32((cosTheta ^ Float32(2.0)) * Float32(Float32(-1.0) + t_1))))
end

Derivation

1. Initial program 96.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. +-commutative 96.9%

      \[\leadsto \frac{1}{\color{blue}{\left(c + 1\right)} + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. associate-+l+ 96.9%

      \[\leadsto \frac{1}{\color{blue}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   3. *-commutative 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)} \]

   4. associate-*l* 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}\right)} \]

   5. /-rgt-identity 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{1}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)\right)} \]

   6. associate-/r/ 96.8%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{\frac{1}{e^{\left(-cosTheta\right) \cdot cosTheta}}}}\right)} \]

   7. exp-neg 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{\color{blue}{e^{-\left(-cosTheta\right) \cdot cosTheta}}}\right)} \]

   8. distribute-rgt-neg-out 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{e^{\color{blue}{\left(-cosTheta\right) \cdot \left(-cosTheta\right)}}}\right)} \]

3. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)}} \]

4. Taylor expanded in cosTheta around 0 97.8%

   \[\leadsto \color{blue}{-1 \cdot \left({cosTheta}^{2} \cdot \left(\pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) + \left(-1 \cdot \left({cosTheta}^{3} \cdot \left(-1 \cdot \left(\sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}\right) + \pi \cdot \left(-1 \cdot \sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) + \left(-1 \cdot \left({cosTheta}^{4} \cdot \left(-1 \cdot \left(\sqrt{\pi} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \left(-1 \cdot \left(\sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}\right) + \pi \cdot \left(-1 \cdot \sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) + \left(-1 \cdot \left(\sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \left(-1 \cdot \sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) + \pi \cdot \left(\sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) + cosTheta \cdot \sqrt{\pi}\right)\right)} \]

6. Simplified 97.8%

   \[\leadsto \color{blue}{\left(\left(cosTheta \cdot \sqrt{\pi} - {cosTheta}^{4} \cdot \left(\left(0.5 \cdot \left(\pi \cdot \sqrt{\frac{1}{\pi}}\right) - \left(1 + \left(c - \sqrt{\frac{1}{\pi}}\right)\right) \cdot \left(\left(\sqrt{\frac{1}{\pi}} \cdot -1.5\right) \cdot \sqrt{{\pi}^{3}}\right)\right) - \mathsf{fma}\left(\pi, \sqrt{\frac{1}{\pi}} \cdot -1.5, {\left(1 + \left(c - \sqrt{\frac{1}{\pi}}\right)\right)}^{2} \cdot \left(-\sqrt{{\pi}^{3}}\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(1 + \left(c - \sqrt{\frac{1}{\pi}}\right)\right)\right)\right)\right) - {cosTheta}^{3} \cdot \mathsf{fma}\left(\pi, \sqrt{\frac{1}{\pi}} \cdot -1.5, {\left(1 + \left(c - \sqrt{\frac{1}{\pi}}\right)\right)}^{2} \cdot \left(-\sqrt{{\pi}^{3}}\right)\right)\right) - \pi \cdot \left(\left(1 + \left(c - \sqrt{\frac{1}{\pi}}\right)\right) \cdot {cosTheta}^{2}\right)} \]

7. Final simplification 97.8%

   \[\leadsto \left(\left(cosTheta \cdot \sqrt{\pi} + {cosTheta}^{4} \cdot \left(\mathsf{fma}\left(\pi, \sqrt{\frac{1}{\pi}} \cdot -1.5, {\left(1 - \left(\sqrt{\frac{1}{\pi}} - c\right)\right)}^{2} \cdot \left(-\sqrt{{\pi}^{3}}\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(1 - \left(\sqrt{\frac{1}{\pi}} - c\right)\right)\right) + \left(\left(1 - \left(\sqrt{\frac{1}{\pi}} - c\right)\right) \cdot \left(\left(\sqrt{\frac{1}{\pi}} \cdot -1.5\right) \cdot \sqrt{{\pi}^{3}}\right) - 0.5 \cdot \left(\pi \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(\pi, \sqrt{\frac{1}{\pi}} \cdot -1.5, {\left(1 - \left(\sqrt{\frac{1}{\pi}} - c\right)\right)}^{2} \cdot \left(-\sqrt{{\pi}^{3}}\right)\right) \cdot {cosTheta}^{3}\right) + \pi \cdot \left({cosTheta}^{2} \cdot \left(-1 + \left(\sqrt{\frac{1}{\pi}} - c\right)\right)\right) \]

Alternative 2: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{cosTheta \cdot \sqrt{\pi}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   c
   (+
    1.0
    (/
     (/ (sqrt (+ 1.0 (* cosTheta -2.0))) (* cosTheta (sqrt PI)))
     (pow (exp cosTheta) cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (c + (1.0f + ((sqrtf((1.0f + (cosTheta * -2.0f))) / (cosTheta * sqrtf(((float) M_PI)))) / powf(expf(cosTheta), cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(c + Float32(Float32(1.0) + Float32(Float32(sqrt(Float32(Float32(1.0) + Float32(cosTheta * Float32(-2.0)))) / Float32(cosTheta * sqrt(Float32(pi)))) / (exp(cosTheta) ^ cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (c + (single(1.0) + ((sqrt((single(1.0) + (cosTheta * single(-2.0)))) / (cosTheta * sqrt(single(pi)))) / (exp(cosTheta) ^ cosTheta))));
end

Derivation

1. Initial program 96.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. +-commutative 96.9%

      \[\leadsto \frac{1}{\color{blue}{\left(c + 1\right)} + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. associate-+l+ 96.9%

      \[\leadsto \frac{1}{\color{blue}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   3. *-commutative 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)} \]

   4. associate-*l* 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}\right)} \]

   5. /-rgt-identity 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{1}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)\right)} \]

   6. associate-/r/ 96.8%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{\frac{1}{e^{\left(-cosTheta\right) \cdot cosTheta}}}}\right)} \]

   7. exp-neg 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{\color{blue}{e^{-\left(-cosTheta\right) \cdot cosTheta}}}\right)} \]

   8. distribute-rgt-neg-out 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{e^{\color{blue}{\left(-cosTheta\right) \cdot \left(-cosTheta\right)}}}\right)} \]

3. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)}} \]

4. Final simplification 97.7%

   \[\leadsto \frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{cosTheta \cdot \sqrt{\pi}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \]

Alternative 3: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot e^{{cosTheta}^{2}}}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (/
    (sqrt (/ (fma cosTheta -2.0 1.0) PI))
    (* cosTheta (exp (pow cosTheta 2.0)))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (sqrtf((fmaf(cosTheta, -2.0f, 1.0f) / ((float) M_PI))) / (cosTheta * expf(powf(cosTheta, 2.0f)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi))) / Float32(cosTheta * exp((cosTheta ^ Float32(2.0)))))))
end

Derivation

1. Initial program 96.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. +-commutative 96.9%

      \[\leadsto \frac{1}{\color{blue}{\left(c + 1\right)} + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. associate-+l+ 96.9%

      \[\leadsto \frac{1}{\color{blue}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   3. *-commutative 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)} \]

   4. associate-*l* 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}\right)} \]

   5. /-rgt-identity 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{1}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)\right)} \]

   6. associate-/r/ 96.8%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{\frac{1}{e^{\left(-cosTheta\right) \cdot cosTheta}}}}\right)} \]

   7. exp-neg 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{\color{blue}{e^{-\left(-cosTheta\right) \cdot cosTheta}}}\right)} \]

   8. distribute-rgt-neg-out 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{e^{\color{blue}{\left(-cosTheta\right) \cdot \left(-cosTheta\right)}}}\right)} \]

3. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)}} \]

4. Taylor expanded in c around 0 96.8%

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}}} \]
5. Step-by-step derivation

   1. +-commutative 96.8%

      \[\leadsto \frac{1}{1 + \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}} \cdot \sqrt{\frac{\color{blue}{-2 \cdot cosTheta + 1}}{\pi}}} \]

   2. *-commutative 96.8%

      \[\leadsto \frac{1}{1 + \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}} \cdot \sqrt{\frac{\color{blue}{cosTheta \cdot -2} + 1}{\pi}}} \]

   3. fma-udef 96.8%

      \[\leadsto \frac{1}{1 + \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}} \cdot \sqrt{\frac{\color{blue}{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\pi}}} \]

   4. associate-*l/ 97.2%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1 \cdot \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot e^{{cosTheta}^{2}}}}} \]

   5. *-lft-identity 97.2%

      \[\leadsto \frac{1}{1 + \frac{\color{blue}{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}}{cosTheta \cdot e^{{cosTheta}^{2}}}} \]

6. Simplified 97.2%

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot e^{{cosTheta}^{2}}}}} \]

7. Final simplification 97.2%

   \[\leadsto \frac{1}{1 + \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot e^{{cosTheta}^{2}}}} \]

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + \frac{e^{-{cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+
    c
    (*
     (/ (exp (- (pow cosTheta 2.0))) cosTheta)
     (sqrt (/ (+ 1.0 (* cosTheta -2.0)) PI)))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + ((expf(-powf(cosTheta, 2.0f)) / cosTheta) * sqrtf(((1.0f + (cosTheta * -2.0f)) / ((float) M_PI))))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(Float32(exp(Float32(-(cosTheta ^ Float32(2.0)))) / cosTheta) * sqrt(Float32(Float32(Float32(1.0) + Float32(cosTheta * Float32(-2.0))) / Float32(pi)))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((exp(-(cosTheta ^ single(2.0))) / cosTheta) * sqrt(((single(1.0) + (cosTheta * single(-2.0))) / single(pi))))));
end

Derivation

1. Initial program 96.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. +-commutative 96.9%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta} + \left(1 + c\right)}} \]

   2. fma-def 96.9%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}, e^{\left(-cosTheta\right) \cdot cosTheta}, 1 + c\right)}} \]

3. Simplified 97.4%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{cosTheta}}{\sqrt{\pi}}, {\left(e^{cosTheta}\right)}^{\left(-cosTheta\right)}, 1 + c\right)}} \]

4. Taylor expanded in c around 0 97.0%

   \[\leadsto \frac{1}{\color{blue}{1 + \left(c + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}\right)}} \]

5. Final simplification 97.0%

   \[\leadsto \frac{1}{1 + \left(c + \frac{e^{-{cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}}\right)} \]

Alternative 5: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (*
    (sqrt (/ (+ 1.0 (* cosTheta -2.0)) PI))
    (/ 1.0 (* cosTheta (exp (pow cosTheta 2.0))))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (sqrtf(((1.0f + (cosTheta * -2.0f)) / ((float) M_PI))) * (1.0f / (cosTheta * expf(powf(cosTheta, 2.0f))))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(sqrt(Float32(Float32(Float32(1.0) + Float32(cosTheta * Float32(-2.0))) / Float32(pi))) * Float32(Float32(1.0) / Float32(cosTheta * exp((cosTheta ^ Float32(2.0))))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (sqrt(((single(1.0) + (cosTheta * single(-2.0))) / single(pi))) * (single(1.0) / (cosTheta * exp((cosTheta ^ single(2.0)))))));
end

Derivation

1. Initial program 96.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. +-commutative 96.9%

      \[\leadsto \frac{1}{\color{blue}{\left(c + 1\right)} + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. associate-+l+ 96.9%

      \[\leadsto \frac{1}{\color{blue}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   3. *-commutative 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)} \]

   4. associate-*l* 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}\right)} \]

   5. /-rgt-identity 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{1}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)\right)} \]

   6. associate-/r/ 96.8%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{\frac{1}{e^{\left(-cosTheta\right) \cdot cosTheta}}}}\right)} \]

   7. exp-neg 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{\color{blue}{e^{-\left(-cosTheta\right) \cdot cosTheta}}}\right)} \]

   8. distribute-rgt-neg-out 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{e^{\color{blue}{\left(-cosTheta\right) \cdot \left(-cosTheta\right)}}}\right)} \]

3. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)}} \]

4. Taylor expanded in c around 0 96.8%

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}}} \]

5. Final simplification 96.8%

   \[\leadsto \frac{1}{1 + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}}} \]

Alternative 6: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{c + \left(1 + \frac{\frac{1 - cosTheta}{cosTheta \cdot \sqrt{\pi}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   c
   (+
    1.0
    (/
     (/ (- 1.0 cosTheta) (* cosTheta (sqrt PI)))
     (pow (exp cosTheta) cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (c + (1.0f + (((1.0f - cosTheta) / (cosTheta * sqrtf(((float) M_PI)))) / powf(expf(cosTheta), cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(c + Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - cosTheta) / Float32(cosTheta * sqrt(Float32(pi)))) / (exp(cosTheta) ^ cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (c + (single(1.0) + (((single(1.0) - cosTheta) / (cosTheta * sqrt(single(pi)))) / (exp(cosTheta) ^ cosTheta))));
end

Derivation

1. Initial program 96.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. +-commutative 96.9%

      \[\leadsto \frac{1}{\color{blue}{\left(c + 1\right)} + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. associate-+l+ 96.9%

      \[\leadsto \frac{1}{\color{blue}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   3. *-commutative 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)} \]

   4. associate-*l* 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}\right)} \]

   5. /-rgt-identity 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{1}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)\right)} \]

   6. associate-/r/ 96.8%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{\frac{1}{e^{\left(-cosTheta\right) \cdot cosTheta}}}}\right)} \]

   7. exp-neg 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{\color{blue}{e^{-\left(-cosTheta\right) \cdot cosTheta}}}\right)} \]

   8. distribute-rgt-neg-out 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{e^{\color{blue}{\left(-cosTheta\right) \cdot \left(-cosTheta\right)}}}\right)} \]

3. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)}} \]

4. Taylor expanded in cosTheta around 0 95.4%

   \[\leadsto \frac{1}{c + \left(1 + \frac{\frac{\color{blue}{1 + -1 \cdot cosTheta}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \]
5. Step-by-step derivation

   1. mul-1-neg 95.4%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\frac{1 + \color{blue}{\left(-cosTheta\right)}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \]

   2. unsub-neg 95.4%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\frac{\color{blue}{1 - cosTheta}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \]

6. Simplified 95.4%

   \[\leadsto \frac{1}{c + \left(1 + \frac{\frac{\color{blue}{1 - cosTheta}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \]

7. Final simplification 95.4%

   \[\leadsto \frac{1}{c + \left(1 + \frac{\frac{1 - cosTheta}{cosTheta \cdot \sqrt{\pi}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \]

Alternative 7: 95.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{c + \left(1 + \left(\sqrt{\frac{1}{\pi}} \cdot \left(1 - {cosTheta}^{2}\right)\right) \cdot \left(-1 + \frac{1}{cosTheta}\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   c
   (+
    1.0
    (*
     (* (sqrt (/ 1.0 PI)) (- 1.0 (pow cosTheta 2.0)))
     (+ -1.0 (/ 1.0 cosTheta)))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (c + (1.0f + ((sqrtf((1.0f / ((float) M_PI))) * (1.0f - powf(cosTheta, 2.0f))) * (-1.0f + (1.0f / cosTheta)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(c + Float32(Float32(1.0) + Float32(Float32(sqrt(Float32(Float32(1.0) / Float32(pi))) * Float32(Float32(1.0) - (cosTheta ^ Float32(2.0)))) * Float32(Float32(-1.0) + Float32(Float32(1.0) / cosTheta))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (c + (single(1.0) + ((sqrt((single(1.0) / single(pi))) * (single(1.0) - (cosTheta ^ single(2.0)))) * (single(-1.0) + (single(1.0) / cosTheta)))));
end

Derivation

1. Initial program 96.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. +-commutative 96.9%

      \[\leadsto \frac{1}{\color{blue}{\left(c + 1\right)} + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. associate-+l+ 96.9%

      \[\leadsto \frac{1}{\color{blue}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   3. *-commutative 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)} \]

   4. associate-*l* 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}\right)} \]

   5. /-rgt-identity 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{1}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)\right)} \]

   6. associate-/r/ 96.8%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{\frac{1}{e^{\left(-cosTheta\right) \cdot cosTheta}}}}\right)} \]

   7. exp-neg 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{\color{blue}{e^{-\left(-cosTheta\right) \cdot cosTheta}}}\right)} \]

   8. distribute-rgt-neg-out 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{e^{\color{blue}{\left(-cosTheta\right) \cdot \left(-cosTheta\right)}}}\right)} \]

3. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)}} \]

4. Taylor expanded in cosTheta around 0 94.6%

   \[\leadsto \frac{1}{c + \left(1 + \frac{\color{blue}{-1 \cdot \sqrt{\frac{1}{\pi}} + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \]
5. Step-by-step derivation

   1. distribute-rgt-out 94.6%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(-1 + \frac{1}{cosTheta}\right)}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \]

6. Simplified 94.6%

   \[\leadsto \frac{1}{c + \left(1 + \frac{\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(-1 + \frac{1}{cosTheta}\right)}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \]
7. Step-by-step derivation

   1. *-un-lft-identity 94.6%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{1 \cdot \frac{\sqrt{\frac{1}{\pi}} \cdot \left(-1 + \frac{1}{cosTheta}\right)}{{\left(e^{cosTheta}\right)}^{cosTheta}}}\right)} \]

   2. associate-/l* 94.4%

      \[\leadsto \frac{1}{c + \left(1 + 1 \cdot \color{blue}{\frac{\sqrt{\frac{1}{\pi}}}{\frac{{\left(e^{cosTheta}\right)}^{cosTheta}}{-1 + \frac{1}{cosTheta}}}}\right)} \]

   3. pow1/2 94.4%

      \[\leadsto \frac{1}{c + \left(1 + 1 \cdot \frac{\color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}}{\frac{{\left(e^{cosTheta}\right)}^{cosTheta}}{-1 + \frac{1}{cosTheta}}}\right)} \]

   4. inv-pow 94.4%

      \[\leadsto \frac{1}{c + \left(1 + 1 \cdot \frac{{\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}}{\frac{{\left(e^{cosTheta}\right)}^{cosTheta}}{-1 + \frac{1}{cosTheta}}}\right)} \]

   5. pow-pow 94.4%

      \[\leadsto \frac{1}{c + \left(1 + 1 \cdot \frac{\color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}}{\frac{{\left(e^{cosTheta}\right)}^{cosTheta}}{-1 + \frac{1}{cosTheta}}}\right)} \]

   6. metadata-eval 94.4%

      \[\leadsto \frac{1}{c + \left(1 + 1 \cdot \frac{{\pi}^{\color{blue}{-0.5}}}{\frac{{\left(e^{cosTheta}\right)}^{cosTheta}}{-1 + \frac{1}{cosTheta}}}\right)} \]

   7. pow-exp 94.4%

      \[\leadsto \frac{1}{c + \left(1 + 1 \cdot \frac{{\pi}^{-0.5}}{\frac{\color{blue}{e^{cosTheta \cdot cosTheta}}}{-1 + \frac{1}{cosTheta}}}\right)} \]

   8. unpow2 94.4%

      \[\leadsto \frac{1}{c + \left(1 + 1 \cdot \frac{{\pi}^{-0.5}}{\frac{e^{\color{blue}{{cosTheta}^{2}}}}{-1 + \frac{1}{cosTheta}}}\right)} \]

8. Applied egg-rr 94.4%

   \[\leadsto \frac{1}{c + \left(1 + \color{blue}{1 \cdot \frac{{\pi}^{-0.5}}{\frac{e^{{cosTheta}^{2}}}{-1 + \frac{1}{cosTheta}}}}\right)} \]
9. Step-by-step derivation

   1. *-lft-identity 94.4%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\frac{{\pi}^{-0.5}}{\frac{e^{{cosTheta}^{2}}}{-1 + \frac{1}{cosTheta}}}}\right)} \]

   2. associate-/r/ 94.5%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\frac{{\pi}^{-0.5}}{e^{{cosTheta}^{2}}} \cdot \left(-1 + \frac{1}{cosTheta}\right)}\right)} \]

10. Simplified 94.5%

    \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\frac{{\pi}^{-0.5}}{e^{{cosTheta}^{2}}} \cdot \left(-1 + \frac{1}{cosTheta}\right)}\right)} \]

11. Taylor expanded in cosTheta around 0 94.5%

    \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\left(\sqrt{\frac{1}{\pi}} + -1 \cdot \left({cosTheta}^{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \cdot \left(-1 + \frac{1}{cosTheta}\right)\right)} \]
12. Step-by-step derivation

    1. associate-*r* 94.5%

       \[\leadsto \frac{1}{c + \left(1 + \left(\sqrt{\frac{1}{\pi}} + \color{blue}{\left(-1 \cdot {cosTheta}^{2}\right) \cdot \sqrt{\frac{1}{\pi}}}\right) \cdot \left(-1 + \frac{1}{cosTheta}\right)\right)} \]

    2. neg-mul-1 94.5%

       \[\leadsto \frac{1}{c + \left(1 + \left(\sqrt{\frac{1}{\pi}} + \color{blue}{\left(-{cosTheta}^{2}\right)} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(-1 + \frac{1}{cosTheta}\right)\right)} \]

    3. distribute-rgt1-in 94.5%

       \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\left(\left(\left(-{cosTheta}^{2}\right) + 1\right) \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left(-1 + \frac{1}{cosTheta}\right)\right)} \]

13. Simplified 94.5%

    \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\left(\left(\left(-{cosTheta}^{2}\right) + 1\right) \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left(-1 + \frac{1}{cosTheta}\right)\right)} \]

14. Final simplification 94.5%

    \[\leadsto \frac{1}{c + \left(1 + \left(\sqrt{\frac{1}{\pi}} \cdot \left(1 - {cosTheta}^{2}\right)\right) \cdot \left(-1 + \frac{1}{cosTheta}\right)\right)} \]

Alternative 8: 94.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \sqrt{\frac{1}{\pi}} \cdot \left(-1 + \frac{1}{cosTheta}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/ 1.0 (+ 1.0 (* (sqrt (/ 1.0 PI)) (+ -1.0 (/ 1.0 cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (sqrtf((1.0f / ((float) M_PI))) * (-1.0f + (1.0f / cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(sqrt(Float32(Float32(1.0) / Float32(pi))) * Float32(Float32(-1.0) + Float32(Float32(1.0) / cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (sqrt((single(1.0) / single(pi))) * (single(-1.0) + (single(1.0) / cosTheta))));
end

Derivation

1. Initial program 96.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

2. Taylor expanded in c around 0 96.8%

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}}} \]

3. Taylor expanded in cosTheta around 0 94.1%

   \[\leadsto \frac{1}{1 + \color{blue}{\left(-1 \cdot \sqrt{\frac{1}{\pi}} + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)}} \]
4. Step-by-step derivation

   1. distribute-rgt-out 94.1%

      \[\leadsto \frac{1}{1 + \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(-1 + \frac{1}{cosTheta}\right)}} \]

5. Simplified 94.1%

   \[\leadsto \frac{1}{1 + \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(-1 + \frac{1}{cosTheta}\right)}} \]

6. Final simplification 94.1%

   \[\leadsto \frac{1}{1 + \sqrt{\frac{1}{\pi}} \cdot \left(-1 + \frac{1}{cosTheta}\right)} \]

Alternative 9: 94.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \frac{\sqrt{\frac{1}{\pi}} \cdot \left(1 - cosTheta\right)}{cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/ 1.0 (+ 1.0 (/ (* (sqrt (/ 1.0 PI)) (- 1.0 cosTheta)) cosTheta))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + ((sqrtf((1.0f / ((float) M_PI))) * (1.0f - cosTheta)) / cosTheta));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(sqrt(Float32(Float32(1.0) / Float32(pi))) * Float32(Float32(1.0) - cosTheta)) / cosTheta)))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((sqrt((single(1.0) / single(pi))) * (single(1.0) - cosTheta)) / cosTheta));
end

Derivation

1. Initial program 96.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

2. Taylor expanded in c around 0 96.8%

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}}} \]
3. Step-by-step derivation

   1. associate-*l/ 97.2%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{-1 \cdot {cosTheta}^{2}} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}}{cosTheta}}} \]

   2. mul-1-neg 97.2%

      \[\leadsto \frac{1}{1 + \frac{e^{\color{blue}{-{cosTheta}^{2}}} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}}{cosTheta}} \]

   3. cancel-sign-sub-inv 97.2%

      \[\leadsto \frac{1}{1 + \frac{e^{-{cosTheta}^{2}} \cdot \sqrt{\frac{\color{blue}{1 + \left(-2\right) \cdot cosTheta}}{\pi}}}{cosTheta}} \]

   4. metadata-eval 97.2%

      \[\leadsto \frac{1}{1 + \frac{e^{-{cosTheta}^{2}} \cdot \sqrt{\frac{1 + \color{blue}{-2} \cdot cosTheta}{\pi}}}{cosTheta}} \]

   5. *-commutative 97.2%

      \[\leadsto \frac{1}{1 + \frac{e^{-{cosTheta}^{2}} \cdot \sqrt{\frac{1 + \color{blue}{cosTheta \cdot -2}}{\pi}}}{cosTheta}} \]

   6. +-commutative 97.2%

      \[\leadsto \frac{1}{1 + \frac{e^{-{cosTheta}^{2}} \cdot \sqrt{\frac{\color{blue}{cosTheta \cdot -2 + 1}}{\pi}}}{cosTheta}} \]

   7. fma-def 97.2%

      \[\leadsto \frac{1}{1 + \frac{e^{-{cosTheta}^{2}} \cdot \sqrt{\frac{\color{blue}{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\pi}}}{cosTheta}} \]

4. Applied egg-rr 97.2%

   \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{-{cosTheta}^{2}} \cdot \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}}} \]

5. Taylor expanded in cosTheta around 0 94.5%

   \[\leadsto \frac{1}{1 + \frac{\color{blue}{\sqrt{\frac{1}{\pi}} + -1 \cdot \left(cosTheta \cdot \sqrt{\frac{1}{\pi}}\right)}}{cosTheta}} \]
6. Step-by-step derivation

   1. associate-*r* 94.5%

      \[\leadsto \frac{1}{1 + \frac{\sqrt{\frac{1}{\pi}} + \color{blue}{\left(-1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\pi}}}}{cosTheta}} \]

   2. distribute-rgt1-in 94.5%

      \[\leadsto \frac{1}{1 + \frac{\color{blue}{\left(-1 \cdot cosTheta + 1\right) \cdot \sqrt{\frac{1}{\pi}}}}{cosTheta}} \]

   3. mul-1-neg 94.5%

      \[\leadsto \frac{1}{1 + \frac{\left(\color{blue}{\left(-cosTheta\right)} + 1\right) \cdot \sqrt{\frac{1}{\pi}}}{cosTheta}} \]

7. Simplified 94.5%

   \[\leadsto \frac{1}{1 + \frac{\color{blue}{\left(\left(-cosTheta\right) + 1\right) \cdot \sqrt{\frac{1}{\pi}}}}{cosTheta}} \]

8. Final simplification 94.5%

   \[\leadsto \frac{1}{1 + \frac{\sqrt{\frac{1}{\pi}} \cdot \left(1 - cosTheta\right)}{cosTheta}} \]

Alternative 10: 92.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ cosTheta \cdot \sqrt{\pi} \end{array} \]

FPCore

(FPCore (cosTheta c) :precision binary32 (* cosTheta (sqrt PI)))

C

float code(float cosTheta, float c) {
	return cosTheta * sqrtf(((float) M_PI));
}

Julia

function code(cosTheta, c)
	return Float32(cosTheta * sqrt(Float32(pi)))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * sqrt(single(pi));
end

Derivation

1. Initial program 96.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

   1. +-commutative 96.9%

      \[\leadsto \frac{1}{\color{blue}{\left(c + 1\right)} + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. associate-+l+ 96.9%

      \[\leadsto \frac{1}{\color{blue}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]

   3. *-commutative 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)} \]

   4. associate-*l* 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}\right)} \]

   5. /-rgt-identity 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(\color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{1}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)\right)} \]

   6. associate-/r/ 96.8%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{\frac{1}{e^{\left(-cosTheta\right) \cdot cosTheta}}}}\right)} \]

   7. exp-neg 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{\color{blue}{e^{-\left(-cosTheta\right) \cdot cosTheta}}}\right)} \]

   8. distribute-rgt-neg-out 96.9%

      \[\leadsto \frac{1}{c + \left(1 + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{e^{\color{blue}{\left(-cosTheta\right) \cdot \left(-cosTheta\right)}}}\right)} \]

3. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)}} \]

4. Taylor expanded in cosTheta around 0 92.3%

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\pi}} \]

5. Final simplification 92.3%

   \[\leadsto cosTheta \cdot \sqrt{\pi} \]

Alternative 11: 10.9% accurate, 421.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (cosTheta c) :precision binary32 1.0)

C

float code(float cosTheta, float c) {
	return 1.0f;
}

Fortran

real(4) function code(costheta, c)
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = 1.0e0
end function

Julia

function code(cosTheta, c)
	return Float32(1.0)
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0);
end

Derivation

1. Initial program 96.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

2. Taylor expanded in c around 0 96.8%

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}}} \]

3. Taylor expanded in cosTheta around inf 10.8%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 10.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023310 
(FPCore (cosTheta c)
  :name "Beckmann Sample, normalization factor"
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999)) (and (< -1.0 c) (< c 1.0)))
  (/ 1.0 (+ (+ 1.0 c) (* (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta)) (exp (* (- cosTheta) cosTheta))))))