Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 13.8s

Alternatives: 12

Speedup: 0.8×

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: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + ((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(Float32(Float32(1.0) + c) + 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) / ((single(1.0) + c) + ((sqrt((single(1.0) + (cosTheta * single(-2.0)))) / (cosTheta * sqrt(single(pi)))) / (exp(cosTheta) ^ cosTheta)));
end

Derivation

1. Initial program 97.8%

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

   1. distribute-lft-neg-out 97.8%

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

   2. exp-neg 97.8%

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

   3. associate-*r/ 97.8%

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

   4. associate-/l* 97.8%

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

   5. /-rgt-identity 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. times-frac 98.3%

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

   9. *-lft-identity 98.3%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. distribute-lft-neg-out 97.8%

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

   2. exp-neg 97.8%

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

   3. associate-*r/ 97.8%

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

   4. associate-/l* 97.8%

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

   5. /-rgt-identity 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. times-frac 98.3%

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

   9. *-lft-identity 98.3%

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

3. Simplified 98.4%

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

5. Taylor expanded in c around 0 97.8%

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

6. Final simplification 97.8%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (sqrtf(((-2.0f * (cosTheta / ((float) M_PI))) + (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(Float32(Float32(-2.0) * Float32(cosTheta / Float32(pi))) + Float32(Float32(1.0) / Float32(pi)))) / Float32(cosTheta * exp((cosTheta ^ Float32(2.0)))))))
end

MATLAB

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in c around 0 97.8%

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

   1. cancel-sign-sub-inv 97.8%

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

   2. metadata-eval 97.8%

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

   3. mul-1-neg 97.8%

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

   4. exp-neg 97.8%

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

   5. associate-/r* 97.8%

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

   6. *-commutative 97.8%

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

5. Simplified 97.9%

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

6. Taylor expanded in cosTheta around 0 97.9%

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

7. Final simplification 97.9%

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

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (*
    (/ (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 + ((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(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) + ((exp(-(cosTheta ^ single(2.0))) / cosTheta) * sqrt(((single(1.0) - (cosTheta * single(2.0))) / single(pi)))));
end

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in c around 0 97.8%

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

4. Final simplification 97.8%

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

Alternative 5: 96.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \frac{1}{1 + \left(t_0 \cdot \left(-1 + \frac{1}{cosTheta}\right) + t_0 \cdot \left(cosTheta \cdot -1.5\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (/
    1.0
    (+ 1.0 (+ (* t_0 (+ -1.0 (/ 1.0 cosTheta))) (* t_0 (* cosTheta -1.5)))))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	return 1.0f / (1.0f + ((t_0 * (-1.0f + (1.0f / cosTheta))) + (t_0 * (cosTheta * -1.5f))));
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(t_0 * Float32(Float32(-1.0) + Float32(Float32(1.0) / cosTheta))) + Float32(t_0 * Float32(cosTheta * Float32(-1.5))))))
end

MATLAB

function tmp = code(cosTheta, c)
	t_0 = sqrt((single(1.0) / single(pi)));
	tmp = single(1.0) / (single(1.0) + ((t_0 * (single(-1.0) + (single(1.0) / cosTheta))) + (t_0 * (cosTheta * single(-1.5)))));
end

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in c around 0 97.8%

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

   1. cancel-sign-sub-inv 97.8%

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

   2. metadata-eval 97.8%

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

   3. mul-1-neg 97.8%

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

   4. exp-neg 97.8%

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

   5. associate-/r* 97.8%

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

   6. *-commutative 97.8%

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

5. Simplified 97.9%

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

6. Taylor expanded in cosTheta around 0 96.5%

   \[\leadsto \frac{1}{1 + \color{blue}{\left(-1 \cdot \sqrt{\frac{1}{\pi}} + \left(cosTheta \cdot \left(-1 \cdot \sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right) + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative 96.5%

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

   2. associate-+r+ 96.5%

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

   3. distribute-rgt-out 96.5%

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

   4. *-commutative 96.5%

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

   5. distribute-rgt-out 96.5%

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

   6. associate-*l* 96.5%

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

   7. metadata-eval 96.5%

      \[\leadsto \frac{1}{1 + \left(\sqrt{\frac{1}{\pi}} \cdot \left(-1 + \frac{1}{cosTheta}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{-1.5} \cdot cosTheta\right)\right)} \]

8. Simplified 96.5%

   \[\leadsto \frac{1}{1 + \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(-1 + \frac{1}{cosTheta}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(-1.5 \cdot cosTheta\right)\right)}} \]

9. Final simplification 96.5%

   \[\leadsto \frac{1}{1 + \left(\sqrt{\frac{1}{\pi}} \cdot \left(-1 + \frac{1}{cosTheta}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(cosTheta \cdot -1.5\right)\right)} \]
10. Add Preprocessing

Alternative 6: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \left(\left(-1 + \frac{1}{cosTheta}\right) \cdot {\pi}^{-0.5}\right) \cdot e^{cosTheta \cdot \left(-cosTheta\right)}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (+ -1.0 (/ 1.0 cosTheta)) (pow PI -0.5))
    (exp (* cosTheta (- cosTheta)))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((-1.0f + (1.0f / cosTheta)) * powf(((float) M_PI), -0.5f)) * 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) + Float32(Float32(1.0) / cosTheta)) * (Float32(pi) ^ Float32(-0.5))) * exp(Float32(cosTheta * Float32(-cosTheta))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(-1.0) + (single(1.0) / cosTheta)) * (single(pi) ^ single(-0.5))) * exp((cosTheta * -cosTheta))));
end

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in cosTheta around 0 95.7%

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

   1. +-commutative 95.7%

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

   2. distribute-rgt-out 95.7%

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

5. Simplified 95.7%

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

   1. pow1/2 95.7%

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

   2. inv-pow 95.7%

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

   3. pow-pow 95.7%

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

   4. metadata-eval 95.7%

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

   5. expm1-log1p-u 95.7%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5}\right)\right)} \cdot \left(\frac{1}{cosTheta} + -1\right)\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. expm1-udef 95.7%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\frac{1}{cosTheta} + -1\right)\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

7. Applied egg-rr 95.7%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\frac{1}{cosTheta} + -1\right)\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
8. Step-by-step derivation

   1. expm1-def 95.7%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5}\right)\right)} \cdot \left(\frac{1}{cosTheta} + -1\right)\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. expm1-log1p 95.7%

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

9. Simplified 95.7%

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

10. Final simplification 95.7%

    \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(-1 + \frac{1}{cosTheta}\right) \cdot {\pi}^{-0.5}\right) \cdot e^{cosTheta \cdot \left(-cosTheta\right)}} \]
11. Add Preprocessing

Alternative 7: 95.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \frac{1}{\left(1 + c\right) + \left(\frac{t_0}{cosTheta} + t_0 \cdot \left(-1 - cosTheta\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (/ 1.0 (+ (+ 1.0 c) (+ (/ t_0 cosTheta) (* t_0 (- -1.0 cosTheta)))))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	return 1.0f / ((1.0f + c) + ((t_0 / cosTheta) + (t_0 * (-1.0f - cosTheta))));
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(t_0 / cosTheta) + Float32(t_0 * Float32(Float32(-1.0) - cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	t_0 = sqrt((single(1.0) / single(pi)));
	tmp = single(1.0) / ((single(1.0) + c) + ((t_0 / cosTheta) + (t_0 * (single(-1.0) - cosTheta))));
end

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in cosTheta around 0 95.7%

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

   1. +-commutative 95.7%

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

   2. distribute-rgt-out 95.7%

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

5. Simplified 95.7%

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

6. Taylor expanded in cosTheta around 0 95.8%

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

   1. associate-+r+ 95.8%

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

   2. +-commutative 95.8%

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

   3. associate-*l/ 95.9%

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

   4. *-lft-identity 95.9%

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

   5. associate-*r* 95.9%

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

   6. distribute-rgt-out 95.9%

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

   7. mul-1-neg 95.9%

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

8. Simplified 95.9%

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

9. Final simplification 95.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{\sqrt{\frac{1}{\pi}}}{cosTheta} + \sqrt{\frac{1}{\pi}} \cdot \left(-1 - cosTheta\right)\right)} \]
10. Add Preprocessing

Alternative 8: 94.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \frac{1}{1 + \left(t_0 \cdot \frac{1}{cosTheta} - t_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (/ 1.0 (+ 1.0 (- (* t_0 (/ 1.0 cosTheta)) t_0)))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	return 1.0f / (1.0f + ((t_0 * (1.0f / cosTheta)) - t_0));
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(t_0 * Float32(Float32(1.0) / cosTheta)) - t_0)))
end

MATLAB

function tmp = code(cosTheta, c)
	t_0 = sqrt((single(1.0) / single(pi)));
	tmp = single(1.0) / (single(1.0) + ((t_0 * (single(1.0) / cosTheta)) - t_0));
end

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in c around 0 97.8%

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

   1. cancel-sign-sub-inv 97.8%

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

   2. metadata-eval 97.8%

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

   3. mul-1-neg 97.8%

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

   4. exp-neg 97.8%

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

   5. associate-/r* 97.8%

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

   6. *-commutative 97.8%

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

5. Simplified 97.9%

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

6. Taylor expanded in cosTheta around 0 95.4%

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

7. Final simplification 95.4%

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

Alternative 9: 94.7% accurate, 2.8× 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 97.8%

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

3. Taylor expanded in c around 0 97.8%

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

   1. cancel-sign-sub-inv 97.8%

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

   2. metadata-eval 97.8%

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

   3. mul-1-neg 97.8%

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

   4. exp-neg 97.8%

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

   5. associate-/r* 97.8%

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

   6. *-commutative 97.8%

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

5. Simplified 97.9%

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

6. Taylor expanded in cosTheta around 0 95.4%

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

   1. distribute-rgt-out 95.4%

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

8. Simplified 95.4%

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

9. Final simplification 95.4%

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

Alternative 10: 92.6% accurate, 3.1× 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 97.8%

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

3. Taylor expanded in cosTheta around 0 93.8%

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

4. Final simplification 93.8%

   \[\leadsto cosTheta \cdot \sqrt{\pi} \]
5. Add Preprocessing

Alternative 11: 10.8% accurate, 107.3× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 (- 1.0 c))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in cosTheta around inf 10.7%

   \[\leadsto \color{blue}{\frac{1}{1 + c}} \]

4. Taylor expanded in c around 0 10.7%

   \[\leadsto \color{blue}{1 + -1 \cdot c} \]
5. Step-by-step derivation

   1. mul-1-neg 10.7%

      \[\leadsto 1 + \color{blue}{\left(-c\right)} \]

   2. unsub-neg 10.7%

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

6. Simplified 10.7%

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

7. Final simplification 10.7%

   \[\leadsto 1 - c \]
8. Add Preprocessing

Alternative 12: 10.8% accurate, 322.0× 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 97.8%

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

3. Taylor expanded in c around 0 97.8%

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

   1. cancel-sign-sub-inv 97.8%

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

   2. metadata-eval 97.8%

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

   3. mul-1-neg 97.8%

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

   4. exp-neg 97.8%

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

   5. associate-/r* 97.8%

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

   6. *-commutative 97.8%

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

5. Simplified 97.9%

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

6. Taylor expanded in cosTheta around inf 10.7%

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

7. Final simplification 10.7%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(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))))))