Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.5%

Time: 15.2s

Alternatives: 12

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: 97.9% 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, 1.0× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. frac-times 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. associate--l- 98.4%

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

   4. *-commutative 98.4%

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 98.0% 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 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-*l* 97.5%

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

   5. fma-def 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in c around 0 97.6%

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

6. Final simplification 97.6%

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

Alternative 3: 97.6% 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 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-*l* 97.5%

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

   5. fma-def 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in c around 0 96.9%

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

6. Final simplification 96.9%

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

Alternative 4: 96.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (expf((cosTheta * -cosTheta)) * ((((cosTheta * -0.5f) + (1.0f / cosTheta)) + -1.0f) / sqrtf(((float) M_PI)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(exp(Float32(cosTheta * Float32(-cosTheta))) * Float32(Float32(Float32(Float32(cosTheta * Float32(-0.5)) + Float32(Float32(1.0) / cosTheta)) + Float32(-1.0)) / sqrt(Float32(pi))))))
end

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-*l/ 97.9%

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

   2. *-un-lft-identity 97.9%

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

   3. associate--l- 98.0%

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

4. Applied egg-rr 98.0%

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

5. Taylor expanded in cosTheta around 0 95.7%

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

6. Final simplification 95.7%

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

Alternative 5: 96.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \frac{1}{1 + \left(t\_0 \cdot \left(\frac{1}{cosTheta} + -1\right) + cosTheta \cdot \left(t\_0 \cdot -1.5\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (/
    1.0
    (+ 1.0 (+ (* t_0 (+ (/ 1.0 cosTheta) -1.0)) (* cosTheta (* t_0 -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 / cosTheta) + -1.0f)) + (cosTheta * (t_0 * -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(Float32(1.0) / cosTheta) + Float32(-1.0))) + Float32(cosTheta * Float32(t_0 * 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) / cosTheta) + single(-1.0))) + (cosTheta * (t_0 * single(-1.5)))));
end

Derivation

1. Initial program 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-*l* 97.5%

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

   5. fma-def 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in c around 0 96.9%

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

6. Taylor expanded in cosTheta around 0 95.1%

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

   1. +-commutative 95.1%

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

   2. associate-+r+ 95.1%

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

   3. distribute-rgt-out 95.1%

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

   4. distribute-rgt-out 95.1%

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

   5. metadata-eval 95.1%

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

8. Simplified 95.1%

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

9. Final simplification 95.1%

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

Alternative 6: 96.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. frac-times 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. associate--l- 98.4%

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

   4. *-commutative 98.4%

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in cosTheta around 0 94.9%

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

   1. mul-1-neg 94.9%

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

   2. unsub-neg 94.9%

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

7. Simplified 94.9%

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

8. Final simplification 94.9%

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

Alternative 7: 95.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + fmaf(powf(((float) M_PI), -0.5f), ((1.0f / cosTheta) + -1.0f), c));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + fma((Float32(pi) ^ Float32(-0.5)), Float32(Float32(Float32(1.0) / cosTheta) + Float32(-1.0)), c)))
end

Derivation

1. Initial program 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-*l* 97.5%

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

   5. fma-def 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in cosTheta around 0 93.8%

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

   1. +-commutative 93.8%

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

   2. distribute-rgt-out 93.7%

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

   3. fma-def 93.7%

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

7. Simplified 93.7%

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

   1. expm1-log1p-u 93.7%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}}\right)\right)}, -1 + \frac{1}{cosTheta}, c\right)} \]

   2. expm1-udef 93.7%

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

   3. pow1/2 93.7%

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

   4. inv-pow 93.7%

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

   5. pow-pow 93.7%

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

   6. metadata-eval 93.7%

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

9. Applied egg-rr 93.7%

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

    1. expm1-def 93.7%

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

    2. expm1-log1p 93.7%

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

11. Simplified 93.7%

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

12. Final simplification 93.7%

    \[\leadsto \frac{1}{1 + \mathsf{fma}\left({\pi}^{-0.5}, \frac{1}{cosTheta} + -1, c\right)} \]
13. Add Preprocessing

Alternative 8: 95.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + (((1.0f / cosTheta) + -1.0f) * powf(((float) M_PI), -0.5f))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(Float32(Float32(Float32(1.0) / cosTheta) + Float32(-1.0)) * (Float32(pi) ^ Float32(-0.5))))))
end

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-*l* 97.5%

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

   5. fma-def 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in cosTheta around 0 93.8%

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

   1. +-commutative 93.8%

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

   2. distribute-rgt-out 93.7%

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

   3. fma-def 93.7%

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

7. Simplified 93.7%

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

   1. fma-udef 93.7%

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

   2. pow1/2 93.7%

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

   3. inv-pow 93.7%

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

   4. pow-pow 93.7%

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

   5. metadata-eval 93.7%

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

9. Applied egg-rr 93.7%

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

10. Final simplification 93.7%

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

Alternative 9: 95.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-*l* 97.5%

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

   5. fma-def 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in cosTheta around 0 93.8%

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

   1. +-commutative 93.8%

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

   2. distribute-rgt-out 93.7%

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

   3. fma-def 93.7%

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

7. Simplified 93.7%

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

8. Taylor expanded in c around 0 93.5%

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

9. Final simplification 93.5%

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

Alternative 10: 93.0% 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 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-*l* 97.5%

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

   5. fma-def 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in cosTheta around 0 92.4%

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

6. Final simplification 92.4%

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

Alternative 11: 10.7% 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 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-*l* 97.5%

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

   5. fma-def 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in cosTheta around inf 10.9%

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

6. Taylor expanded in c around 0 10.9%

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

   1. mul-1-neg 10.9%

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

   2. unsub-neg 10.9%

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

8. Simplified 10.9%

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

9. Final simplification 10.9%

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

Alternative 12: 10.7% 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 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-*l* 97.5%

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

   5. fma-def 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in cosTheta around inf 10.9%

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

6. Taylor expanded in c around 0 10.9%

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

7. Final simplification 10.9%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))))))