GTR1 distribution

Percentage Accurate: 98.5% → 98.9%

Time: 25.1s

Alternatives: 15

Speedup: 1.0×

Specification

\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (* (* PI (log (* alpha alpha))) (+ 1.0 (* (* t_0 cosTheta) cosTheta))))))

C

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (* (* PI (log (* alpha alpha))) (+ 1.0 (* (* t_0 cosTheta) cosTheta))))))

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \langle \left( \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\left(\pi \cdot \left(\log \alpha \cdot 2\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (/
    (fma alpha alpha -1.0)
    (*
     (* PI (* (log alpha) 2.0))
     (fma (fma alpha alpha -1.0) (* cosTheta cosTheta) 1.0))))))

C

float code(float cosTheta, float alpha) {
	double tmp = fma(alpha, alpha, -1.0) / ((((double) M_PI) * (log(alpha) * 2.0)) * fma(fma(alpha, alpha, -1.0), (((double) cosTheta) * ((double) cosTheta)), 1.0));
	return (float) tmp;
}

Julia

function code(cosTheta, alpha)
	tmp = Float64(fma(alpha, alpha, -1.0) / Float64(Float64(pi * Float64(log(alpha) * 2.0)) * fma(fma(alpha, alpha, -1.0), Float64(Float64(cosTheta) * Float64(cosTheta)), 1.0)))
	return Float32(tmp)
end

Derivation

1. Initial program 98.9%

   \[\langle \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\left(\pi \cdot \left(\log \alpha \cdot 2\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \rangle_{\text{binary64}} \]

2. Final simplification 98.9%

   \[\leadsto \langle \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\left(\pi \cdot \left(\log \alpha \cdot 2\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \rangle_{\text{binary64}} \]

Alternative 2: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \alpha \cdot \alpha\\ \frac{t_0}{\langle \left( \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \cdot \left(1 + cosTheta \cdot \left(cosTheta \cdot t_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ -1.0 (* alpha alpha))))
   (/
    t_0
    (*
     (cast (! :precision binary64 (* PI (log (* alpha alpha)))))
     (+ 1.0 (* cosTheta (* cosTheta t_0)))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = -1.0f + (alpha * alpha);
	double tmp = ((double) M_PI) * log((((double) alpha) * ((double) alpha)));
	return t_0 / (((float) tmp) * (1.0f + (cosTheta * (cosTheta * t_0))));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(-1.0) + Float32(alpha * alpha))
	tmp = Float64(pi * log(Float64(Float64(alpha) * Float64(alpha))))
	return Float32(t_0 / Float32(Float32(tmp) * Float32(Float32(1.0) + Float32(cosTheta * Float32(cosTheta * t_0)))))
end

MATLAB

function tmp_2 = code(cosTheta, alpha)
	t_0 = single(-1.0) + (alpha * alpha);
	tmp = pi * log((double(alpha) * double(alpha)));
	tmp_2 = t_0 / single((single(tmp) * double((single(1.0) + (cosTheta * (cosTheta * t_0))))));
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Step-by-step derivation

   1. rewrite-binary32/binary64 98.7%

      \[\leadsto \color{blue}{\frac{\alpha \cdot \alpha - 1}{\langle \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \rangle_{\text{binary64}} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

3. Applied rewrite-once 98.7%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\langle \color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \rangle_{\text{binary64}}} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

4. Final simplification 98.7%

   \[\leadsto \frac{-1 + \alpha \cdot \alpha}{\langle \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \rangle_{\text{binary64}} \cdot \left(1 + cosTheta \cdot \left(cosTheta \cdot \left(-1 + \alpha \cdot \alpha\right)\right)\right)} \]

Alternative 3: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\pi}{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}}} \cdot \frac{\frac{1}{\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)}}{2} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (*
  (/ 1.0 (/ PI (/ (fma alpha alpha -1.0) (log alpha))))
  (/ (/ 1.0 (fma cosTheta (* cosTheta (fma alpha alpha -1.0)) 1.0)) 2.0)))

C

float code(float cosTheta, float alpha) {
	return (1.0f / (((float) M_PI) / (fmaf(alpha, alpha, -1.0f) / logf(alpha)))) * ((1.0f / fmaf(cosTheta, (cosTheta * fmaf(alpha, alpha, -1.0f)), 1.0f)) / 2.0f);
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(1.0) / Float32(Float32(pi) / Float32(fma(alpha, alpha, Float32(-1.0)) / log(alpha)))) * Float32(Float32(Float32(1.0) / fma(cosTheta, Float32(cosTheta * fma(alpha, alpha, Float32(-1.0))), Float32(1.0))) / Float32(2.0)))
end

Derivation

1. Initial program 98.5%

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

3. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{\pi}{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}}} \cdot \frac{\frac{1}{\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)}}{2}} \]

4. Final simplification 98.6%

   \[\leadsto \frac{1}{\frac{\pi}{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}}} \cdot \frac{\frac{1}{\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)}}{2} \]

Alternative 4: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot -2}}{\pi \cdot \left(-1 - \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (/ (fma alpha alpha -1.0) (* (log alpha) -2.0))
  (* PI (- -1.0 (* (fma alpha alpha -1.0) (* cosTheta cosTheta))))))

C

float code(float cosTheta, float alpha) {
	return (fmaf(alpha, alpha, -1.0f) / (logf(alpha) * -2.0f)) / (((float) M_PI) * (-1.0f - (fmaf(alpha, alpha, -1.0f) * (cosTheta * cosTheta))));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(log(alpha) * Float32(-2.0))) / Float32(Float32(pi) * Float32(Float32(-1.0) - Float32(fma(alpha, alpha, Float32(-1.0)) * Float32(cosTheta * cosTheta)))))
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around inf 98.3%

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

4. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\frac{0.5}{\pi \cdot \left(\log \alpha \cdot \left(-1 - \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)\right)\right)} \cdot \left(1 - \alpha \cdot \alpha\right)} \]

6. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot -2}}{\pi \cdot \left(-1 - \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)\right)}} \]

7. Final simplification 98.6%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot -2}}{\pi \cdot \left(-1 - \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)\right)} \]

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (*
  (/
   0.5
   (*
    PI
    (*
     (log alpha)
     (- -1.0 (* (* cosTheta cosTheta) (+ -1.0 (* alpha alpha)))))))
  (- 1.0 (* alpha alpha))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around inf 98.3%

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

4. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\frac{0.5}{\pi \cdot \left(\log \alpha \cdot \left(-1 - \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)\right)\right)} \cdot \left(1 - \alpha \cdot \alpha\right)} \]

5. Taylor expanded in alpha around 0 98.3%

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

   1. distribute-rgt-out 98.3%

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

   2. unpow2 98.3%

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

   3. unpow2 98.3%

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

7. Simplified 98.3%

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

8. Final simplification 98.3%

   \[\leadsto \frac{0.5}{\pi \cdot \left(\log \alpha \cdot \left(-1 - \left(cosTheta \cdot cosTheta\right) \cdot \left(-1 + \alpha \cdot \alpha\right)\right)\right)} \cdot \left(1 - \alpha \cdot \alpha\right) \]

Alternative 6: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \alpha \cdot \alpha\\ \frac{t_0}{\left(1 + cosTheta \cdot \left(cosTheta \cdot t_0\right)\right) \cdot \left(2 \cdot \left(\pi \cdot \log \alpha\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ -1.0 (* alpha alpha))))
   (/
    t_0
    (* (+ 1.0 (* cosTheta (* cosTheta t_0))) (* 2.0 (* PI (log alpha)))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = -1.0f + (alpha * alpha);
	return t_0 / ((1.0f + (cosTheta * (cosTheta * t_0))) * (2.0f * (((float) M_PI) * logf(alpha))));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(-1.0) + Float32(alpha * alpha))
	return Float32(t_0 / Float32(Float32(Float32(1.0) + Float32(cosTheta * Float32(cosTheta * t_0))) * Float32(Float32(2.0) * Float32(Float32(pi) * log(alpha)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = single(-1.0) + (alpha * alpha);
	tmp = t_0 / ((single(1.0) + (cosTheta * (cosTheta * t_0))) * (single(2.0) * (single(pi) * log(alpha))));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0 98.5%

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

3. Final simplification 98.5%

   \[\leadsto \frac{-1 + \alpha \cdot \alpha}{\left(1 + cosTheta \cdot \left(cosTheta \cdot \left(-1 + \alpha \cdot \alpha\right)\right)\right) \cdot \left(2 \cdot \left(\pi \cdot \log \alpha\right)\right)} \]

Alternative 7: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ -1.0 (* alpha alpha))))
   (/
    t_0
    (* (+ 1.0 (* cosTheta (* cosTheta t_0))) (* PI (log (* alpha alpha)))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Final simplification 98.5%

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

Alternative 8: 97.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (* (/ (+ alpha 1.0) (* PI (- 1.0 (* cosTheta cosTheta)))) (- 1.0 alpha))
  (* (log alpha) -2.0)))

C

float code(float cosTheta, float alpha) {
	return (((alpha + 1.0f) / (((float) M_PI) * (1.0f - (cosTheta * cosTheta)))) * (1.0f - alpha)) / (logf(alpha) * -2.0f);
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(Float32(alpha + Float32(1.0)) / Float32(Float32(pi) * Float32(Float32(1.0) - Float32(cosTheta * cosTheta)))) * Float32(Float32(1.0) - alpha)) / Float32(log(alpha) * Float32(-2.0)))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (((alpha + single(1.0)) / (single(pi) * (single(1.0) - (cosTheta * cosTheta)))) * (single(1.0) - alpha)) / (log(alpha) * single(-2.0));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0 98.2%

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

   1. clear-num 98.0%

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

   2. inv-pow 98.0%

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

   3. *-commutative 98.0%

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

   4. difference-of-sqr-1 97.9%

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

   5. times-frac 97.7%

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

   6. *-commutative 97.7%

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

   7. pow2 97.7%

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

   8. log-pow 97.9%

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

   9. associate-*r* 97.9%

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

   10. *-commutative 97.9%

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

4. Applied egg-rr 97.9%

   \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(cosTheta, -cosTheta, 1\right)}{\alpha - -1}\right)}^{-1} \cdot {\left(\frac{\pi \cdot \left(\log \alpha \cdot 2\right)}{\alpha + -1}\right)}^{-1}} \]
5. Step-by-step derivation

   1. *-commutative 97.9%

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

   2. unpow-1 97.9%

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

   3. *-commutative 97.9%

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

   4. count-2 97.9%

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

   5. log-prod 97.8%

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

   6. associate-/l* 97.9%

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

   7. +-commutative 97.9%

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

   8. log-prod 98.0%

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

   9. count-2 98.0%

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

   10. *-commutative 98.0%

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

   11. unpow-1 98.0%

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

   12. sub-neg 98.0%

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

   13. metadata-eval 98.0%

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

   14. +-commutative 98.0%

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

6. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\pi}{\frac{-1 + \alpha}{\log \alpha \cdot 2}}} \cdot \frac{1}{\frac{\mathsf{fma}\left(cosTheta, -cosTheta, 1\right)}{1 + \alpha}}} \]
7. Step-by-step derivation

   1. associate-*l/ 98.1%

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

   2. div-inv 98.1%

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

   3. associate-/r/ 98.0%

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

   4. associate-/r* 98.0%

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

   5. frac-2neg 98.0%

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

   6. clear-num 97.9%

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

   7. +-commutative 97.9%

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

   8. fma-udef 97.9%

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

   9. distribute-rgt-neg-in 97.9%

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

   10. +-commutative 97.9%

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

   11. unsub-neg 97.9%

       \[\leadsto \frac{-\frac{\frac{\alpha + 1}{\color{blue}{1 - cosTheta \cdot cosTheta}}}{\frac{\pi}{-1 + \alpha}}}{-\log \alpha \cdot 2} \]

   12. +-commutative 97.9%

       \[\leadsto \frac{-\frac{\frac{\alpha + 1}{1 - cosTheta \cdot cosTheta}}{\frac{\pi}{\color{blue}{\alpha + -1}}}}{-\log \alpha \cdot 2} \]

   13. distribute-rgt-neg-in 97.9%

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

   14. metadata-eval 97.9%

       \[\leadsto \frac{-\frac{\frac{\alpha + 1}{1 - cosTheta \cdot cosTheta}}{\frac{\pi}{\alpha + -1}}}{\log \alpha \cdot \color{blue}{-2}} \]

8. Applied egg-rr 97.9%

   \[\leadsto \color{blue}{\frac{-\frac{\frac{\alpha + 1}{1 - cosTheta \cdot cosTheta}}{\frac{\pi}{\alpha + -1}}}{\log \alpha \cdot -2}} \]
9. Step-by-step derivation

   1. associate-/r/ 98.1%

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

   2. distribute-rgt-neg-in 98.1%

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

   3. associate-/l/ 98.1%

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

   4. +-commutative 98.1%

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

   5. distribute-neg-in 98.1%

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

   6. metadata-eval 98.1%

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

   7. sub-neg 98.1%

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

10. Simplified 98.1%

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

11. Final simplification 98.1%

    \[\leadsto \frac{\frac{\alpha + 1}{\pi \cdot \left(1 - cosTheta \cdot cosTheta\right)} \cdot \left(1 - \alpha\right)}{\log \alpha \cdot -2} \]

Alternative 9: 97.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (*
  (- 1.0 (* alpha alpha))
  (/ 0.5 (* PI (* (log alpha) (+ -1.0 (* cosTheta cosTheta)))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around inf 98.3%

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

4. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\frac{0.5}{\pi \cdot \left(\log \alpha \cdot \left(-1 - \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)\right)\right)} \cdot \left(1 - \alpha \cdot \alpha\right)} \]

5. Taylor expanded in alpha around 0 98.0%

   \[\leadsto \frac{0.5}{\pi \cdot \left(\log \alpha \cdot \left(-1 - \color{blue}{-1 \cdot {cosTheta}^{2}}\right)\right)} \cdot \left(1 - \alpha \cdot \alpha\right) \]
6. Step-by-step derivation

   1. unpow2 98.0%

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

   2. neg-mul-1 98.0%

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

   3. distribute-rgt-neg-in 98.0%

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

7. Simplified 98.0%

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

8. Final simplification 98.0%

   \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{0.5}{\pi \cdot \left(\log \alpha \cdot \left(-1 + cosTheta \cdot cosTheta\right)\right)} \]

Alternative 10: 97.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (+ -1.0 (* alpha alpha))
  (* (* PI (log (* alpha alpha))) (- 1.0 (* cosTheta cosTheta)))))

C

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

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(-1.0) + Float32(alpha * alpha)) / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * Float32(Float32(1.0) - Float32(cosTheta * cosTheta))))
end

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0 98.2%

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

3. Final simplification 98.2%

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

Alternative 11: 95.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(1 - \alpha \cdot \alpha\right) \cdot \frac{-0.5}{\pi \cdot \log \alpha} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (* (- 1.0 (* alpha alpha)) (/ -0.5 (* PI (log alpha)))))

C

float code(float cosTheta, float alpha) {
	return (1.0f - (alpha * alpha)) * (-0.5f / (((float) M_PI) * logf(alpha)));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(1.0) - Float32(alpha * alpha)) * Float32(Float32(-0.5) / Float32(Float32(pi) * log(alpha))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(1.0) - (alpha * alpha)) * (single(-0.5) / (single(pi) * log(alpha)));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around inf 98.3%

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

4. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\frac{0.5}{\pi \cdot \left(\log \alpha \cdot \left(-1 - \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)\right)\right)} \cdot \left(1 - \alpha \cdot \alpha\right)} \]

5. Taylor expanded in cosTheta around 0 95.7%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \log \alpha}} \cdot \left(1 - \alpha \cdot \alpha\right) \]

6. Final simplification 95.7%

   \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{-0.5}{\pi \cdot \log \alpha} \]

Alternative 12: 95.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1 + \alpha \cdot \alpha}{\pi \cdot \left(2 \cdot \log \alpha\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ (+ -1.0 (* alpha alpha)) (* PI (* 2.0 (log alpha)))))

C

float code(float cosTheta, float alpha) {
	return (-1.0f + (alpha * alpha)) / (((float) M_PI) * (2.0f * logf(alpha)));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(-1.0) + Float32(alpha * alpha)) / Float32(Float32(pi) * Float32(Float32(2.0) * log(alpha))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-1.0) + (alpha * alpha)) / (single(pi) * (single(2.0) * log(alpha)));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0 98.2%

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

3. Taylor expanded in cosTheta around 0 95.9%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\pi \cdot \log \left({\alpha}^{2}\right)}} \]
4. Step-by-step derivation

   1. log-pow 95.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\pi \cdot \color{blue}{\left(2 \cdot \log \alpha\right)}} \]

   2. *-commutative 95.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\pi \cdot \color{blue}{\left(\log \alpha \cdot 2\right)}} \]

5. Simplified 95.8%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\pi \cdot \left(\log \alpha \cdot 2\right)}} \]

6. Final simplification 95.8%

   \[\leadsto \frac{-1 + \alpha \cdot \alpha}{\pi \cdot \left(2 \cdot \log \alpha\right)} \]

Alternative 13: 26.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(\alpha + -2\right) \cdot \frac{-\alpha}{\pi} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (* (+ alpha -2.0) (/ (- alpha) PI)))

C

float code(float cosTheta, float alpha) {
	return (alpha + -2.0f) * (-alpha / ((float) M_PI));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(alpha + Float32(-2.0)) * Float32(Float32(-alpha) / Float32(pi)))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (alpha + single(-2.0)) * (-alpha / single(pi));
end

Derivation

1. Initial program 98.5%

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

3. Applied egg-rr -0.0%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\frac{0}{0} \cdot \mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)} \cdot \frac{1 - \alpha}{\pi}} \]

5. Simplified 26.4%

   \[\leadsto \color{blue}{\left(\alpha + -2\right) \cdot \frac{-\alpha}{\pi}} \]

6. Final simplification 26.4%

   \[\leadsto \left(\alpha + -2\right) \cdot \frac{-\alpha}{\pi} \]

Alternative 14: 25.7% accurate, 44.4× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot \left(2 - \alpha\right) \end{array} \]

FPCore

(FPCore (cosTheta alpha) :precision binary32 (* alpha (- 2.0 alpha)))

C

float code(float cosTheta, float alpha) {
	return alpha * (2.0f - alpha);
}

Fortran

real(4) function code(costheta, alpha)
    real(4), intent (in) :: costheta
    real(4), intent (in) :: alpha
    code = alpha * (2.0e0 - alpha)
end function

Julia

function code(cosTheta, alpha)
	return Float32(alpha * Float32(Float32(2.0) - alpha))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = alpha * (single(2.0) - alpha);
end

Derivation

1. Initial program 98.5%

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

3. Applied egg-rr -0.0%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\pi \cdot \frac{0}{0}} \cdot \frac{1 - \alpha}{\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)}} \]

5. Simplified 25.5%

   \[\leadsto \color{blue}{\left(\alpha + -2\right) \cdot \left(-\alpha\right)} \]
6. Step-by-step derivation

   1. add-exp-log_binary32 25.5%

      \[\leadsto \color{blue}{e^{\log \left(\left(\alpha + -2\right) \cdot \left(-\alpha\right)\right)}} \]

7. Applied rewrite-once 25.5%

   \[\leadsto \color{blue}{e^{\log \left(\left(\alpha + -2\right) \cdot \left(-\alpha\right)\right)}} \]

8. Taylor expanded in alpha around 0 25.5%

   \[\leadsto \color{blue}{-1 \cdot {\alpha}^{2} + 2 \cdot \alpha} \]
9. Step-by-step derivation

   1. +-commutative 25.5%

      \[\leadsto \color{blue}{2 \cdot \alpha + -1 \cdot {\alpha}^{2}} \]

   2. *-commutative 25.5%

      \[\leadsto \color{blue}{\alpha \cdot 2} + -1 \cdot {\alpha}^{2} \]

   3. mul-1-neg 25.5%

      \[\leadsto \alpha \cdot 2 + \color{blue}{\left(-{\alpha}^{2}\right)} \]

   4. unpow2 25.5%

      \[\leadsto \alpha \cdot 2 + \left(-\color{blue}{\alpha \cdot \alpha}\right) \]

   5. sub-neg 25.5%

      \[\leadsto \color{blue}{\alpha \cdot 2 - \alpha \cdot \alpha} \]

   6. distribute-lft-out-- 25.5%

      \[\leadsto \color{blue}{\alpha \cdot \left(2 - \alpha\right)} \]

10. Simplified 25.5%

    \[\leadsto \color{blue}{\alpha \cdot \left(2 - \alpha\right)} \]

11. Final simplification 25.5%

    \[\leadsto \alpha \cdot \left(2 - \alpha\right) \]

Alternative 15: 25.5% accurate, 74.0× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot 2 \end{array} \]

FPCore

(FPCore (cosTheta alpha) :precision binary32 (* alpha 2.0))

C

float code(float cosTheta, float alpha) {
	return alpha * 2.0f;
}

Fortran

real(4) function code(costheta, alpha)
    real(4), intent (in) :: costheta
    real(4), intent (in) :: alpha
    code = alpha * 2.0e0
end function

Julia

function code(cosTheta, alpha)
	return Float32(alpha * Float32(2.0))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = alpha * single(2.0);
end

Derivation

1. Initial program 98.5%

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

3. Applied egg-rr -0.0%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\pi \cdot \frac{0}{0}} \cdot \frac{1 - \alpha}{\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)}} \]

5. Simplified 25.5%

   \[\leadsto \color{blue}{\left(\alpha + -2\right) \cdot \left(-\alpha\right)} \]

6. Taylor expanded in alpha around 0 25.3%

   \[\leadsto \color{blue}{2 \cdot \alpha} \]
7. Step-by-step derivation

   1. *-commutative 25.3%

      \[\leadsto \color{blue}{\alpha \cdot 2} \]

8. Simplified 25.3%

   \[\leadsto \color{blue}{\alpha \cdot 2} \]

9. Final simplification 25.3%

   \[\leadsto \alpha \cdot 2 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (cosTheta alpha)
  :name "GTR1 distribution"
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0)) (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (- (* alpha alpha) 1.0) (* (* PI (log (* alpha alpha))) (+ 1.0 (* (* (- (* alpha alpha) 1.0) cosTheta) cosTheta)))))