HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 22.5s

Alternatives: 13

Speedup: 0.3×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\\ t_1 := \sqrt[3]{e^{\left(\frac{t_0}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}}\\ t_2 := \sqrt[3]{\frac{0.5}{v} \cdot e^{0.6931 + \frac{t_0 + -1}{v}}}\\ t_3 := \sqrt[3]{\sqrt[3]{t_2}}\\ t_1 \cdot \left(\left({\left({t_2}^{2}\right)}^{0.3333333333333333} \cdot \left(t_3 \cdot \left(t_3 \cdot t_3\right)\right)\right) \cdot t_1\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)))
        (t_1
         (cbrt (exp (+ (+ (/ t_0 v) (+ 0.6931 (/ -1.0 v))) (log (/ 0.5 v))))))
        (t_2 (cbrt (* (/ 0.5 v) (exp (+ 0.6931 (/ (+ t_0 -1.0) v))))))
        (t_3 (cbrt (cbrt t_2))))
   (*
    t_1
    (* (* (pow (pow t_2 2.0) 0.3333333333333333) (* t_3 (* t_3 t_3))) t_1))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O);
	float t_1 = cbrtf(expf((((t_0 / v) + (0.6931f + (-1.0f / v))) + logf((0.5f / v)))));
	float t_2 = cbrtf(((0.5f / v) * expf((0.6931f + ((t_0 + -1.0f) / v)))));
	float t_3 = cbrtf(cbrtf(t_2));
	return t_1 * ((powf(powf(t_2, 2.0f), 0.3333333333333333f) * (t_3 * (t_3 * t_3))) * t_1);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O))
	t_1 = cbrt(exp(Float32(Float32(Float32(t_0 / v) + Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))) + log(Float32(Float32(0.5) / v)))))
	t_2 = cbrt(Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) + Float32(Float32(t_0 + Float32(-1.0)) / v)))))
	t_3 = cbrt(cbrt(t_2))
	return Float32(t_1 * Float32(Float32(((t_2 ^ Float32(2.0)) ^ Float32(0.3333333333333333)) * Float32(t_3 * Float32(t_3 * t_3))) * t_1))
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Step-by-step derivation

   1. add-cube-cbrt 99.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}} \]
6. Step-by-step derivation

   1. pow1/3 99.8%

      \[\leadsto \left(\color{blue}{{\left(e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   2. add-cube-cbrt 99.8%

      \[\leadsto \left({\color{blue}{\left(\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   3. unpow-prod-down 99.8%

      \[\leadsto \left(\color{blue}{\left({\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right)}^{0.3333333333333333}\right)} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

7. Applied egg-rr 99.9%

   \[\leadsto \left(\color{blue}{\left({\left({\left(\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931}}}\right)} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]
8. Step-by-step derivation

   1. add-cube-cbrt 99.9%

      \[\leadsto \left(\left({\left({\left(\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931}}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931}}}}\right)}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

9. Applied egg-rr 99.9%

   \[\leadsto \left(\left({\left({\left(\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v} + 0.6931}}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v} + 0.6931}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v} + 0.6931}}}}\right)}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

10. Final simplification 99.9%

    \[\leadsto \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \left(\left({\left({\left(\sqrt[3]{\frac{0.5}{v} \cdot e^{0.6931 + \frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v}}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\frac{0.5}{v} \cdot e^{0.6931 + \frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v}}}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\frac{0.5}{v} \cdot e^{0.6931 + \frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v}}}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\frac{0.5}{v} \cdot e^{0.6931 + \frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v}}}}}\right)\right)\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \]

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\\ t_1 := \sqrt[3]{e^{\left(\frac{t_0}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}}\\ t_1 \cdot \left(t_1 \cdot {\left(\sqrt{\sqrt[3]{\frac{0.5}{v} \cdot e^{0.6931 + \frac{t_0 + -1}{v}}}}\right)}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)))
        (t_1
         (cbrt (exp (+ (+ (/ t_0 v) (+ 0.6931 (/ -1.0 v))) (log (/ 0.5 v)))))))
   (*
    t_1
    (*
     t_1
     (pow
      (sqrt (cbrt (* (/ 0.5 v) (exp (+ 0.6931 (/ (+ t_0 -1.0) v))))))
      2.0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O);
	float t_1 = cbrtf(expf((((t_0 / v) + (0.6931f + (-1.0f / v))) + logf((0.5f / v)))));
	return t_1 * (t_1 * powf(sqrtf(cbrtf(((0.5f / v) * expf((0.6931f + ((t_0 + -1.0f) / v)))))), 2.0f));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O))
	t_1 = cbrt(exp(Float32(Float32(Float32(t_0 / v) + Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))) + log(Float32(Float32(0.5) / v)))))
	return Float32(t_1 * Float32(t_1 * (sqrt(cbrt(Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) + Float32(Float32(t_0 + Float32(-1.0)) / v)))))) ^ Float32(2.0))))
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Step-by-step derivation

   1. add-cube-cbrt 99.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 99.8%

      \[\leadsto \left(\color{blue}{\left(\sqrt{\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}} \cdot \sqrt{\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}}\right)} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   2. pow2 99.8%

      \[\leadsto \left(\color{blue}{{\left(\sqrt{\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}}\right)}^{2}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

7. Applied egg-rr 99.9%

   \[\leadsto \left(\color{blue}{{\left(\sqrt{\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931}}}\right)}^{2}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

8. Final simplification 99.9%

   \[\leadsto \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot {\left(\sqrt{\sqrt[3]{\frac{0.5}{v} \cdot e^{0.6931 + \frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v}}}}\right)}^{2}\right) \]

Alternative 3: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\\ t_1 := \sqrt[3]{e^{\left(\frac{t_0}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}}\\ t_1 \cdot \left(t_1 \cdot {\left(\frac{0.5}{v} \cdot e^{0.6931 + \frac{t_0 + -1}{v}}\right)}^{0.3333333333333333}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)))
        (t_1
         (cbrt (exp (+ (+ (/ t_0 v) (+ 0.6931 (/ -1.0 v))) (log (/ 0.5 v)))))))
   (*
    t_1
    (*
     t_1
     (pow
      (* (/ 0.5 v) (exp (+ 0.6931 (/ (+ t_0 -1.0) v))))
      0.3333333333333333)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O);
	float t_1 = cbrtf(expf((((t_0 / v) + (0.6931f + (-1.0f / v))) + logf((0.5f / v)))));
	return t_1 * (t_1 * powf(((0.5f / v) * expf((0.6931f + ((t_0 + -1.0f) / v)))), 0.3333333333333333f));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O))
	t_1 = cbrt(exp(Float32(Float32(Float32(t_0 / v) + Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))) + log(Float32(Float32(0.5) / v)))))
	return Float32(t_1 * Float32(t_1 * (Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) + Float32(Float32(t_0 + Float32(-1.0)) / v)))) ^ Float32(0.3333333333333333))))
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Step-by-step derivation

   1. add-cube-cbrt 99.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}} \]
6. Step-by-step derivation

   1. pow1/3 99.8%

      \[\leadsto \left(\color{blue}{{\left(e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   2. +-commutative 99.8%

      \[\leadsto \left({\left(e^{\color{blue}{\log \left(\frac{0.5}{v}\right) + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right)}}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   3. exp-sum 99.8%

      \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   4. add-exp-log 99.8%

      \[\leadsto \left({\left(\color{blue}{\frac{0.5}{v}} \cdot e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   5. associate--r- 99.8%

      \[\leadsto \left({\left(\frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right) + 0.6931}}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   6. sub-div 99.8%

      \[\leadsto \left({\left(\frac{0.5}{v} \cdot e^{\color{blue}{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v}} + 0.6931}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

7. Applied egg-rr 99.8%

   \[\leadsto \left(\color{blue}{{\left(\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

8. Final simplification 99.8%

   \[\leadsto \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot {\left(\frac{0.5}{v} \cdot e^{0.6931 + \frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v}}\right)}^{0.3333333333333333}\right) \]

Alternative 4: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\\ t_1 := \sqrt[3]{e^{\left(\frac{t_0}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}}\\ t_1 \cdot \left(\sqrt[3]{\frac{0.5}{v} \cdot e^{0.6931 + \frac{t_0 + -1}{v}}} \cdot t_1\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)))
        (t_1
         (cbrt (exp (+ (+ (/ t_0 v) (+ 0.6931 (/ -1.0 v))) (log (/ 0.5 v)))))))
   (* t_1 (* (cbrt (* (/ 0.5 v) (exp (+ 0.6931 (/ (+ t_0 -1.0) v))))) t_1))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O);
	float t_1 = cbrtf(expf((((t_0 / v) + (0.6931f + (-1.0f / v))) + logf((0.5f / v)))));
	return t_1 * (cbrtf(((0.5f / v) * expf((0.6931f + ((t_0 + -1.0f) / v))))) * t_1);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O))
	t_1 = cbrt(exp(Float32(Float32(Float32(t_0 / v) + Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))) + log(Float32(Float32(0.5) / v)))))
	return Float32(t_1 * Float32(cbrt(Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) + Float32(Float32(t_0 + Float32(-1.0)) / v))))) * t_1))
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Step-by-step derivation

   1. add-cube-cbrt 99.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}} \]
6. Step-by-step derivation

   1. pow1/3 99.8%

      \[\leadsto \left(\color{blue}{{\left(e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   2. +-commutative 99.8%

      \[\leadsto \left({\left(e^{\color{blue}{\log \left(\frac{0.5}{v}\right) + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right)}}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   3. exp-sum 99.8%

      \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   4. add-exp-log 99.8%

      \[\leadsto \left({\left(\color{blue}{\frac{0.5}{v}} \cdot e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   5. associate--r- 99.8%

      \[\leadsto \left({\left(\frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right) + 0.6931}}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   6. sub-div 99.8%

      \[\leadsto \left({\left(\frac{0.5}{v} \cdot e^{\color{blue}{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v}} + 0.6931}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

7. Applied egg-rr 99.8%

   \[\leadsto \left(\color{blue}{{\left(\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]
8. Step-by-step derivation

   1. pow1/3 99.8%

      \[\leadsto \left(\color{blue}{\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931}}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   2. sub-neg 99.8%

      \[\leadsto \left(\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\color{blue}{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + \left(-1\right)}}{v} + 0.6931}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   3. metadata-eval 99.8%

      \[\leadsto \left(\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + \color{blue}{-1}}{v} + 0.6931}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

9. Applied egg-rr 99.8%

   \[\leadsto \left(\color{blue}{\sqrt[3]{\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v} + 0.6931}}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

10. Final simplification 99.8%

    \[\leadsto \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \left(\sqrt[3]{\frac{0.5}{v} \cdot e^{0.6931 + \frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v}}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \]

Alternative 5: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{0.5}{v}\right)\\ t_1 := cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\\ \sqrt[3]{e^{\left(\frac{t_1}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + t_0}} \cdot \left({\left(\frac{0.5}{v} \cdot e^{0.6931 + \frac{t_1 + -1}{v}}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(0.6931 + \left(t_0 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) + \frac{-1}{v}}}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (log (/ 0.5 v)))
        (t_1 (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O))))
   (*
    (cbrt (exp (+ (+ (/ t_1 v) (+ 0.6931 (/ -1.0 v))) t_0)))
    (*
     (pow (* (/ 0.5 v) (exp (+ 0.6931 (/ (+ t_1 -1.0) v)))) 0.3333333333333333)
     (cbrt
      (exp
       (+ (+ 0.6931 (+ t_0 (/ (* cosTheta_i cosTheta_O) v))) (/ -1.0 v))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = logf((0.5f / v));
	float t_1 = (cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O);
	return cbrtf(expf((((t_1 / v) + (0.6931f + (-1.0f / v))) + t_0))) * (powf(((0.5f / v) * expf((0.6931f + ((t_1 + -1.0f) / v)))), 0.3333333333333333f) * cbrtf(expf(((0.6931f + (t_0 + ((cosTheta_i * cosTheta_O) / v))) + (-1.0f / v)))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = log(Float32(Float32(0.5) / v))
	t_1 = Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O))
	return Float32(cbrt(exp(Float32(Float32(Float32(t_1 / v) + Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))) + t_0))) * Float32((Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) + Float32(Float32(t_1 + Float32(-1.0)) / v)))) ^ Float32(0.3333333333333333)) * cbrt(exp(Float32(Float32(Float32(0.6931) + Float32(t_0 + Float32(Float32(cosTheta_i * cosTheta_O) / v))) + Float32(Float32(-1.0) / v))))))
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Step-by-step derivation

   1. add-cube-cbrt 99.8%

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left(\sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}} \]
6. Step-by-step derivation

   1. pow1/3 99.8%

      \[\leadsto \left(\color{blue}{{\left(e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   2. +-commutative 99.8%

      \[\leadsto \left({\left(e^{\color{blue}{\log \left(\frac{0.5}{v}\right) + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right)}}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   3. exp-sum 99.8%

      \[\leadsto \left({\color{blue}{\left(e^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   4. add-exp-log 99.8%

      \[\leadsto \left({\left(\color{blue}{\frac{0.5}{v}} \cdot e^{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   5. associate--r- 99.8%

      \[\leadsto \left({\left(\frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right) + 0.6931}}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

   6. sub-div 99.8%

      \[\leadsto \left({\left(\frac{0.5}{v} \cdot e^{\color{blue}{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v}} + 0.6931}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

7. Applied egg-rr 99.8%

   \[\leadsto \left(\color{blue}{{\left(\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

8. Taylor expanded in sinTheta_i around 0 99.8%

   \[\leadsto \left({\left(\frac{0.5}{v} \cdot e^{\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) - 1}{v} + 0.6931}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\color{blue}{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}}}}\right) \cdot \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

9. Final simplification 99.8%

   \[\leadsto \sqrt[3]{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \left({\left(\frac{0.5}{v} \cdot e^{0.6931 + \frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v}}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) + \frac{-1}{v}}}\right) \]

Alternative 6: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.6931 + \frac{-1}{v}\\ t_1 := \log \left(\frac{0.5}{v}\right)\\ \sqrt{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + t_0\right) + t_1}} \cdot \sqrt{e^{t_1 + \left(\frac{cosTheta_i \cdot cosTheta_O}{v} + t_0\right)}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ 0.6931 (/ -1.0 v))) (t_1 (log (/ 0.5 v))))
   (*
    (sqrt
     (exp
      (+
       (+ (/ (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)) v) t_0)
       t_1)))
    (sqrt (exp (+ t_1 (+ (/ (* cosTheta_i cosTheta_O) v) t_0)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = 0.6931f + (-1.0f / v);
	float t_1 = logf((0.5f / v));
	return sqrtf(expf((((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) / v) + t_0) + t_1))) * sqrtf(expf((t_1 + (((cosTheta_i * cosTheta_O) / v) + t_0))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: t_1
    t_0 = 0.6931e0 + ((-1.0e0) / v)
    t_1 = log((0.5e0 / v))
    code = sqrt(exp((((((costheta_i * costheta_o) - (sintheta_i * sintheta_o)) / v) + t_0) + t_1))) * sqrt(exp((t_1 + (((costheta_i * costheta_o) / v) + t_0))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))
	t_1 = log(Float32(Float32(0.5) / v))
	return Float32(sqrt(exp(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O)) / v) + t_0) + t_1))) * sqrt(exp(Float32(t_1 + Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) + t_0)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = single(0.6931) + (single(-1.0) / v);
	t_1 = log((single(0.5) / v));
	tmp = sqrt(exp((((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) / v) + t_0) + t_1))) * sqrt(exp((t_1 + (((cosTheta_i * cosTheta_O) / v) + t_0))));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 99.8%

      \[\leadsto \color{blue}{\sqrt{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot \sqrt{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

   2. associate-+r+ 99.8%

      \[\leadsto \sqrt{e^{\color{blue}{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + 0.6931\right) + \log \left(\frac{0.5}{v}\right)}}} \cdot \sqrt{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

   3. associate-/l* 99.8%

      \[\leadsto \sqrt{e^{\left(\left(\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + 0.6931\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

   4. associate--l- 99.8%

      \[\leadsto \sqrt{e^{\left(\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + 0.6931\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

   5. associate-+l- 99.8%

      \[\leadsto \sqrt{e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \left(\frac{1}{v} - 0.6931\right)\right)} + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

   6. sub-div 99.8%

      \[\leadsto \sqrt{e^{\left(\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v}} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\sqrt{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}}} \]

6. Taylor expanded in cosTheta_i around inf 99.8%

   \[\leadsto \sqrt{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt{e^{\left(\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} - \left(\frac{1}{v} - 0.6931\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

7. Final simplification 99.8%

   \[\leadsto \sqrt{e^{\left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}} \cdot \sqrt{e^{\log \left(\frac{0.5}{v}\right) + \left(\frac{cosTheta_i \cdot cosTheta_O}{v} + \left(0.6931 + \frac{-1}{v}\right)\right)}} \]

Alternative 7: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{-1}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (+ 0.6931 (+ (log (/ 0.5 v)) (/ -1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((0.6931f + (logf((0.5f / v)) + (-1.0f / v))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp((0.6931e0 + (log((0.5e0 / v)) + ((-1.0e0) / v))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(0.6931) + Float32(log(Float32(Float32(0.5) / v)) + Float32(Float32(-1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(0.6931) + (log((single(0.5) / v)) + (single(-1.0) / v))));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

2. Taylor expanded in sinTheta_i around 0 99.8%

   \[\leadsto e^{\color{blue}{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}}} \]
3. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto e^{\color{blue}{0.6931 + \left(\left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}\right)}} \]

4. Simplified 99.8%

   \[\leadsto e^{\color{blue}{0.6931 + \left(\left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}\right)}} \]

5. Taylor expanded in cosTheta_i around 0 99.8%

   \[\leadsto e^{0.6931 + \color{blue}{\left(\log \left(\frac{0.5}{v}\right) - \frac{1}{v}\right)}} \]

6. Final simplification 99.8%

   \[\leadsto e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{-1}{v}\right)} \]

Alternative 8: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ -1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf((0.6931f + (-1.0f / v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * exp((0.6931e0 + ((-1.0e0) / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp((single(0.6931) + (single(-1.0) / v)));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.8%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v}} \]

4. Taylor expanded in sinTheta_i around 0 99.8%

   \[\leadsto \color{blue}{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}} \cdot \frac{0.5}{v} \]

5. Taylor expanded in cosTheta_i around 0 99.8%

   \[\leadsto e^{\color{blue}{0.6931} - \frac{1}{v}} \cdot \frac{0.5}{v} \]

6. Final simplification 99.8%

   \[\leadsto \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \]

Alternative 9: 13.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{0.6931 + cosTheta_O \cdot \frac{cosTheta_i}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (+ 0.6931 (* cosTheta_O (/ cosTheta_i v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((0.6931f + (cosTheta_O * (cosTheta_i / v))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp((0.6931e0 + (costheta_o * (costheta_i / v))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(0.6931) + Float32(cosTheta_O * Float32(cosTheta_i / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(0.6931) + (cosTheta_O * (cosTheta_i / v))));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

2. Taylor expanded in sinTheta_i around 0 99.8%

   \[\leadsto e^{\color{blue}{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}}} \]
3. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto e^{\color{blue}{0.6931 + \left(\left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}\right)}} \]

4. Simplified 99.8%

   \[\leadsto e^{\color{blue}{0.6931 + \left(\left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}\right)}} \]

5. Taylor expanded in cosTheta_i around inf 12.1%

   \[\leadsto e^{0.6931 + \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}}} \]
6. Step-by-step derivation

   1. associate-*l/ 12.1%

      \[\leadsto e^{0.6931 + \color{blue}{\frac{cosTheta_i}{v} \cdot cosTheta_O}} \]

   2. *-commutative 12.1%

      \[\leadsto e^{0.6931 + \color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}}} \]

7. Simplified 12.1%

   \[\leadsto e^{0.6931 + \color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}}} \]

8. Final simplification 12.1%

   \[\leadsto e^{0.6931 + cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

Alternative 10: 13.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{0.6931 + \frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (+ 0.6931 (/ cosTheta_i (/ v cosTheta_O)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((0.6931f + (cosTheta_i / (v / cosTheta_O))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp((0.6931e0 + (costheta_i / (v / costheta_o))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(0.6931) + Float32(cosTheta_i / Float32(v / cosTheta_O))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(0.6931) + (cosTheta_i / (v / cosTheta_O))));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

2. Taylor expanded in sinTheta_i around 0 99.8%

   \[\leadsto e^{\color{blue}{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}}} \]
3. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto e^{\color{blue}{0.6931 + \left(\left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}\right)}} \]

4. Simplified 99.8%

   \[\leadsto e^{\color{blue}{0.6931 + \left(\left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}\right)}} \]

5. Taylor expanded in cosTheta_i around inf 12.1%

   \[\leadsto e^{0.6931 + \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}}} \]
6. Step-by-step derivation

   1. associate-/l* 12.1%

      \[\leadsto e^{0.6931 + \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}} \]

7. Simplified 12.1%

   \[\leadsto e^{0.6931 + \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}} \]

8. Final simplification 12.1%

   \[\leadsto e^{0.6931 + \frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]

Alternative 11: 13.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (+ 0.6931 (/ (* cosTheta_i cosTheta_O) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((0.6931f + ((cosTheta_i * cosTheta_O) / v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp((0.6931e0 + ((costheta_i * costheta_o) / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(0.6931) + Float32(Float32(cosTheta_i * cosTheta_O) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(0.6931) + ((cosTheta_i * cosTheta_O) / v)));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

2. Taylor expanded in sinTheta_i around 0 99.8%

   \[\leadsto e^{\color{blue}{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}}} \]
3. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto e^{\color{blue}{0.6931 + \left(\left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}\right)}} \]

4. Simplified 99.8%

   \[\leadsto e^{\color{blue}{0.6931 + \left(\left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}\right)}} \]

5. Taylor expanded in cosTheta_i around inf 12.1%

   \[\leadsto e^{0.6931 + \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}}} \]

6. Final simplification 12.1%

   \[\leadsto e^{0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}} \]

Alternative 12: 4.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{0.6931} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp 0.6931)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf(0.6931f);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * exp(0.6931e0)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(Float32(0.6931)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp(single(0.6931));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Taylor expanded in v around inf 4.7%

   \[\leadsto \color{blue}{e^{\log \left(\frac{1}{v}\right) + \left(0.6931 + \log 0.5\right)}} \]
5. Step-by-step derivation

   1. +-commutative 4.7%

      \[\leadsto e^{\color{blue}{\left(0.6931 + \log 0.5\right) + \log \left(\frac{1}{v}\right)}} \]

   2. associate-+l+ 4.7%

      \[\leadsto e^{\color{blue}{0.6931 + \left(\log 0.5 + \log \left(\frac{1}{v}\right)\right)}} \]

   3. log-rec 4.7%

      \[\leadsto e^{0.6931 + \left(\log 0.5 + \color{blue}{\left(-\log v\right)}\right)} \]

   4. sub-neg 4.7%

      \[\leadsto e^{0.6931 + \color{blue}{\left(\log 0.5 - \log v\right)}} \]

   5. log-div 4.7%

      \[\leadsto e^{0.6931 + \color{blue}{\log \left(\frac{0.5}{v}\right)}} \]

   6. exp-sum 4.7%

      \[\leadsto \color{blue}{e^{0.6931} \cdot e^{\log \left(\frac{0.5}{v}\right)}} \]

   7. rem-exp-log 4.7%

      \[\leadsto e^{0.6931} \cdot \color{blue}{\frac{0.5}{v}} \]

6. Simplified 4.7%

   \[\leadsto \color{blue}{e^{0.6931} \cdot \frac{0.5}{v}} \]

7. Final simplification 4.7%

   \[\leadsto \frac{0.5}{v} \cdot e^{0.6931} \]

Alternative 13: 4.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot e^{0.6931}}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* 0.5 (exp 0.6931)) v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f * expf(0.6931f)) / v;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 * exp(0.6931e0)) / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) * exp(Float32(0.6931))) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) * exp(single(0.6931))) / v;
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.8%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Taylor expanded in v around inf 4.7%

   \[\leadsto \color{blue}{e^{\log \left(\frac{1}{v}\right) + \left(0.6931 + \log 0.5\right)}} \]
5. Step-by-step derivation

   1. +-commutative 4.7%

      \[\leadsto e^{\color{blue}{\left(0.6931 + \log 0.5\right) + \log \left(\frac{1}{v}\right)}} \]

   2. associate-+l+ 4.7%

      \[\leadsto e^{\color{blue}{0.6931 + \left(\log 0.5 + \log \left(\frac{1}{v}\right)\right)}} \]

   3. log-rec 4.7%

      \[\leadsto e^{0.6931 + \left(\log 0.5 + \color{blue}{\left(-\log v\right)}\right)} \]

   4. sub-neg 4.7%

      \[\leadsto e^{0.6931 + \color{blue}{\left(\log 0.5 - \log v\right)}} \]

   5. log-div 4.7%

      \[\leadsto e^{0.6931 + \color{blue}{\log \left(\frac{0.5}{v}\right)}} \]

   6. exp-sum 4.7%

      \[\leadsto \color{blue}{e^{0.6931} \cdot e^{\log \left(\frac{0.5}{v}\right)}} \]

   7. rem-exp-log 4.7%

      \[\leadsto e^{0.6931} \cdot \color{blue}{\frac{0.5}{v}} \]

6. Simplified 4.7%

   \[\leadsto \color{blue}{e^{0.6931} \cdot \frac{0.5}{v}} \]
7. Step-by-step derivation

   1. associate-*r/ 4.7%

      \[\leadsto \color{blue}{\frac{e^{0.6931} \cdot 0.5}{v}} \]

8. Applied egg-rr 4.7%

   \[\leadsto \color{blue}{\frac{e^{0.6931} \cdot 0.5}{v}} \]

9. Final simplification 4.7%

   \[\leadsto \frac{0.5 \cdot e^{0.6931}}{v} \]

Reproduce

herbie shell --seed 2023229 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))