HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 21.3s

Alternatives: 12

Speedup: 2.0×

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: 99.6% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{0.5}{v}\right)\\ {\left({\left(e^{\left(0.6931 + \left(t_0 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) + \frac{-1}{v}}\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\left(0.6931 + t_0\right) + \frac{-1}{v}\right)}\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (log (/ 0.5 v))))
   (pow
    (*
     (pow
      (exp (+ (+ 0.6931 (+ t_0 (/ (* cosTheta_i cosTheta_O) v))) (/ -1.0 v)))
      0.3333333333333333)
     (exp (* 0.16666666666666666 (+ (+ 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 = logf((0.5f / v));
	return powf((powf(expf(((0.6931f + (t_0 + ((cosTheta_i * cosTheta_O) / v))) + (-1.0f / v))), 0.3333333333333333f) * expf((0.16666666666666666f * ((0.6931f + t_0) + (-1.0f / v))))), 2.0f);
}

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
    t_0 = log((0.5e0 / v))
    code = ((exp(((0.6931e0 + (t_0 + ((costheta_i * costheta_o) / v))) + ((-1.0e0) / v))) ** 0.3333333333333333e0) * exp((0.16666666666666666e0 * ((0.6931e0 + t_0) + ((-1.0e0) / v))))) ** 2.0e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = log(Float32(Float32(0.5) / v))
	return Float32((exp(Float32(Float32(Float32(0.6931) + Float32(t_0 + Float32(Float32(cosTheta_i * cosTheta_O) / v))) + Float32(Float32(-1.0) / v))) ^ Float32(0.3333333333333333)) * exp(Float32(Float32(0.16666666666666666) * Float32(Float32(Float32(0.6931) + t_0) + Float32(Float32(-1.0) / v))))) ^ Float32(2.0)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = log((single(0.5) / v));
	tmp = ((exp(((single(0.6931) + (t_0 + ((cosTheta_i * cosTheta_O) / v))) + (single(-1.0) / v))) ^ single(0.3333333333333333)) * exp((single(0.16666666666666666) * ((single(0.6931) + t_0) + (single(-1.0) / v))))) ^ single(2.0);
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

   1. add-sqr-sqrt 99.5%

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

   2. pow2 99.5%

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

   3. associate-/r/ 99.5%

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

   4. associate-*l/ 99.5%

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

   5. div-inv 99.5%

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

   6. clear-num 99.5%

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

   7. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

   1. add-cube-cbrt 99.6%

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in cosTheta_O around 0 99.6%

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

   1. unpow1/3 99.6%

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

   2. exp-diff 99.5%

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

   3. +-commutative 99.5%

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

   4. associate-*r/ 99.5%

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

   5. fma-udef 99.5%

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

   6. div-exp 99.6%

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

   7. fma-udef 99.6%

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

   8. associate-*r/ 99.6%

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

   9. +-commutative 99.6%

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

   10. *-lft-identity 99.6%

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

   11. metadata-eval 99.6%

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

   12. cancel-sign-sub-inv 99.6%

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

   13. associate-*r/ 99.6%

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

   14. div-sub 99.6%

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

   15. neg-mul-1 99.6%

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

   16. distribute-lft-neg-in 99.6%

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

   17. cancel-sign-sub 99.6%

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

10. Simplified 99.6%

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

11. Taylor expanded in sinTheta_i around 0 99.6%

    \[\leadsto {\color{blue}{\left({\left(1 \cdot e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}}\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}\right)}\right)}}^{2} \]

12. Final simplification 99.6%

    \[\leadsto {\left({\left(e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) + \frac{-1}{v}}\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{-1}{v}\right)}\right)}^{2} \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

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) - ((sinTheta_i * sinTheta_O) / v)) + (-1.0f / v))) + logf((1.0f / (v * 2.0f)))));
}

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) - ((sintheta_i * sintheta_o) / v)) + ((-1.0e0) / v))) + log((1.0e0 / (v * 2.0e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Final simplification 99.5%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (- (+ 0.6931 (log (/ 0.5 v))) (/ (+ 1.0 (* sinTheta_i sinTheta_O)) 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 + (sinTheta_i * sinTheta_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 + log((0.5e0 / v))) - ((1.0e0 + (sintheta_i * sintheta_o)) / v)))
end function

Julia

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in cosTheta_O around 0 99.5%

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

   1. associate--l+ 99.5%

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

   2. associate-*r/ 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-udef 99.5%

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

6. Applied egg-rr 99.5%

   \[\leadsto e^{\color{blue}{0.6931 + \left(\log \left(\frac{0.5}{v}\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right)}} \]
7. Step-by-step derivation

   1. associate-+r- 99.5%

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

   2. fma-udef 99.5%

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

   3. associate-*r/ 99.5%

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

   4. +-commutative 99.5%

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

   5. *-lft-identity 99.5%

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

   6. metadata-eval 99.5%

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

   7. cancel-sign-sub-inv 99.5%

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

   8. associate-*r/ 99.5%

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

   9. div-sub 99.5%

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

   10. neg-mul-1 99.5%

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

   11. distribute-lft-neg-in 99.5%

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

   12. cancel-sign-sub 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ (- -1.0 (* sinTheta_i sinTheta_O)) 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 - (sinTheta_i * sinTheta_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 = (0.5e0 / v) * exp((0.6931e0 + (((-1.0e0) - (sintheta_i * sintheta_o)) / 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(Float32(-1.0) - Float32(sinTheta_i * sinTheta_O)) / 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) - (sinTheta_i * sinTheta_O)) / v)));
end

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.4%

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

3. Simplified 99.5%

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

   1. add-cbrt-cube 97.9%

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

   2. pow1/3 97.9%

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

   3. pow3 97.9%

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

   4. associate-*l/ 97.9%

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

   5. *-commutative 97.9%

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

   6. +-commutative 97.9%

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

5. Applied egg-rr 97.9%

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

6. Taylor expanded in cosTheta_O around 0 99.5%

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

   1. *-lft-identity 99.5%

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

   2. metadata-eval 99.5%

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

   3. cancel-sign-sub-inv 99.5%

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

   4. associate-*r/ 99.5%

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

   5. div-sub 99.5%

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

   6. neg-mul-1 99.5%

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

   7. distribute-lft-neg-in 99.5%

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

   8. cancel-sign-sub 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 5: 44.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq 4.0000000126843074 \cdot 10^{-30}:\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{-sinTheta_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 4.0000000126843074e-30)
   (/ (* sinTheta_O (- sinTheta_i)) v)
   (exp (* sinTheta_O (/ (- sinTheta_i) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 4.0000000126843074e-30f) {
		tmp = (sinTheta_O * -sinTheta_i) / v;
	} else {
		tmp = expf((sinTheta_O * (-sinTheta_i / v)));
	}
	return tmp;
}

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) :: tmp
    if ((sintheta_i * sintheta_o) <= 4.0000000126843074e-30) then
        tmp = (sintheta_o * -sintheta_i) / v
    else
        tmp = exp((sintheta_o * (-sintheta_i / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(4.0000000126843074e-30))
		tmp = Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v);
	else
		tmp = exp(Float32(sinTheta_O * Float32(Float32(-sinTheta_i) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(4.0000000126843074e-30))
		tmp = (sinTheta_O * -sinTheta_i) / v;
	else
		tmp = exp((sinTheta_O * (-sinTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 4e-30

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in sinTheta_i around inf 6.2%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   5. Step-by-step derivation

      1. mul-1-neg 6.2%

         \[\leadsto e^{\color{blue}{-\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

      2. associate-*l/ 6.2%

         \[\leadsto e^{-\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}} \]

      3. distribute-lft-neg-in 6.2%

         \[\leadsto e^{\color{blue}{\left(-\frac{sinTheta_i}{v}\right) \cdot sinTheta_O}} \]

      4. *-commutative 6.2%

         \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

      5. distribute-neg-frac 6.2%

         \[\leadsto e^{sinTheta_O \cdot \color{blue}{\frac{-sinTheta_i}{v}}} \]

   6. Simplified 6.2%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \frac{-sinTheta_i}{v}}} \]

   7. Taylor expanded in sinTheta_O around 0 6.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v} + 1} \]
   8. Step-by-step derivation

      1. +-commutative 6.4%

         \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

      2. mul-1-neg 6.4%

         \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_i \cdot sinTheta_O}{v}\right)} \]

      3. unsub-neg 6.4%

         \[\leadsto \color{blue}{1 - \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

      4. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

   9. Simplified 6.4%

      \[\leadsto \color{blue}{1 - sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

   10. Taylor expanded in sinTheta_i around inf 41.6%

       \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

   if 4e-30 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in sinTheta_i around inf 57.9%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   5. Step-by-step derivation

      1. mul-1-neg 57.9%

         \[\leadsto e^{\color{blue}{-\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

      2. associate-*l/ 57.9%

         \[\leadsto e^{-\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}} \]

      3. distribute-lft-neg-in 57.9%

         \[\leadsto e^{\color{blue}{\left(-\frac{sinTheta_i}{v}\right) \cdot sinTheta_O}} \]

      4. *-commutative 57.9%

         \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

      5. distribute-neg-frac 57.9%

         \[\leadsto e^{sinTheta_O \cdot \color{blue}{\frac{-sinTheta_i}{v}}} \]

   6. Simplified 57.9%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \frac{-sinTheta_i}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq 4.0000000126843074 \cdot 10^{-30}:\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{-sinTheta_i}{v}}\\ \end{array} \]

Alternative 6: 44.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq 4.0000000126843074 \cdot 10^{-30}:\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta_O}{\frac{v}{-sinTheta_i}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 4.0000000126843074e-30)
   (/ (* sinTheta_O (- sinTheta_i)) v)
   (exp (/ sinTheta_O (/ v (- sinTheta_i))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 4.0000000126843074e-30f) {
		tmp = (sinTheta_O * -sinTheta_i) / v;
	} else {
		tmp = expf((sinTheta_O / (v / -sinTheta_i)));
	}
	return tmp;
}

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) :: tmp
    if ((sintheta_i * sintheta_o) <= 4.0000000126843074e-30) then
        tmp = (sintheta_o * -sintheta_i) / v
    else
        tmp = exp((sintheta_o / (v / -sintheta_i)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(4.0000000126843074e-30))
		tmp = Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v);
	else
		tmp = exp(Float32(sinTheta_O / Float32(v / Float32(-sinTheta_i))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(4.0000000126843074e-30))
		tmp = (sinTheta_O * -sinTheta_i) / v;
	else
		tmp = exp((sinTheta_O / (v / -sinTheta_i)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 4e-30

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in sinTheta_i around inf 6.2%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   5. Step-by-step derivation

      1. mul-1-neg 6.2%

         \[\leadsto e^{\color{blue}{-\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

      2. associate-*l/ 6.2%

         \[\leadsto e^{-\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}} \]

      3. distribute-lft-neg-in 6.2%

         \[\leadsto e^{\color{blue}{\left(-\frac{sinTheta_i}{v}\right) \cdot sinTheta_O}} \]

      4. *-commutative 6.2%

         \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

      5. distribute-neg-frac 6.2%

         \[\leadsto e^{sinTheta_O \cdot \color{blue}{\frac{-sinTheta_i}{v}}} \]

   6. Simplified 6.2%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \frac{-sinTheta_i}{v}}} \]

   7. Taylor expanded in sinTheta_O around 0 6.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v} + 1} \]
   8. Step-by-step derivation

      1. +-commutative 6.4%

         \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

      2. mul-1-neg 6.4%

         \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_i \cdot sinTheta_O}{v}\right)} \]

      3. unsub-neg 6.4%

         \[\leadsto \color{blue}{1 - \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

      4. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

   9. Simplified 6.4%

      \[\leadsto \color{blue}{1 - sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

   10. Taylor expanded in sinTheta_i around inf 41.6%

       \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

   if 4e-30 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in sinTheta_i around inf 57.9%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   5. Step-by-step derivation

      1. mul-1-neg 57.9%

         \[\leadsto e^{\color{blue}{-\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

      2. associate-*l/ 57.9%

         \[\leadsto e^{-\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}} \]

      3. distribute-lft-neg-in 57.9%

         \[\leadsto e^{\color{blue}{\left(-\frac{sinTheta_i}{v}\right) \cdot sinTheta_O}} \]

      4. *-commutative 57.9%

         \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

      5. distribute-neg-frac 57.9%

         \[\leadsto e^{sinTheta_O \cdot \color{blue}{\frac{-sinTheta_i}{v}}} \]

   6. Simplified 57.9%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \frac{-sinTheta_i}{v}}} \]

   7. Taylor expanded in sinTheta_O around 0 57.9%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 57.9%

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

      2. neg-mul-1 57.9%

         \[\leadsto e^{\frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v}} \]

      3. distribute-lft-neg-in 57.9%

         \[\leadsto e^{\frac{\color{blue}{\left(-sinTheta_i\right) \cdot sinTheta_O}}{v}} \]

      4. *-commutative 57.9%

         \[\leadsto e^{\frac{\color{blue}{sinTheta_O \cdot \left(-sinTheta_i\right)}}{v}} \]

      5. associate-/l* 57.9%

         \[\leadsto e^{\color{blue}{\frac{sinTheta_O}{\frac{v}{-sinTheta_i}}}} \]

   9. Simplified 57.9%

      \[\leadsto e^{\color{blue}{\frac{sinTheta_O}{\frac{v}{-sinTheta_i}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq 4.0000000126843074 \cdot 10^{-30}:\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta_O}{\frac{v}{-sinTheta_i}}}\\ \end{array} \]

Alternative 7: 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 99.5%

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

   1. exp-sum 99.4%

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

3. Simplified 99.5%

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

   1. add-cbrt-cube 97.9%

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

   2. pow1/3 97.9%

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

   3. pow3 97.9%

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

   4. associate-*l/ 97.9%

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

   5. *-commutative 97.9%

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

   6. +-commutative 97.9%

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

5. Applied egg-rr 97.9%

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

6. Taylor expanded in cosTheta_O around 0 99.5%

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

   1. *-lft-identity 99.5%

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

   2. metadata-eval 99.5%

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

   3. cancel-sign-sub-inv 99.5%

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

   4. associate-*r/ 99.5%

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

   5. div-sub 99.5%

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

   6. neg-mul-1 99.5%

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

   7. distribute-lft-neg-in 99.5%

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

   8. cancel-sign-sub 99.5%

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

8. Simplified 99.5%

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

9. Taylor expanded in sinTheta_i around 0 99.4%

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

10. Final simplification 99.4%

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

Alternative 8: 97.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((((cosTheta_i * cosTheta_O) + -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((((costheta_i * costheta_o) + (-1.0e0)) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in v around 0 95.9%

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

   1. associate--r+ 95.9%

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

6. Simplified 95.9%

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

7. Taylor expanded in sinTheta_i around 0 95.9%

   \[\leadsto \color{blue}{e^{\frac{cosTheta_i \cdot cosTheta_O - 1}{v}}} \]

8. Final simplification 95.9%

   \[\leadsto e^{\frac{cosTheta_i \cdot cosTheta_O + -1}{v}} \]

Alternative 9: 97.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\frac{-1 - sinTheta_i \cdot sinTheta_O}{v}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((-1.0f - (sinTheta_i * sinTheta_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((((-1.0e0) - (sintheta_i * sintheta_o)) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in v around 0 95.9%

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

   1. associate--r+ 95.9%

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

6. Simplified 95.9%

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

7. Taylor expanded in cosTheta_i around 0 95.9%

   \[\leadsto \color{blue}{e^{-1 \cdot \frac{1 + sinTheta_i \cdot sinTheta_O}{v}}} \]
8. Step-by-step derivation

   1. mul-1-neg 95.9%

      \[\leadsto e^{\color{blue}{-\frac{1 + sinTheta_i \cdot sinTheta_O}{v}}} \]

9. Simplified 95.9%

   \[\leadsto \color{blue}{e^{-\frac{1 + sinTheta_i \cdot sinTheta_O}{v}}} \]

10. Final simplification 95.9%

    \[\leadsto e^{\frac{-1 - sinTheta_i \cdot sinTheta_O}{v}} \]

Alternative 10: 20.4% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \left(-sinTheta_i\right) \cdot \frac{sinTheta_O}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (- sinTheta_i) (/ sinTheta_O v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return -sinTheta_i * (sinTheta_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 = -sintheta_i * (sintheta_o / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(-sinTheta_i) * Float32(sinTheta_O / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = -sinTheta_i * (sinTheta_O / v);
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in sinTheta_i around inf 13.2%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
5. Step-by-step derivation

   1. mul-1-neg 13.2%

      \[\leadsto e^{\color{blue}{-\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

   2. associate-*l/ 13.2%

      \[\leadsto e^{-\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}} \]

   3. distribute-lft-neg-in 13.2%

      \[\leadsto e^{\color{blue}{\left(-\frac{sinTheta_i}{v}\right) \cdot sinTheta_O}} \]

   4. *-commutative 13.2%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

   5. distribute-neg-frac 13.2%

      \[\leadsto e^{sinTheta_O \cdot \color{blue}{\frac{-sinTheta_i}{v}}} \]

6. Simplified 13.2%

   \[\leadsto e^{\color{blue}{sinTheta_O \cdot \frac{-sinTheta_i}{v}}} \]

7. Taylor expanded in sinTheta_O around 0 6.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v} + 1} \]
8. Step-by-step derivation

   1. +-commutative 6.4%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

   2. mul-1-neg 6.4%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_i \cdot sinTheta_O}{v}\right)} \]

   3. unsub-neg 6.4%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

   4. associate-*r/ 6.4%

      \[\leadsto 1 - \color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

9. Simplified 6.4%

   \[\leadsto \color{blue}{1 - sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

10. Taylor expanded in sinTheta_i around inf 36.8%

    \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]
11. Step-by-step derivation

    1. mul-1-neg 36.8%

       \[\leadsto \color{blue}{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \]

    2. associate-*r/ 17.2%

       \[\leadsto -\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

    3. distribute-lft-neg-in 17.2%

       \[\leadsto \color{blue}{\left(-sinTheta_i\right) \cdot \frac{sinTheta_O}{v}} \]

    4. *-commutative 17.2%

       \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]

12. Simplified 17.2%

    \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]

13. Final simplification 17.2%

    \[\leadsto \left(-sinTheta_i\right) \cdot \frac{sinTheta_O}{v} \]

Alternative 11: 38.3% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* sinTheta_O (- sinTheta_i)) v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (sinTheta_O * -sinTheta_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 = (sintheta_o * -sintheta_i) / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sinTheta_O * -sinTheta_i) / v;
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in sinTheta_i around inf 13.2%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
5. Step-by-step derivation

   1. mul-1-neg 13.2%

      \[\leadsto e^{\color{blue}{-\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

   2. associate-*l/ 13.2%

      \[\leadsto e^{-\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}} \]

   3. distribute-lft-neg-in 13.2%

      \[\leadsto e^{\color{blue}{\left(-\frac{sinTheta_i}{v}\right) \cdot sinTheta_O}} \]

   4. *-commutative 13.2%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

   5. distribute-neg-frac 13.2%

      \[\leadsto e^{sinTheta_O \cdot \color{blue}{\frac{-sinTheta_i}{v}}} \]

6. Simplified 13.2%

   \[\leadsto e^{\color{blue}{sinTheta_O \cdot \frac{-sinTheta_i}{v}}} \]

7. Taylor expanded in sinTheta_O around 0 6.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v} + 1} \]
8. Step-by-step derivation

   1. +-commutative 6.4%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

   2. mul-1-neg 6.4%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_i \cdot sinTheta_O}{v}\right)} \]

   3. unsub-neg 6.4%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

   4. associate-*r/ 6.4%

      \[\leadsto 1 - \color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

9. Simplified 6.4%

   \[\leadsto \color{blue}{1 - sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

10. Taylor expanded in sinTheta_i around inf 36.8%

    \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

11. Final simplification 36.8%

    \[\leadsto \frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v} \]

Alternative 12: 6.4% accurate, 223.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 1.0)

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f;
}

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 = 1.0e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(1.0)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in v around 0 95.9%

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

   1. associate--r+ 95.9%

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

6. Simplified 95.9%

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

7. Taylor expanded in v around inf 6.6%

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

8. Final simplification 6.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023268 
(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))))))