HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 16.1s

Alternatives: 11

Speedup: N/A×

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, 0.3× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ 0.6931 (/ -1.0 v))))
   (*
    (pow
     (*
      (pow (pow (cbrt (exp t_0)) 2.0) 0.3333333333333333)
      (pow
       (cbrt
        (exp
         (+
          (- (* cosTheta_O (/ cosTheta_i v)) (* sinTheta_O (/ sinTheta_i v)))
          t_0)))
       0.3333333333333333))
     3.0)
    (/ 0.5 v))))

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);
	return powf((powf(powf(cbrtf(expf(t_0)), 2.0f), 0.3333333333333333f) * powf(cbrtf(expf((((cosTheta_O * (cosTheta_i / v)) - (sinTheta_O * (sinTheta_i / v))) + t_0))), 0.3333333333333333f)), 3.0f) * (0.5f / v);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))
	return Float32((Float32(((cbrt(exp(t_0)) ^ Float32(2.0)) ^ Float32(0.3333333333333333)) * (cbrt(exp(Float32(Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) - Float32(sinTheta_O * Float32(sinTheta_i / v))) + t_0))) ^ Float32(0.3333333333333333))) ^ Float32(3.0)) * Float32(Float32(0.5) / 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 99.9%

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

3. Simplified 99.9%

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

   1. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

   3. *-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. +-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

   1. pow1/3 99.9%

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

   2. add-cube-cbrt 99.9%

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

   3. unpow-prod-down 99.9%

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

7. Applied egg-rr 99.9%

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

8. Taylor expanded in cosTheta_O around 0 99.8%

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

   1. unpow1/3 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-*r/ 99.9%

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

10. Simplified 99.9%

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

11. Taylor expanded in sinTheta_O around 0 99.8%

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

    1. unpow1/3 99.9%

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

13. Simplified 99.9%

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

14. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot {\left(e^{0.3333333333333333 \cdot \left(0.6931 + \frac{-1}{v}\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (pow (exp (* 0.3333333333333333 (+ 0.6931 (/ -1.0 v)))) 3.0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * powf(expf((0.3333333333333333f * (0.6931f + (-1.0f / v)))), 3.0f);
}

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 = (0.5e0 / v) * (exp((0.3333333333333333e0 * (0.6931e0 + ((-1.0e0) / v)))) ** 3.0e0)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * (exp(Float32(Float32(0.3333333333333333) * Float32(Float32(0.6931) + Float32(Float32(-1.0) / v)))) ^ Float32(3.0)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * (exp((single(0.3333333333333333) * (single(0.6931) + (single(-1.0) / v)))) ^ single(3.0));
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. exp-sum 99.9%

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

3. Simplified 99.9%

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

   1. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

   3. *-commutative 99.9%

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

   4. *-commutative 99.9%

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

   5. +-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in sinTheta_O around 0 99.9%

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

7. Taylor expanded in cosTheta_i around 0 99.9%

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

   1. add-exp-log 99.9%

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

   2. log-pow 99.9%

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

   3. add-log-exp 99.9%

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

   4. sub-neg 99.9%

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

   5. distribute-neg-frac 99.9%

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

   6. metadata-eval 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

    \[\leadsto \frac{0.5}{v} \cdot {\left(e^{0.3333333333333333 \cdot \left(0.6931 + \frac{-1}{v}\right)}\right)}^{3} \]

Alternative 3: 45.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_O \cdot sinTheta_i \leq 5.400000054739634 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_O \cdot sinTheta_i}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_i \cdot \frac{-sinTheta_O}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_O sinTheta_i) 5.400000054739634e-31)
   (/ 1.0 (/ v (* sinTheta_O sinTheta_i)))
   (exp (* sinTheta_i (/ (- sinTheta_O) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_O * sinTheta_i) <= 5.400000054739634e-31f) {
		tmp = 1.0f / (v / (sinTheta_O * sinTheta_i));
	} else {
		tmp = expf((sinTheta_i * (-sinTheta_O / 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_o * sintheta_i) <= 5.400000054739634e-31) then
        tmp = 1.0e0 / (v / (sintheta_o * sintheta_i))
    else
        tmp = exp((sintheta_i * (-sintheta_o / 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_O * sinTheta_i) <= Float32(5.400000054739634e-31))
		tmp = Float32(Float32(1.0) / Float32(v / Float32(sinTheta_O * sinTheta_i)));
	else
		tmp = exp(Float32(sinTheta_i * Float32(Float32(-sinTheta_O) / 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_O * sinTheta_i) <= single(5.400000054739634e-31))
		tmp = single(1.0) / (v / (sinTheta_O * sinTheta_i));
	else
		tmp = exp((sinTheta_i * (-sinTheta_O / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 5.40000005e-31

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l- 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. associate-/l* 99.9%

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

      8. associate-/r* 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in 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. associate-*r/ 6.2%

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

      2. neg-mul-1 6.2%

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

   6. Simplified 6.2%

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

   7. Taylor expanded in sinTheta_i 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. *-commutative 6.4%

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

      4. associate-*r/ 6.4%

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

      5. unsub-neg 6.4%

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

   9. Simplified 6.4%

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

   10. Taylor expanded in sinTheta_O around inf 44.7%

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

       1. mul-1-neg 44.7%

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

       2. *-commutative 44.7%

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

       3. associate-*r/ 25.6%

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

       4. distribute-rgt-neg-in 25.6%

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

       5. distribute-neg-frac 25.6%

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

   12. Simplified 25.6%

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

       1. add-sqr-sqrt 20.3%

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

       2. sqrt-unprod 51.1%

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

       3. swap-sqr 61.4%

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

       4. distribute-frac-neg 61.4%

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

       5. distribute-frac-neg 61.4%

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

       6. sqr-neg 61.4%

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

       7. swap-sqr 51.1%

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

       8. sqrt-unprod 17.3%

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

       9. add-sqr-sqrt 25.6%

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

       10. associate-*r/ 44.6%

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

       11. clear-num 45.1%

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

   14. Applied egg-rr 45.1%

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

   if 5.40000005e-31 < (*.f32 sinTheta_i sinTheta_O)

   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 sinTheta_i around inf 42.6%

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

      1. associate-*r/ 42.6%

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

      2. associate-*l/ 42.6%

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

      3. metadata-eval 42.6%

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

      4. distribute-neg-frac 42.6%

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

      5. distribute-lft-neg-in 42.6%

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

      6. *-commutative 42.6%

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

      7. associate-*l* 42.6%

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

      8. distribute-rgt-neg-in 42.6%

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

      9. associate-*r/ 42.6%

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

      10. *-commutative 42.6%

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

      11. *-lft-identity 42.6%

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

   6. Simplified 42.6%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_O \cdot sinTheta_i \leq 5.400000054739634 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_O \cdot sinTheta_i}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_i \cdot \frac{-sinTheta_O}{v}}\\ \end{array} \]

Alternative 4: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((0.6931f + (-1.0f / v))) * (0.5f / 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 + ((-1.0e0) / v))) * (0.5e0 / v)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(0.6931) + (single(-1.0) / v))) * (single(0.5) / 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 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in sinTheta_i around 0 99.9%

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

5. Taylor expanded in cosTheta_i around 0 99.9%

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

6. Final simplification 99.9%

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

Alternative 5: 98.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((single(-1.0) + (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. 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 sinTheta_i around 0 99.8%

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

5. Taylor expanded in v around 0 98.5%

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

6. Final simplification 98.5%

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

Alternative 6: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp((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 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in sinTheta_i around 0 99.9%

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

5. Taylor expanded in cosTheta_i around 0 99.9%

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

6. Taylor expanded in v around 0 98.6%

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

7. Final simplification 98.6%

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

Alternative 7: 38.9% accurate, 31.9× speedup

Math

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

FPCore

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

C

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

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 / (v / (sintheta_o * sintheta_i))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / (v / (sinTheta_O * sinTheta_i));
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 sinTheta_i around inf 11.7%

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

   1. associate-*r/ 11.7%

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

   2. neg-mul-1 11.7%

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

6. Simplified 11.7%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. +-commutative 6.3%

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

   2. mul-1-neg 6.3%

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

   3. *-commutative 6.3%

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

   4. associate-*r/ 6.3%

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

   5. unsub-neg 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_O around inf 38.9%

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

    1. mul-1-neg 38.9%

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

    2. *-commutative 38.9%

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

    3. associate-*r/ 22.7%

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

    4. distribute-rgt-neg-in 22.7%

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

    5. distribute-neg-frac 22.7%

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

12. Simplified 22.7%

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

    1. add-sqr-sqrt 17.2%

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

    2. sqrt-unprod 44.7%

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

    3. swap-sqr 53.7%

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

    4. distribute-frac-neg 53.7%

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

    5. distribute-frac-neg 53.7%

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

    6. sqr-neg 53.7%

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

    7. swap-sqr 44.7%

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

    8. sqrt-unprod 15.7%

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

    9. add-sqr-sqrt 22.7%

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

    10. associate-*r/ 38.9%

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

    11. clear-num 39.2%

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

14. Applied egg-rr 39.2%

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

15. Final simplification 39.2%

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

Alternative 8: 38.6% 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 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 sinTheta_i around inf 11.7%

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

   1. associate-*r/ 11.7%

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

   2. neg-mul-1 11.7%

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

6. Simplified 11.7%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. +-commutative 6.3%

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

   2. mul-1-neg 6.3%

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

   3. *-commutative 6.3%

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

   4. associate-*r/ 6.3%

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

   5. unsub-neg 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_O around inf 38.9%

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

    1. associate-*r/ 38.9%

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

    2. mul-1-neg 38.9%

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

    3. *-commutative 38.9%

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

    4. distribute-lft-neg-in 38.9%

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

12. Simplified 38.9%

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

13. Final simplification 38.9%

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

Alternative 9: 20.4% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ sinTheta_O \cdot \frac{sinTheta_i}{v} \end{array} \]

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(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 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 sinTheta_i around inf 11.7%

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

   1. associate-*r/ 11.7%

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

   2. neg-mul-1 11.7%

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

6. Simplified 11.7%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. +-commutative 6.3%

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

   2. mul-1-neg 6.3%

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

   3. *-commutative 6.3%

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

   4. associate-*r/ 6.3%

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

   5. unsub-neg 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_O around inf 38.9%

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

    1. mul-1-neg 38.9%

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

    2. *-commutative 38.9%

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

    3. associate-*r/ 22.7%

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

    4. distribute-rgt-neg-in 22.7%

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

    5. distribute-neg-frac 22.7%

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

12. Simplified 22.7%

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

    1. add-sqr-sqrt 17.2%

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

    2. sqrt-unprod 44.7%

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

    3. swap-sqr 53.7%

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

    4. distribute-frac-neg 53.7%

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

    5. distribute-frac-neg 53.7%

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

    6. sqr-neg 53.7%

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

    7. swap-sqr 44.7%

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

    8. sqrt-unprod 15.7%

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

    9. add-sqr-sqrt 22.7%

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

    10. expm1-log1p-u 22.2%

        \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(sinTheta_O \cdot \frac{sinTheta_i}{v}\right)\right)} \]

    11. expm1-udef 73.1%

        \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(sinTheta_O \cdot \frac{sinTheta_i}{v}\right)} - 1} \]

14. Applied egg-rr 73.1%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(sinTheta_O \cdot \frac{sinTheta_i}{v}\right)} - 1} \]
15. Step-by-step derivation

    1. expm1-def 22.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(sinTheta_O \cdot \frac{sinTheta_i}{v}\right)\right)} \]

    2. expm1-log1p 22.7%

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

16. Simplified 22.7%

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

17. Final simplification 22.7%

    \[\leadsto sinTheta_O \cdot \frac{sinTheta_i}{v} \]

Alternative 10: 38.6% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_O \cdot sinTheta_i}{v} \end{array} \]

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 * 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 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 sinTheta_i around inf 11.7%

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

   1. associate-*r/ 11.7%

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

   2. neg-mul-1 11.7%

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

6. Simplified 11.7%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. +-commutative 6.3%

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

   2. mul-1-neg 6.3%

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

   3. *-commutative 6.3%

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

   4. associate-*r/ 6.3%

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

   5. unsub-neg 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_O around inf 38.9%

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

    1. mul-1-neg 38.9%

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

    2. *-commutative 38.9%

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

    3. associate-*r/ 22.7%

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

    4. distribute-rgt-neg-in 22.7%

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

    5. distribute-neg-frac 22.7%

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

12. Simplified 22.7%

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

    1. add-sqr-sqrt 17.2%

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

    2. sqrt-unprod 44.7%

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

    3. swap-sqr 53.7%

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

    4. distribute-frac-neg 53.7%

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

    5. distribute-frac-neg 53.7%

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

    6. sqr-neg 53.7%

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

    7. swap-sqr 44.7%

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

    8. sqrt-unprod 15.7%

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

    9. add-sqr-sqrt 22.7%

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

    10. associate-*r/ 38.9%

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

14. Applied egg-rr 38.9%

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

15. Final simplification 38.9%

    \[\leadsto \frac{sinTheta_O \cdot sinTheta_i}{v} \]

Alternative 11: 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 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 sinTheta_i around inf 11.7%

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

   1. associate-*r/ 11.7%

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

   2. neg-mul-1 11.7%

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

6. Simplified 11.7%

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

7. Taylor expanded in sinTheta_i around 0 6.4%

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

8. Final simplification 6.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023207 
(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))))))