HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 10.1s

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, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(e^{0.5 \cdot \left(\log \left(\frac{0.5}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)}\right)}^{2} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return powf(expf((0.5f * (logf((0.5f / v)) + (0.6931f + (-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((0.5e0 * (log((0.5e0 / v)) + (0.6931e0 + ((-1.0e0) / v))))) ** 2.0e0
end function

Julia

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

MATLAB

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

Derivation

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 0 99.9%

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

5. Taylor expanded in cosTheta_i around 0 99.9%

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

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

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

   3. +-commutative 100.0%

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

   4. associate--l+ 100.0%

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

7. Applied egg-rr 100.0%

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

   1. pow1/2 100.0%

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

   2. pow-exp 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 0 99.9%

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

5. Taylor expanded in cosTheta_i around 0 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 99.7% 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 99.9%

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

   1. exp-sum 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in sinTheta_i around 0 99.8%

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

5. Taylor expanded in cosTheta_i around 0 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 98.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 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 = exp(((-1.0e0) / v))
end function

Julia

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

MATLAB

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

Derivation

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 0 99.9%

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

5. Taylor expanded in cosTheta_i around 0 99.9%

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

6. Taylor expanded in v around 0 98.9%

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

7. Final simplification 98.9%

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

Alternative 5: 57.1% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.961817850054744 \cdot 10^{-44}:\\ \;\;\;\;\frac{sinTheta_O \cdot sinTheta_i}{v}\\ \mathbf{elif}\;cosTheta_i \cdot cosTheta_O \leq 0:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* cosTheta_i cosTheta_O) -1.961817850054744e-44)
   (/ (* sinTheta_O sinTheta_i) v)
   (if (<= (* cosTheta_i cosTheta_O) 0.0)
     (/ (* cosTheta_i cosTheta_O) v)
     (/ (* 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 ((cosTheta_i * cosTheta_O) <= -1.961817850054744e-44f) {
		tmp = (sinTheta_O * sinTheta_i) / v;
	} else if ((cosTheta_i * cosTheta_O) <= 0.0f) {
		tmp = (cosTheta_i * cosTheta_O) / v;
	} else {
		tmp = (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 ((costheta_i * costheta_o) <= (-1.961817850054744e-44)) then
        tmp = (sintheta_o * sintheta_i) / v
    else if ((costheta_i * costheta_o) <= 0.0e0) then
        tmp = (costheta_i * costheta_o) / v
    else
        tmp = (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(cosTheta_i * cosTheta_O) <= Float32(-1.961817850054744e-44))
		tmp = Float32(Float32(sinTheta_O * sinTheta_i) / v);
	elseif (Float32(cosTheta_i * cosTheta_O) <= Float32(0.0))
		tmp = Float32(Float32(cosTheta_i * cosTheta_O) / v);
	else
		tmp = Float32(Float32(sinTheta_O * 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 ((cosTheta_i * cosTheta_O) <= single(-1.961817850054744e-44))
		tmp = (sinTheta_O * sinTheta_i) / v;
	elseif ((cosTheta_i * cosTheta_O) <= single(0.0))
		tmp = (cosTheta_i * cosTheta_O) / v;
	else
		tmp = (sinTheta_O * -sinTheta_i) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -1.96182e-44

   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 15.3%

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

      1. associate-*r/ 15.3%

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

      2. associate-*l/ 15.3%

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

      3. metadata-eval 15.3%

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

      4. distribute-neg-frac 15.3%

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

      5. distribute-lft-neg-in 15.3%

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

      6. *-commutative 15.3%

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

      7. associate-*l* 15.3%

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

      8. distribute-lft-neg-in 15.3%

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

      9. associate-*r/ 15.3%

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

      10. *-commutative 15.3%

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

      11. *-lft-identity 15.3%

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

   6. Simplified 15.3%

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

   7. Taylor expanded in sinTheta_i around 0 6.2%

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

      1. mul-1-neg 6.2%

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

      2. associate-*r/ 6.2%

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

      3. distribute-lft-neg-in 6.2%

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

      4. +-commutative 6.2%

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

      5. distribute-lft-neg-in 6.2%

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

      6. associate-*r/ 6.2%

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

      7. unsub-neg 6.2%

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

      8. *-commutative 6.2%

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

      9. associate-*r/ 6.2%

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

   9. Simplified 6.2%

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

   10. Taylor expanded in sinTheta_O around inf 50.7%

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

       1. mul-1-neg 50.7%

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

       2. *-commutative 50.7%

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

       3. associate-*r/ 24.2%

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

       4. distribute-lft-neg-in 24.2%

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

   12. Simplified 24.2%

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

       1. associate-*r/ 50.7%

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

       2. add-sqr-sqrt 25.4%

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

       3. sqrt-unprod 64.8%

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

       4. sqr-neg 64.8%

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

       5. sqrt-unprod 25.3%

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

       6. add-sqr-sqrt 50.7%

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

   14. Applied egg-rr 50.7%

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

   if -1.96182e-44 < (*.f32 cosTheta_i cosTheta_O) < 0.0

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l- 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. associate-/l* 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in sinTheta_i around 0 100.0%

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

   5. Taylor expanded in cosTheta_i around inf 6.3%

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

      1. associate-*l/ 6.3%

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

      2. *-commutative 6.3%

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

   7. Simplified 6.3%

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

   8. Taylor expanded in cosTheta_O around 0 6.3%

      \[\leadsto \color{blue}{1 + \frac{cosTheta_i \cdot cosTheta_O}{v}} \]
   9. Step-by-step derivation

      1. *-commutative 6.3%

         \[\leadsto 1 + \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \]

      2. associate-*r/ 6.3%

         \[\leadsto 1 + \color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

   10. Simplified 6.3%

       \[\leadsto \color{blue}{1 + cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

   11. Taylor expanded in cosTheta_O around inf 96.3%

       \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} \]

   if 0.0 < (*.f32 cosTheta_i cosTheta_O)

   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 17.0%

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

      1. associate-*r/ 17.0%

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

      2. associate-*l/ 17.0%

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

      3. metadata-eval 17.0%

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

      4. distribute-neg-frac 17.0%

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

      5. distribute-lft-neg-in 17.0%

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

      6. *-commutative 17.0%

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

      7. associate-*l* 17.0%

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

      8. distribute-lft-neg-in 17.0%

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

      9. associate-*r/ 17.0%

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

      10. *-commutative 17.0%

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

      11. *-lft-identity 17.0%

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

   6. Simplified 17.0%

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

   7. Taylor expanded in sinTheta_i around 0 6.1%

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

      1. mul-1-neg 6.1%

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

      2. associate-*r/ 6.1%

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

      3. distribute-lft-neg-in 6.1%

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

      4. +-commutative 6.1%

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

      5. distribute-lft-neg-in 6.1%

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

      6. associate-*r/ 6.1%

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

      7. unsub-neg 6.1%

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

      8. *-commutative 6.1%

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

      9. associate-*r/ 6.1%

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

   9. Simplified 6.1%

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

   10. Taylor expanded in sinTheta_O around inf 35.1%

       \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.961817850054744 \cdot 10^{-44}:\\ \;\;\;\;\frac{sinTheta_O \cdot sinTheta_i}{v}\\ \mathbf{elif}\;cosTheta_i \cdot cosTheta_O \leq 0:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \end{array} \]

Alternative 6: 57.2% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.961817850054744 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_O \cdot sinTheta_i}}\\ \mathbf{elif}\;cosTheta_i \cdot cosTheta_O \leq 0:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* cosTheta_i cosTheta_O) -1.961817850054744e-44)
   (/ 1.0 (/ v (* sinTheta_O sinTheta_i)))
   (if (<= (* cosTheta_i cosTheta_O) 0.0)
     (/ (* cosTheta_i cosTheta_O) v)
     (/ (* 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 ((cosTheta_i * cosTheta_O) <= -1.961817850054744e-44f) {
		tmp = 1.0f / (v / (sinTheta_O * sinTheta_i));
	} else if ((cosTheta_i * cosTheta_O) <= 0.0f) {
		tmp = (cosTheta_i * cosTheta_O) / v;
	} else {
		tmp = (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 ((costheta_i * costheta_o) <= (-1.961817850054744e-44)) then
        tmp = 1.0e0 / (v / (sintheta_o * sintheta_i))
    else if ((costheta_i * costheta_o) <= 0.0e0) then
        tmp = (costheta_i * costheta_o) / v
    else
        tmp = (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(cosTheta_i * cosTheta_O) <= Float32(-1.961817850054744e-44))
		tmp = Float32(Float32(1.0) / Float32(v / Float32(sinTheta_O * sinTheta_i)));
	elseif (Float32(cosTheta_i * cosTheta_O) <= Float32(0.0))
		tmp = Float32(Float32(cosTheta_i * cosTheta_O) / v);
	else
		tmp = Float32(Float32(sinTheta_O * 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 ((cosTheta_i * cosTheta_O) <= single(-1.961817850054744e-44))
		tmp = single(1.0) / (v / (sinTheta_O * sinTheta_i));
	elseif ((cosTheta_i * cosTheta_O) <= single(0.0))
		tmp = (cosTheta_i * cosTheta_O) / v;
	else
		tmp = (sinTheta_O * -sinTheta_i) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -1.96182e-44

   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 15.3%

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

      1. associate-*r/ 15.3%

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

      2. associate-*l/ 15.3%

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

      3. metadata-eval 15.3%

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

      4. distribute-neg-frac 15.3%

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

      5. distribute-lft-neg-in 15.3%

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

      6. *-commutative 15.3%

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

      7. associate-*l* 15.3%

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

      8. distribute-lft-neg-in 15.3%

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

      9. associate-*r/ 15.3%

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

      10. *-commutative 15.3%

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

      11. *-lft-identity 15.3%

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

   6. Simplified 15.3%

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

   7. Taylor expanded in sinTheta_i around 0 6.2%

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

      1. mul-1-neg 6.2%

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

      2. associate-*r/ 6.2%

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

      3. distribute-lft-neg-in 6.2%

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

      4. +-commutative 6.2%

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

      5. distribute-lft-neg-in 6.2%

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

      6. associate-*r/ 6.2%

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

      7. unsub-neg 6.2%

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

      8. *-commutative 6.2%

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

      9. associate-*r/ 6.2%

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

   9. Simplified 6.2%

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

   10. Taylor expanded in sinTheta_O around inf 50.7%

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

       1. mul-1-neg 50.7%

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

       2. *-commutative 50.7%

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

       3. associate-*r/ 24.2%

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

       4. distribute-lft-neg-in 24.2%

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

   12. Simplified 24.2%

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

       1. associate-*r/ 50.7%

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

       2. clear-num 51.4%

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

       3. add-sqr-sqrt 26.2%

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

       4. sqrt-unprod 65.6%

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

       5. sqr-neg 65.6%

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

       6. sqrt-unprod 25.3%

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

       7. add-sqr-sqrt 51.4%

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

   14. Applied egg-rr 51.4%

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

   if -1.96182e-44 < (*.f32 cosTheta_i cosTheta_O) < 0.0

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l- 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. associate-/l* 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in sinTheta_i around 0 100.0%

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

   5. Taylor expanded in cosTheta_i around inf 6.3%

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

      1. associate-*l/ 6.3%

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

      2. *-commutative 6.3%

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

   7. Simplified 6.3%

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

   8. Taylor expanded in cosTheta_O around 0 6.3%

      \[\leadsto \color{blue}{1 + \frac{cosTheta_i \cdot cosTheta_O}{v}} \]
   9. Step-by-step derivation

      1. *-commutative 6.3%

         \[\leadsto 1 + \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \]

      2. associate-*r/ 6.3%

         \[\leadsto 1 + \color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

   10. Simplified 6.3%

       \[\leadsto \color{blue}{1 + cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

   11. Taylor expanded in cosTheta_O around inf 96.3%

       \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} \]

   if 0.0 < (*.f32 cosTheta_i cosTheta_O)

   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 17.0%

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

      1. associate-*r/ 17.0%

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

      2. associate-*l/ 17.0%

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

      3. metadata-eval 17.0%

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

      4. distribute-neg-frac 17.0%

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

      5. distribute-lft-neg-in 17.0%

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

      6. *-commutative 17.0%

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

      7. associate-*l* 17.0%

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

      8. distribute-lft-neg-in 17.0%

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

      9. associate-*r/ 17.0%

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

      10. *-commutative 17.0%

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

      11. *-lft-identity 17.0%

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

   6. Simplified 17.0%

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

   7. Taylor expanded in sinTheta_i around 0 6.1%

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

      1. mul-1-neg 6.1%

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

      2. associate-*r/ 6.1%

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

      3. distribute-lft-neg-in 6.1%

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

      4. +-commutative 6.1%

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

      5. distribute-lft-neg-in 6.1%

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

      6. associate-*r/ 6.1%

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

      7. unsub-neg 6.1%

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

      8. *-commutative 6.1%

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

      9. associate-*r/ 6.1%

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

   9. Simplified 6.1%

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

   10. Taylor expanded in sinTheta_O around inf 35.1%

       \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.961817850054744 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_O \cdot sinTheta_i}}\\ \mathbf{elif}\;cosTheta_i \cdot cosTheta_O \leq 0:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \end{array} \]

Alternative 7: 57.1% accurate, 16.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.961817850054744 \cdot 10^{-44} \lor \neg \left(cosTheta_i \cdot cosTheta_O \leq 0\right):\\ \;\;\;\;\frac{sinTheta_O \cdot sinTheta_i}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (or (<= (* cosTheta_i cosTheta_O) -1.961817850054744e-44)
         (not (<= (* cosTheta_i cosTheta_O) 0.0)))
   (/ (* sinTheta_O sinTheta_i) v)
   (/ (* cosTheta_i cosTheta_O) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (((cosTheta_i * cosTheta_O) <= -1.961817850054744e-44f) || !((cosTheta_i * cosTheta_O) <= 0.0f)) {
		tmp = (sinTheta_O * sinTheta_i) / v;
	} else {
		tmp = (cosTheta_i * cosTheta_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 (((costheta_i * costheta_o) <= (-1.961817850054744e-44)) .or. (.not. ((costheta_i * costheta_o) <= 0.0e0))) then
        tmp = (sintheta_o * sintheta_i) / v
    else
        tmp = (costheta_i * costheta_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(cosTheta_i * cosTheta_O) <= Float32(-1.961817850054744e-44)) || !(Float32(cosTheta_i * cosTheta_O) <= Float32(0.0)))
		tmp = Float32(Float32(sinTheta_O * sinTheta_i) / v);
	else
		tmp = Float32(Float32(cosTheta_i * cosTheta_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 (((cosTheta_i * cosTheta_O) <= single(-1.961817850054744e-44)) || ~(((cosTheta_i * cosTheta_O) <= single(0.0))))
		tmp = (sinTheta_O * sinTheta_i) / v;
	else
		tmp = (cosTheta_i * cosTheta_O) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -1.96182e-44 or 0.0 < (*.f32 cosTheta_i cosTheta_O)

   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 16.2%

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

      1. associate-*r/ 16.2%

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

      2. associate-*l/ 16.2%

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

      3. metadata-eval 16.2%

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

      4. distribute-neg-frac 16.2%

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

      5. distribute-lft-neg-in 16.2%

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

      6. *-commutative 16.2%

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

      7. associate-*l* 16.2%

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

      8. distribute-lft-neg-in 16.2%

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

      9. associate-*r/ 16.2%

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

      10. *-commutative 16.2%

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

      11. *-lft-identity 16.2%

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

   6. Simplified 16.2%

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

   7. Taylor expanded in sinTheta_i around 0 6.2%

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

      1. mul-1-neg 6.2%

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

      2. associate-*r/ 6.2%

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

      3. distribute-lft-neg-in 6.2%

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

      4. +-commutative 6.2%

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

      5. distribute-lft-neg-in 6.2%

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

      6. associate-*r/ 6.2%

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

      7. unsub-neg 6.2%

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

      8. *-commutative 6.2%

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

      9. associate-*r/ 6.2%

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

   9. Simplified 6.2%

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

   10. Taylor expanded in sinTheta_O around inf 42.7%

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

       1. mul-1-neg 42.7%

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

       2. *-commutative 42.7%

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

       3. associate-*r/ 23.3%

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

       4. distribute-lft-neg-in 23.3%

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

   12. Simplified 23.3%

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

       1. associate-*r/ 42.7%

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

       2. add-sqr-sqrt 20.9%

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

       3. sqrt-unprod 58.8%

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

       4. sqr-neg 58.8%

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

       5. sqrt-unprod 21.7%

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

       6. add-sqr-sqrt 42.7%

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

   14. Applied egg-rr 42.7%

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

   if -1.96182e-44 < (*.f32 cosTheta_i cosTheta_O) < 0.0

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l- 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. associate-/l* 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in sinTheta_i around 0 100.0%

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

   5. Taylor expanded in cosTheta_i around inf 6.3%

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

      1. associate-*l/ 6.3%

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

      2. *-commutative 6.3%

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

   7. Simplified 6.3%

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

   8. Taylor expanded in cosTheta_O around 0 6.3%

      \[\leadsto \color{blue}{1 + \frac{cosTheta_i \cdot cosTheta_O}{v}} \]
   9. Step-by-step derivation

      1. *-commutative 6.3%

         \[\leadsto 1 + \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \]

      2. associate-*r/ 6.3%

         \[\leadsto 1 + \color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

   10. Simplified 6.3%

       \[\leadsto \color{blue}{1 + cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

   11. Taylor expanded in cosTheta_O around inf 96.3%

       \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.961817850054744 \cdot 10^{-44} \lor \neg \left(cosTheta_i \cdot cosTheta_O \leq 0\right):\\ \;\;\;\;\frac{sinTheta_O \cdot sinTheta_i}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \end{array} \]

Alternative 8: 45.3% accurate, 16.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.961817850054744 \cdot 10^{-44}:\\ \;\;\;\;\frac{sinTheta_O}{\frac{v}{sinTheta_i}}\\ \mathbf{elif}\;cosTheta_i \cdot cosTheta_O \leq 1.999933168284379 \cdot 10^{-41}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;sinTheta_O \cdot \frac{sinTheta_i}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* cosTheta_i cosTheta_O) -1.961817850054744e-44)
   (/ sinTheta_O (/ v sinTheta_i))
   (if (<= (* cosTheta_i cosTheta_O) 1.999933168284379e-41)
     (/ (* cosTheta_i cosTheta_O) v)
     (* 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 ((cosTheta_i * cosTheta_O) <= -1.961817850054744e-44f) {
		tmp = sinTheta_O / (v / sinTheta_i);
	} else if ((cosTheta_i * cosTheta_O) <= 1.999933168284379e-41f) {
		tmp = (cosTheta_i * cosTheta_O) / v;
	} else {
		tmp = 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 ((costheta_i * costheta_o) <= (-1.961817850054744e-44)) then
        tmp = sintheta_o / (v / sintheta_i)
    else if ((costheta_i * costheta_o) <= 1.999933168284379e-41) then
        tmp = (costheta_i * costheta_o) / v
    else
        tmp = 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(cosTheta_i * cosTheta_O) <= Float32(-1.961817850054744e-44))
		tmp = Float32(sinTheta_O / Float32(v / sinTheta_i));
	elseif (Float32(cosTheta_i * cosTheta_O) <= Float32(1.999933168284379e-41))
		tmp = Float32(Float32(cosTheta_i * cosTheta_O) / v);
	else
		tmp = Float32(sinTheta_O * 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 ((cosTheta_i * cosTheta_O) <= single(-1.961817850054744e-44))
		tmp = sinTheta_O / (v / sinTheta_i);
	elseif ((cosTheta_i * cosTheta_O) <= single(1.999933168284379e-41))
		tmp = (cosTheta_i * cosTheta_O) / v;
	else
		tmp = sinTheta_O * (sinTheta_i / v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -1.96182e-44

   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 15.3%

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

      1. associate-*r/ 15.3%

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

      2. associate-*l/ 15.3%

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

      3. metadata-eval 15.3%

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

      4. distribute-neg-frac 15.3%

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

      5. distribute-lft-neg-in 15.3%

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

      6. *-commutative 15.3%

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

      7. associate-*l* 15.3%

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

      8. distribute-lft-neg-in 15.3%

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

      9. associate-*r/ 15.3%

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

      10. *-commutative 15.3%

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

      11. *-lft-identity 15.3%

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

   6. Simplified 15.3%

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

   7. Taylor expanded in sinTheta_i around 0 6.2%

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

      1. mul-1-neg 6.2%

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

      2. associate-*r/ 6.2%

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

      3. distribute-lft-neg-in 6.2%

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

      4. +-commutative 6.2%

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

      5. distribute-lft-neg-in 6.2%

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

      6. associate-*r/ 6.2%

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

      7. unsub-neg 6.2%

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

      8. *-commutative 6.2%

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

      9. associate-*r/ 6.2%

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

   9. Simplified 6.2%

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

   10. Taylor expanded in sinTheta_O around inf 50.7%

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

       1. mul-1-neg 50.7%

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

       2. *-commutative 50.7%

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

       3. associate-*r/ 24.2%

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

       4. distribute-lft-neg-in 24.2%

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

   12. Simplified 24.2%

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

       1. expm1-log1p-u 23.7%

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

       2. expm1-udef 72.6%

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

       3. add-sqr-sqrt 35.5%

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

       4. sqrt-unprod 76.2%

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

       5. sqr-neg 76.2%

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

       6. sqrt-unprod 37.2%

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

       7. add-sqr-sqrt 72.7%

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

   14. Applied egg-rr 72.7%

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

       1. expm1-def 23.7%

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

       2. expm1-log1p 24.2%

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

       3. associate-*r/ 50.7%

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

       4. associate-/l* 24.2%

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

   16. Simplified 24.2%

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

   if -1.96182e-44 < (*.f32 cosTheta_i cosTheta_O) < 1.99993e-41

   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 0 99.9%

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

   5. Taylor expanded in cosTheta_i around inf 6.3%

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

      1. associate-*l/ 6.3%

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

      2. *-commutative 6.3%

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

   7. Simplified 6.3%

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

   8. Taylor expanded in cosTheta_O around 0 6.3%

      \[\leadsto \color{blue}{1 + \frac{cosTheta_i \cdot cosTheta_O}{v}} \]
   9. Step-by-step derivation

      1. *-commutative 6.3%

         \[\leadsto 1 + \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \]

      2. associate-*r/ 6.3%

         \[\leadsto 1 + \color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

   10. Simplified 6.3%

       \[\leadsto \color{blue}{1 + cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

   11. Taylor expanded in cosTheta_O around inf 86.7%

       \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} \]

   if 1.99993e-41 < (*.f32 cosTheta_i cosTheta_O)

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l- 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. associate-/l* 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in sinTheta_i around inf 18.7%

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

      1. associate-*r/ 18.7%

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

      2. associate-*l/ 18.7%

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

      3. metadata-eval 18.7%

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

      4. distribute-neg-frac 18.7%

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

      5. distribute-lft-neg-in 18.7%

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

      6. *-commutative 18.7%

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

      7. associate-*l* 18.7%

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

      8. distribute-lft-neg-in 18.7%

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

      9. associate-*r/ 18.7%

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

      10. *-commutative 18.7%

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

      11. *-lft-identity 18.7%

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

   6. Simplified 18.7%

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

   7. Taylor expanded in sinTheta_i around 0 6.0%

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

      1. mul-1-neg 6.0%

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

      2. associate-*r/ 6.0%

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

      3. distribute-lft-neg-in 6.0%

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

      4. +-commutative 6.0%

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

      5. distribute-lft-neg-in 6.0%

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

      6. associate-*r/ 6.0%

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

      7. unsub-neg 6.0%

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

      8. *-commutative 6.0%

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

      9. associate-*r/ 6.0%

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

   9. Simplified 6.0%

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

   10. Taylor expanded in sinTheta_O 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. *-commutative 36.8%

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

       3. associate-*r/ 23.2%

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

       4. distribute-lft-neg-in 23.2%

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

   12. Simplified 23.2%

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

       1. expm1-log1p-u 22.4%

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

       2. expm1-udef 69.9%

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

       3. add-sqr-sqrt 36.8%

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

       4. sqrt-unprod 76.4%

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

       5. sqr-neg 76.4%

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

       6. sqrt-unprod 34.4%

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

       7. add-sqr-sqrt 71.3%

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

   14. Applied egg-rr 71.3%

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

       1. expm1-def 22.6%

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

       2. expm1-log1p 23.2%

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

   16. Simplified 23.2%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.961817850054744 \cdot 10^{-44}:\\ \;\;\;\;\frac{sinTheta_O}{\frac{v}{sinTheta_i}}\\ \mathbf{elif}\;cosTheta_i \cdot cosTheta_O \leq 1.999933168284379 \cdot 10^{-41}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;sinTheta_O \cdot \frac{sinTheta_i}{v}\\ \end{array} \]

Alternative 9: 22.4% accurate, 31.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_O \leq 1.5000000170217692 \cdot 10^{-18}:\\ \;\;\;\;sinTheta_O \cdot \frac{sinTheta_i}{v}\\ \mathbf{else}:\\ \;\;\;\;cosTheta_O \cdot \frac{cosTheta_i}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O 1.5000000170217692e-18)
   (* sinTheta_O (/ sinTheta_i v))
   (* cosTheta_O (/ cosTheta_i v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_O <= 1.5000000170217692e-18f) {
		tmp = sinTheta_O * (sinTheta_i / v);
	} else {
		tmp = cosTheta_O * (cosTheta_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_o <= 1.5000000170217692e-18) then
        tmp = sintheta_o * (sintheta_i / v)
    else
        tmp = costheta_o * (costheta_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 (sinTheta_O <= Float32(1.5000000170217692e-18))
		tmp = Float32(sinTheta_O * Float32(sinTheta_i / v));
	else
		tmp = Float32(cosTheta_O * Float32(cosTheta_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_O <= single(1.5000000170217692e-18))
		tmp = sinTheta_O * (sinTheta_i / v);
	else
		tmp = cosTheta_O * (cosTheta_i / v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 1.50000002e-18

   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 11.1%

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

      1. associate-*r/ 11.1%

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

      2. associate-*l/ 11.1%

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

      3. metadata-eval 11.1%

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

      4. distribute-neg-frac 11.1%

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

      5. distribute-lft-neg-in 11.1%

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

      6. *-commutative 11.1%

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

      7. associate-*l* 11.1%

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

      8. distribute-lft-neg-in 11.1%

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

      9. associate-*r/ 11.1%

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

      10. *-commutative 11.1%

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

      11. *-lft-identity 11.1%

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

   6. Simplified 11.1%

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

   7. Taylor expanded in sinTheta_i around 0 6.2%

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

      1. mul-1-neg 6.2%

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

      2. associate-*r/ 6.2%

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

      3. distribute-lft-neg-in 6.2%

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

      4. +-commutative 6.2%

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

      5. distribute-lft-neg-in 6.2%

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

      6. associate-*r/ 6.2%

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

      7. unsub-neg 6.2%

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

      8. *-commutative 6.2%

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

      9. associate-*r/ 6.2%

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

   9. Simplified 6.2%

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

   10. Taylor expanded in sinTheta_O around inf 48.3%

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

       1. mul-1-neg 48.3%

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

       2. *-commutative 48.3%

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

       3. associate-*r/ 25.6%

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

       4. distribute-lft-neg-in 25.6%

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

   12. Simplified 25.6%

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

       1. expm1-log1p-u 25.3%

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

       2. expm1-udef 79.2%

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

       3. add-sqr-sqrt 46.6%

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

       4. sqrt-unprod 82.8%

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

       5. sqr-neg 82.8%

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

       6. sqrt-unprod 32.1%

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

       7. add-sqr-sqrt 78.1%

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

   14. Applied egg-rr 78.1%

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

       1. expm1-def 25.1%

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

       2. expm1-log1p 25.6%

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

   16. Simplified 25.6%

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

   if 1.50000002e-18 < sinTheta_O

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l- 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. associate-/l* 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in sinTheta_i around 0 100.0%

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

   5. Taylor expanded in cosTheta_i around inf 15.7%

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

      1. associate-*l/ 15.7%

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

      2. *-commutative 15.7%

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

   7. Simplified 15.7%

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

   8. Taylor expanded in cosTheta_O around 0 6.1%

      \[\leadsto \color{blue}{1 + \frac{cosTheta_i \cdot cosTheta_O}{v}} \]
   9. Step-by-step derivation

      1. *-commutative 6.1%

         \[\leadsto 1 + \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \]

      2. associate-*r/ 6.1%

         \[\leadsto 1 + \color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

   10. Simplified 6.1%

       \[\leadsto \color{blue}{1 + cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

   11. Taylor expanded in cosTheta_O around inf 42.1%

       \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} \]
   12. Step-by-step derivation

       1. *-commutative 42.1%

          \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \]

       2. associate-*r/ 19.6%

          \[\leadsto \color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

   13. Simplified 19.6%

       \[\leadsto \color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_O \leq 1.5000000170217692 \cdot 10^{-18}:\\ \;\;\;\;sinTheta_O \cdot \frac{sinTheta_i}{v}\\ \mathbf{else}:\\ \;\;\;\;cosTheta_O \cdot \frac{cosTheta_i}{v}\\ \end{array} \]

Alternative 10: 19.9% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ cosTheta_O \cdot \frac{cosTheta_i}{v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 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 = costheta_o * (costheta_i / v)
end function

Julia

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

MATLAB

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

Derivation

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 0 99.9%

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

5. Taylor expanded in cosTheta_i around inf 16.7%

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

   1. associate-*l/ 16.7%

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

   2. *-commutative 16.7%

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

7. Simplified 16.7%

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

8. Taylor expanded in cosTheta_O around 0 6.1%

   \[\leadsto \color{blue}{1 + \frac{cosTheta_i \cdot cosTheta_O}{v}} \]
9. Step-by-step derivation

   1. *-commutative 6.1%

      \[\leadsto 1 + \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \]

   2. associate-*r/ 6.1%

      \[\leadsto 1 + \color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

10. Simplified 6.1%

    \[\leadsto \color{blue}{1 + cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

11. Taylor expanded in cosTheta_O around inf 38.7%

    \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} \]
12. Step-by-step derivation

    1. *-commutative 38.7%

       \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \]

    2. associate-*r/ 18.3%

       \[\leadsto \color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

13. Simplified 18.3%

    \[\leadsto \color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} \]

14. Final simplification 18.3%

    \[\leadsto cosTheta_O \cdot \frac{cosTheta_i}{v} \]

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 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 13.8%

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

   1. associate-*r/ 13.8%

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

   2. associate-*l/ 13.8%

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

   3. metadata-eval 13.8%

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

   4. distribute-neg-frac 13.8%

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

   5. distribute-lft-neg-in 13.8%

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

   6. *-commutative 13.8%

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

   7. associate-*l* 13.8%

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

   8. distribute-lft-neg-in 13.8%

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

   9. associate-*r/ 13.8%

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

   10. *-commutative 13.8%

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

   11. *-lft-identity 13.8%

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

6. Simplified 13.8%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

8. Final simplification 6.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023230 
(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))))))