HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 21.1s

Alternatives: 11

Speedup: 2.0×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.7% 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, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \left(\mathsf{neg}\left(\log \left(2 \cdot v\right)\right)\right)} \]

   2. unsub-neg N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \log \left(2 \cdot v\right)} \]

   3. exp-diff N/A

      \[\leadsto \frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{\color{blue}{e^{\log \left(2 \cdot v\right)}}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}\right), \color{blue}{\left(e^{\log \left(2 \cdot v\right)}\right)}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{e^{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_O around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right), \color{blue}{v}\right) \]

7. Simplified 99.8%

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

8. Taylor expanded in sinTheta_O around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{\frac{6931}{10000} - \frac{1}{v}}}{v}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot e^{\frac{6931}{10000} - \frac{1}{v}}}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot e^{\frac{6931}{10000} - \frac{1}{v}}\right), \color{blue}{v}\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(e^{\frac{6931}{10000} - \frac{1}{v}}\right)\right), v\right) \]

   4. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right), v\right) \]

   5. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\frac{6931}{10000}, \left(\frac{1}{v}\right)\right)\right)\right), v\right) \]

   6. /-lowering-/.f32 99.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(1, v\right)\right)\right)\right), v\right) \]

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \left(\mathsf{neg}\left(\log \left(2 \cdot v\right)\right)\right)} \]

   2. unsub-neg N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \log \left(2 \cdot v\right)} \]

   3. exp-diff N/A

      \[\leadsto \frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{\color{blue}{e^{\log \left(2 \cdot v\right)}}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}\right), \color{blue}{\left(e^{\log \left(2 \cdot v\right)}\right)}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{e^{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_O around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right), \color{blue}{v}\right) \]

7. Simplified 99.8%

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

8. Taylor expanded in sinTheta_O around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{\frac{6931}{10000} - \frac{1}{v}}}{v}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot e^{\frac{6931}{10000} - \frac{1}{v}}}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot e^{\frac{6931}{10000} - \frac{1}{v}}\right), \color{blue}{v}\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(e^{\frac{6931}{10000} - \frac{1}{v}}\right)\right), v\right) \]

   4. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right), v\right) \]

   5. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\frac{6931}{10000}, \left(\frac{1}{v}\right)\right)\right)\right), v\right) \]

   6. /-lowering-/.f32 99.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(1, v\right)\right)\right)\right), v\right) \]

10. Simplified 99.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot e^{0.6931 - \frac{1}{v}}}{v}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{e^{\frac{6931}{10000} - \frac{1}{v}} \cdot \frac{1}{2}}{v} \]

    2. associate-/l* N/A

       \[\leadsto e^{\frac{6931}{10000} - \frac{1}{v}} \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\left(e^{\frac{6931}{10000} - \frac{1}{v}}\right), \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right) \]

    4. exp-lowering-exp.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\left(\frac{6931}{10000} - \frac{1}{v}\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{v}\right)\right) \]

    5. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{v}\right)\right) \]

    6. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{v}\right)\right) \]

    7. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right), \left(\frac{\frac{1}{2}}{v}\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right), \left(\frac{\frac{1}{2}}{v}\right)\right) \]

    9. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right), \left(\frac{\frac{1}{2}}{v}\right)\right) \]

    10. /-lowering-/.f32 99.4%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right)\right) \]

12. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{e^{0.6931 + \frac{-1}{v}} \cdot \frac{0.5}{v}} \]
13. Add Preprocessing

Alternative 3: 98.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \left(\mathsf{neg}\left(\log \left(2 \cdot v\right)\right)\right)} \]

   2. unsub-neg N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \log \left(2 \cdot v\right)} \]

   3. exp-diff N/A

      \[\leadsto \frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{\color{blue}{e^{\log \left(2 \cdot v\right)}}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}\right), \color{blue}{\left(e^{\log \left(2 \cdot v\right)}\right)}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{e^{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_O around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right), \color{blue}{v}\right) \]

7. Simplified 99.8%

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

8. Taylor expanded in sinTheta_O around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{\frac{6931}{10000} - \frac{1}{v}}}{v}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot e^{\frac{6931}{10000} - \frac{1}{v}}}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot e^{\frac{6931}{10000} - \frac{1}{v}}\right), \color{blue}{v}\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(e^{\frac{6931}{10000} - \frac{1}{v}}\right)\right), v\right) \]

   4. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right), v\right) \]

   5. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\frac{6931}{10000}, \left(\frac{1}{v}\right)\right)\right)\right), v\right) \]

   6. /-lowering-/.f32 99.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(1, v\right)\right)\right)\right), v\right) \]

10. Simplified 99.8%

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

11. Taylor expanded in v around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\color{blue}{\left(\frac{-1}{v}\right)}\right)\right), v\right) \]
12. Step-by-step derivation

    1. /-lowering-/.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(-1, v\right)\right)\right), v\right) \]

13. Simplified 98.6%

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

Alternative 4: 51.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.4999999523982838 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{1 + \left(0.5 \cdot \left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{1}{v \cdot v}\right)\right) + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 1.4999999523982838e-22)
   (/
    1.0
    (+
     1.0
     (+
      (*
       0.5
       (*
        (* sinTheta_O sinTheta_i)
        (* (* sinTheta_O sinTheta_i) (/ 1.0 (* v v)))))
      (/ (* 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 (v <= 1.4999999523982838e-22f) {
		tmp = 1.0f / (1.0f + ((0.5f * ((sinTheta_O * sinTheta_i) * ((sinTheta_O * sinTheta_i) * (1.0f / (v * v))))) + ((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 (v <= 1.4999999523982838e-22) then
        tmp = 1.0e0 / (1.0e0 + ((0.5e0 * ((sintheta_o * sintheta_i) * ((sintheta_o * sintheta_i) * (1.0e0 / (v * v))))) + ((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 (v <= Float32(1.4999999523982838e-22))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(0.5) * Float32(Float32(sinTheta_O * sinTheta_i) * Float32(Float32(sinTheta_O * sinTheta_i) * Float32(Float32(1.0) / Float32(v * v))))) + Float32(Float32(sinTheta_O * sinTheta_i) / v))));
	else
		tmp = Float32(Float32(cosTheta_O * 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 (v <= single(1.4999999523982838e-22))
		tmp = single(1.0) / (single(1.0) + ((single(0.5) * ((sinTheta_O * sinTheta_i) * ((sinTheta_O * sinTheta_i) * (single(1.0) / (v * v))))) + ((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 v < 1.5e-22

   1. Initial program 99.1%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in sinTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      5. *-lowering-*.f32 20.2%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   5. Simplified 20.2%

      \[\leadsto e^{\color{blue}{0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
   6. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{0}}{\color{blue}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

      2. 1-exp N/A

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)}\right) \]

      4. exp-lowering-exp.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      8. *-lowering-*.f32 20.2%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   7. Applied egg-rr 20.2%

      \[\leadsto \color{blue}{\frac{1}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   8. Taylor expanded in v around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}}\right), \color{blue}{\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}}\right)\right), \left(\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}\right)\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot \color{blue}{sinTheta\_i}}{v}\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({sinTheta\_O}^{2}\right), \left({sinTheta\_i}^{2}\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(sinTheta\_O \cdot sinTheta\_O\right), \left({sinTheta\_i}^{2}\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \left({sinTheta\_i}^{2}\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \left(sinTheta\_i \cdot sinTheta\_i\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \left(v \cdot v\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), \color{blue}{v}\right)\right)\right)\right) \]

      13. *-lowering-*.f32 18.2%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

   10. Simplified 18.2%

       \[\leadsto \frac{1}{\color{blue}{1 + \left(0.5 \cdot \frac{\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)}{v \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
   11. Step-by-step derivation

       1. div-inv N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right) \cdot \frac{1}{v \cdot v}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \color{blue}{sinTheta\_i}\right), v\right)\right)\right)\right) \]

       2. unswap-sqr N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right) \cdot \frac{1}{v \cdot v}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right) \cdot \frac{-1 \cdot -1}{v \cdot v}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       4. frac-times N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right) \cdot \left(\frac{-1}{v} \cdot \frac{-1}{v}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(\frac{-1}{v} \cdot \frac{-1}{v}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \color{blue}{sinTheta\_i}\right), v\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), \left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(\frac{-1}{v} \cdot \frac{-1}{v}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \color{blue}{sinTheta\_i}\right), v\right)\right)\right)\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(\frac{-1}{v} \cdot \frac{-1}{v}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \mathsf{*.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), \left(\frac{-1}{v} \cdot \frac{-1}{v}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \left(\frac{-1}{v} \cdot \frac{-1}{v}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       10. frac-times N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \left(\frac{-1 \cdot -1}{v \cdot v}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \left(\frac{1}{v \cdot v}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       12. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \mathsf{/.f32}\left(1, \left(v \cdot v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       13. *-lowering-*.f32 65.0%

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

   12. Applied egg-rr 65.0%

       \[\leadsto \frac{1}{1 + \left(0.5 \cdot \color{blue}{\left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{1}{v \cdot v}\right)\right)} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   if 1.5e-22 < v

   1. Initial program 99.7%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in cosTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right) \]

      3. /-lowering-/.f32 7.9%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right) \]

   5. Simplified 7.9%

      \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

   6. Taylor expanded in cosTheta_O around 0

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right)\right) \]

      3. *-lowering-*.f32 6.5%

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right)\right) \]

   8. Simplified 6.5%

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   9. Taylor expanded in cosTheta_O around inf

      \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right) \]

       2. *-lowering-*.f32 39.8%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right) \]

   11. Simplified 39.8%

       \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 50.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{1 + \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} + 0.5 \cdot \left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v \cdot v}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 5.000000156871975e-23)
   (/
    1.0
    (+
     1.0
     (+
      (/ (* sinTheta_O sinTheta_i) v)
      (*
       0.5
       (* (* sinTheta_O sinTheta_i) (/ (* sinTheta_O sinTheta_i) (* v 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 (v <= 5.000000156871975e-23f) {
		tmp = 1.0f / (1.0f + (((sinTheta_O * sinTheta_i) / v) + (0.5f * ((sinTheta_O * sinTheta_i) * ((sinTheta_O * sinTheta_i) / (v * 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 (v <= 5.000000156871975e-23) then
        tmp = 1.0e0 / (1.0e0 + (((sintheta_o * sintheta_i) / v) + (0.5e0 * ((sintheta_o * sintheta_i) * ((sintheta_o * sintheta_i) / (v * 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 (v <= Float32(5.000000156871975e-23))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(sinTheta_O * sinTheta_i) / v) + Float32(Float32(0.5) * Float32(Float32(sinTheta_O * sinTheta_i) * Float32(Float32(sinTheta_O * sinTheta_i) / Float32(v * v)))))));
	else
		tmp = Float32(Float32(cosTheta_O * 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 (v <= single(5.000000156871975e-23))
		tmp = single(1.0) / (single(1.0) + (((sinTheta_O * sinTheta_i) / v) + (single(0.5) * ((sinTheta_O * sinTheta_i) * ((sinTheta_O * sinTheta_i) / (v * v))))));
	else
		tmp = (cosTheta_O * cosTheta_i) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 5.00000016e-23

   1. Initial program 99.1%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in sinTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      5. *-lowering-*.f32 20.6%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   5. Simplified 20.6%

      \[\leadsto e^{\color{blue}{0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
   6. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{0}}{\color{blue}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

      2. 1-exp N/A

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)}\right) \]

      4. exp-lowering-exp.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      8. *-lowering-*.f32 20.6%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   7. Applied egg-rr 20.6%

      \[\leadsto \color{blue}{\frac{1}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   8. Taylor expanded in v around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}}\right), \color{blue}{\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}}\right)\right), \left(\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}\right)\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot \color{blue}{sinTheta\_i}}{v}\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({sinTheta\_O}^{2}\right), \left({sinTheta\_i}^{2}\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(sinTheta\_O \cdot sinTheta\_O\right), \left({sinTheta\_i}^{2}\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \left({sinTheta\_i}^{2}\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \left(sinTheta\_i \cdot sinTheta\_i\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \left(v \cdot v\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), \color{blue}{v}\right)\right)\right)\right) \]

      13. *-lowering-*.f32 18.5%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

   10. Simplified 18.5%

       \[\leadsto \frac{1}{\color{blue}{1 + \left(0.5 \cdot \frac{\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)}{v \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
   11. Step-by-step derivation

       1. unswap-sqr N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v \cdot v}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v \cdot v}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \color{blue}{sinTheta\_i}\right), v\right)\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v \cdot v}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \color{blue}{sinTheta\_i}\right), v\right)\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v \cdot v}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       5. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), \left(v \cdot v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \left(v \cdot v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       7. *-lowering-*.f32 63.2%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), \mathsf{*.f32}\left(v, v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

   12. Applied egg-rr 63.2%

       \[\leadsto \frac{1}{1 + \left(0.5 \cdot \color{blue}{\left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v \cdot v}\right)} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   if 5.00000016e-23 < v

   1. Initial program 99.7%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in cosTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right) \]

      3. /-lowering-/.f32 7.8%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right) \]

   5. Simplified 7.8%

      \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

   6. Taylor expanded in cosTheta_O around 0

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right)\right) \]

      3. *-lowering-*.f32 6.5%

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right)\right) \]

   8. Simplified 6.5%

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   9. Taylor expanded in cosTheta_O around inf

      \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right) \]

       2. *-lowering-*.f32 39.2%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right) \]

   11. Simplified 39.2%

       \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{1 + \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} + 0.5 \cdot \left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v \cdot v}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.3% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.200000062892823 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{1 + \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} + 0.5 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_i \cdot sinTheta\_i}{v \cdot v}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 1.200000062892823e-22)
   (/
    1.0
    (+
     1.0
     (+
      (/ (* sinTheta_O sinTheta_i) v)
      (*
       0.5
       (* sinTheta_O (* sinTheta_O (/ (* sinTheta_i sinTheta_i) (* v 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 (v <= 1.200000062892823e-22f) {
		tmp = 1.0f / (1.0f + (((sinTheta_O * sinTheta_i) / v) + (0.5f * (sinTheta_O * (sinTheta_O * ((sinTheta_i * sinTheta_i) / (v * 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 (v <= 1.200000062892823e-22) then
        tmp = 1.0e0 / (1.0e0 + (((sintheta_o * sintheta_i) / v) + (0.5e0 * (sintheta_o * (sintheta_o * ((sintheta_i * sintheta_i) / (v * 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 (v <= Float32(1.200000062892823e-22))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(sinTheta_O * sinTheta_i) / v) + Float32(Float32(0.5) * Float32(sinTheta_O * Float32(sinTheta_O * Float32(Float32(sinTheta_i * sinTheta_i) / Float32(v * v))))))));
	else
		tmp = Float32(Float32(cosTheta_O * 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 (v <= single(1.200000062892823e-22))
		tmp = single(1.0) / (single(1.0) + (((sinTheta_O * sinTheta_i) / v) + (single(0.5) * (sinTheta_O * (sinTheta_O * ((sinTheta_i * sinTheta_i) / (v * v)))))));
	else
		tmp = (cosTheta_O * cosTheta_i) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 1.20000006e-22

   1. Initial program 99.1%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in sinTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      5. *-lowering-*.f32 20.3%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   5. Simplified 20.3%

      \[\leadsto e^{\color{blue}{0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
   6. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{0}}{\color{blue}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

      2. 1-exp N/A

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)}\right) \]

      4. exp-lowering-exp.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      8. *-lowering-*.f32 20.3%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   7. Applied egg-rr 20.3%

      \[\leadsto \color{blue}{\frac{1}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   8. Taylor expanded in v around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}}\right), \color{blue}{\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}}\right)\right), \left(\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}\right)\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot \color{blue}{sinTheta\_i}}{v}\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({sinTheta\_O}^{2}\right), \left({sinTheta\_i}^{2}\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(sinTheta\_O \cdot sinTheta\_O\right), \left({sinTheta\_i}^{2}\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \left({sinTheta\_i}^{2}\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \left(sinTheta\_i \cdot sinTheta\_i\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \left(v \cdot v\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), \color{blue}{v}\right)\right)\right)\right) \]

      13. *-lowering-*.f32 18.3%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

   10. Simplified 18.3%

       \[\leadsto \frac{1}{\color{blue}{1 + \left(0.5 \cdot \frac{\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)}{v \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot \frac{sinTheta\_i \cdot sinTheta\_i}{v \cdot v}\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \color{blue}{sinTheta\_i}\right), v\right)\right)\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_i \cdot sinTheta\_i}{v \cdot v}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \color{blue}{sinTheta\_i}\right), v\right)\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(sinTheta\_O, \left(sinTheta\_O \cdot \frac{sinTheta\_i \cdot sinTheta\_i}{v \cdot v}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \color{blue}{sinTheta\_i}\right), v\right)\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_O, \left(\frac{sinTheta\_i \cdot sinTheta\_i}{v \cdot v}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       5. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_i\right), \left(v \cdot v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right), \left(v \cdot v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

       7. *-lowering-*.f32 57.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right), \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

   12. Applied egg-rr 57.9%

       \[\leadsto \frac{1}{1 + \left(0.5 \cdot \color{blue}{\left(sinTheta\_O \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_i \cdot sinTheta\_i}{v \cdot v}\right)\right)} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   if 1.20000006e-22 < v

   1. Initial program 99.7%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in cosTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right) \]

      3. /-lowering-/.f32 7.8%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right) \]

   5. Simplified 7.8%

      \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

   6. Taylor expanded in cosTheta_O around 0

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right)\right) \]

      3. *-lowering-*.f32 6.5%

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right)\right) \]

   8. Simplified 6.5%

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   9. Taylor expanded in cosTheta_O around inf

      \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right) \]

       2. *-lowering-*.f32 39.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right) \]

   11. Simplified 39.6%

       \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.200000062892823 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{1 + \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} + 0.5 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_i \cdot sinTheta\_i}{v \cdot v}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + sinTheta\_i \cdot \left(0.5 \cdot \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v} + \frac{sinTheta\_O}{v}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O 1.999999987845058e-8)
   (/ (* cosTheta_O cosTheta_i) v)
   (/
    1.0
    (+
     1.0
     (*
      sinTheta_i
      (+
       (* 0.5 (/ (* sinTheta_i (* sinTheta_O sinTheta_O)) (* v v)))
       (/ sinTheta_O v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_O <= 1.999999987845058e-8f) {
		tmp = (cosTheta_O * cosTheta_i) / v;
	} else {
		tmp = 1.0f / (1.0f + (sinTheta_i * ((0.5f * ((sinTheta_i * (sinTheta_O * sinTheta_O)) / (v * v))) + (sinTheta_O / v))));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (sintheta_o <= 1.999999987845058e-8) then
        tmp = (costheta_o * costheta_i) / v
    else
        tmp = 1.0e0 / (1.0e0 + (sintheta_i * ((0.5e0 * ((sintheta_i * (sintheta_o * sintheta_o)) / (v * v))) + (sintheta_o / v))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_O <= Float32(1.999999987845058e-8))
		tmp = Float32(Float32(cosTheta_O * cosTheta_i) / v);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(sinTheta_i * Float32(Float32(Float32(0.5) * Float32(Float32(sinTheta_i * Float32(sinTheta_O * sinTheta_O)) / Float32(v * v))) + Float32(sinTheta_O / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_O <= single(1.999999987845058e-8))
		tmp = (cosTheta_O * cosTheta_i) / v;
	else
		tmp = single(1.0) / (single(1.0) + (sinTheta_i * ((single(0.5) * ((sinTheta_i * (sinTheta_O * sinTheta_O)) / (v * v))) + (sinTheta_O / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 1.99999999e-8

   1. Initial program 99.8%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in cosTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right) \]

      3. /-lowering-/.f32 15.1%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right) \]

   5. Simplified 15.1%

      \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

   6. Taylor expanded in cosTheta_O around 0

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right)\right) \]

      3. *-lowering-*.f32 6.3%

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right)\right) \]

   8. Simplified 6.3%

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   9. Taylor expanded in cosTheta_O around inf

      \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right) \]

       2. *-lowering-*.f32 38.2%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right) \]

   11. Simplified 38.2%

       \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   if 1.99999999e-8 < sinTheta_O

   1. Initial program 95.5%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in sinTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      5. *-lowering-*.f32 26.9%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   5. Simplified 26.9%

      \[\leadsto e^{\color{blue}{0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
   6. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{0}}{\color{blue}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

      2. 1-exp N/A

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)}\right) \]

      4. exp-lowering-exp.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      8. *-lowering-*.f32 26.9%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   7. Applied egg-rr 26.9%

      \[\leadsto \color{blue}{\frac{1}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   8. Taylor expanded in sinTheta_i around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(1 + sinTheta\_i \cdot \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{2}} + \frac{sinTheta\_O}{v}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \color{blue}{\left(sinTheta\_i \cdot \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{2}} + \frac{sinTheta\_O}{v}\right)\right)}\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \color{blue}{\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{2}} + \frac{sinTheta\_O}{v}\right)}\right)\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{2}}\right), \color{blue}{\left(\frac{sinTheta\_O}{v}\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{2}}\right)\right), \left(\frac{\color{blue}{sinTheta\_O}}{v}\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left({sinTheta\_O}^{2} \cdot sinTheta\_i\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O}{v}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({sinTheta\_O}^{2}\right), sinTheta\_i\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O}{v}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(sinTheta\_O \cdot sinTheta\_O\right), sinTheta\_i\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O}{v}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), sinTheta\_i\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O}{v}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), sinTheta\_i\right), \left(v \cdot v\right)\right)\right), \left(\frac{sinTheta\_O}{v}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), sinTheta\_i\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \left(\frac{sinTheta\_O}{v}\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f32 57.2%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), sinTheta\_i\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(sinTheta\_O, \color{blue}{v}\right)\right)\right)\right)\right) \]

   10. Simplified 57.2%

       \[\leadsto \frac{1}{\color{blue}{1 + sinTheta\_i \cdot \left(0.5 \cdot \frac{\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot sinTheta\_i}{v \cdot v} + \frac{sinTheta\_O}{v}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + sinTheta\_i \cdot \left(0.5 \cdot \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v} + \frac{sinTheta\_O}{v}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_i \leq -2.6000000763470865 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{1 + sinTheta\_O \cdot \left(0.5 \cdot \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)}{v \cdot v} + \frac{sinTheta\_i}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= cosTheta_i -2.6000000763470865e-11)
   (/
    1.0
    (+
     1.0
     (*
      sinTheta_O
      (+
       (* 0.5 (/ (* sinTheta_O (* sinTheta_i sinTheta_i)) (* v v)))
       (/ 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 (cosTheta_i <= -2.6000000763470865e-11f) {
		tmp = 1.0f / (1.0f + (sinTheta_O * ((0.5f * ((sinTheta_O * (sinTheta_i * sinTheta_i)) / (v * v))) + (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 (costheta_i <= (-2.6000000763470865e-11)) then
        tmp = 1.0e0 / (1.0e0 + (sintheta_o * ((0.5e0 * ((sintheta_o * (sintheta_i * sintheta_i)) / (v * v))) + (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 (cosTheta_i <= Float32(-2.6000000763470865e-11))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(sinTheta_O * Float32(Float32(Float32(0.5) * Float32(Float32(sinTheta_O * Float32(sinTheta_i * sinTheta_i)) / Float32(v * v))) + Float32(sinTheta_i / v)))));
	else
		tmp = Float32(Float32(cosTheta_O * 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 (cosTheta_i <= single(-2.6000000763470865e-11))
		tmp = single(1.0) / (single(1.0) + (sinTheta_O * ((single(0.5) * ((sinTheta_O * (sinTheta_i * sinTheta_i)) / (v * v))) + (sinTheta_i / v))));
	else
		tmp = (cosTheta_O * cosTheta_i) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if cosTheta_i < -2.60000008e-11

   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. Add Preprocessing

   3. Taylor expanded in sinTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      5. *-lowering-*.f32 6.1%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   5. Simplified 6.1%

      \[\leadsto e^{\color{blue}{0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
   6. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{0}}{\color{blue}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

      2. 1-exp N/A

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)}\right) \]

      4. exp-lowering-exp.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      8. *-lowering-*.f32 6.1%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   7. Applied egg-rr 6.1%

      \[\leadsto \color{blue}{\frac{1}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   8. Taylor expanded in sinTheta_O around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(1 + sinTheta\_O \cdot \left(\frac{1}{2} \cdot \frac{sinTheta\_O \cdot {sinTheta\_i}^{2}}{{v}^{2}} + \frac{sinTheta\_i}{v}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \color{blue}{\left(sinTheta\_O \cdot \left(\frac{1}{2} \cdot \frac{sinTheta\_O \cdot {sinTheta\_i}^{2}}{{v}^{2}} + \frac{sinTheta\_i}{v}\right)\right)}\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \color{blue}{\left(\frac{1}{2} \cdot \frac{sinTheta\_O \cdot {sinTheta\_i}^{2}}{{v}^{2}} + \frac{sinTheta\_i}{v}\right)}\right)\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{sinTheta\_O \cdot {sinTheta\_i}^{2}}{{v}^{2}}\right), \color{blue}{\left(\frac{sinTheta\_i}{v}\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{sinTheta\_O \cdot {sinTheta\_i}^{2}}{{v}^{2}}\right)\right), \left(\frac{\color{blue}{sinTheta\_i}}{v}\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot {sinTheta\_i}^{2}\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_i}{v}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \left({sinTheta\_i}^{2}\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_i}{v}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \left(sinTheta\_i \cdot sinTheta\_i\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_i}{v}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_i}{v}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \left(v \cdot v\right)\right)\right), \left(\frac{sinTheta\_i}{v}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \left(\frac{sinTheta\_i}{v}\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f32 7.1%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(sinTheta\_i, \color{blue}{v}\right)\right)\right)\right)\right) \]

   10. Simplified 7.1%

       \[\leadsto \frac{1}{\color{blue}{1 + sinTheta\_O \cdot \left(0.5 \cdot \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)}{v \cdot v} + \frac{sinTheta\_i}{v}\right)}} \]

   if -2.60000008e-11 < cosTheta_i

   1. Initial program 99.4%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in cosTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right) \]

      3. /-lowering-/.f32 13.0%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right) \]

   5. Simplified 13.0%

      \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

   6. Taylor expanded in cosTheta_O around 0

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right)\right) \]

      3. *-lowering-*.f32 6.3%

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right)\right) \]

   8. Simplified 6.3%

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   9. Taylor expanded in cosTheta_O around inf

      \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right) \]

       2. *-lowering-*.f32 42.7%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right) \]

   11. Simplified 42.7%

       \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 39.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_i \leq -2.6000000763470865 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot \left(0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_O}{v \cdot v} + \frac{sinTheta\_O}{v \cdot sinTheta\_i}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= cosTheta_i -2.6000000763470865e-11)
   (/
    1.0
    (*
     (* sinTheta_i sinTheta_i)
     (+
      (* 0.5 (/ (* sinTheta_O sinTheta_O) (* v v)))
      (/ sinTheta_O (* v sinTheta_i)))))
   (/ (* 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 (cosTheta_i <= -2.6000000763470865e-11f) {
		tmp = 1.0f / ((sinTheta_i * sinTheta_i) * ((0.5f * ((sinTheta_O * sinTheta_O) / (v * v))) + (sinTheta_O / (v * sinTheta_i))));
	} 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 (costheta_i <= (-2.6000000763470865e-11)) then
        tmp = 1.0e0 / ((sintheta_i * sintheta_i) * ((0.5e0 * ((sintheta_o * sintheta_o) / (v * v))) + (sintheta_o / (v * sintheta_i))))
    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 (cosTheta_i <= Float32(-2.6000000763470865e-11))
		tmp = Float32(Float32(1.0) / Float32(Float32(sinTheta_i * sinTheta_i) * Float32(Float32(Float32(0.5) * Float32(Float32(sinTheta_O * sinTheta_O) / Float32(v * v))) + Float32(sinTheta_O / Float32(v * sinTheta_i)))));
	else
		tmp = Float32(Float32(cosTheta_O * 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 (cosTheta_i <= single(-2.6000000763470865e-11))
		tmp = single(1.0) / ((sinTheta_i * sinTheta_i) * ((single(0.5) * ((sinTheta_O * sinTheta_O) / (v * v))) + (sinTheta_O / (v * sinTheta_i))));
	else
		tmp = (cosTheta_O * cosTheta_i) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if cosTheta_i < -2.60000008e-11

   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. Add Preprocessing

   3. Taylor expanded in sinTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      5. *-lowering-*.f32 6.1%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   5. Simplified 6.1%

      \[\leadsto e^{\color{blue}{0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
   6. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{0}}{\color{blue}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

      2. 1-exp N/A

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)}\right) \]

      4. exp-lowering-exp.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      8. *-lowering-*.f32 6.1%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   7. Applied egg-rr 6.1%

      \[\leadsto \color{blue}{\frac{1}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   8. Taylor expanded in v around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}}\right), \color{blue}{\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{{sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}}{{v}^{2}}\right)\right), \left(\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}\right)\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot \color{blue}{sinTheta\_i}}{v}\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({sinTheta\_O}^{2}\right), \left({sinTheta\_i}^{2}\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(sinTheta\_O \cdot sinTheta\_O\right), \left({sinTheta\_i}^{2}\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \left({sinTheta\_i}^{2}\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \left(sinTheta\_i \cdot sinTheta\_i\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \left(v \cdot v\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), \color{blue}{v}\right)\right)\right)\right) \]

      13. *-lowering-*.f32 7.1%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right) \]

   10. Simplified 7.1%

       \[\leadsto \frac{1}{\color{blue}{1 + \left(0.5 \cdot \frac{\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)}{v \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   11. Taylor expanded in sinTheta_i around inf

       \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left({sinTheta\_i}^{2} \cdot \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}} + \frac{sinTheta\_O}{sinTheta\_i \cdot v}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\left({sinTheta\_i}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}} + \frac{sinTheta\_O}{sinTheta\_i \cdot v}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\left(sinTheta\_i \cdot sinTheta\_i\right), \left(\color{blue}{\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}} + \frac{sinTheta\_O}{sinTheta\_i \cdot v}\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right), \left(\color{blue}{\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}} + \frac{sinTheta\_O}{sinTheta\_i \cdot v}\right)\right)\right) \]

       4. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right), \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right), \color{blue}{\left(\frac{sinTheta\_O}{sinTheta\_i \cdot v}\right)}\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)\right), \left(\frac{\color{blue}{sinTheta\_O}}{sinTheta\_i \cdot v}\right)\right)\right)\right) \]

       6. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left({sinTheta\_O}^{2}\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O}{sinTheta\_i \cdot v}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_O\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O}{sinTheta\_i \cdot v}\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \left({v}^{2}\right)\right)\right), \left(\frac{sinTheta\_O}{sinTheta\_i \cdot v}\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \left(v \cdot v\right)\right)\right), \left(\frac{sinTheta\_O}{sinTheta\_i \cdot v}\right)\right)\right)\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \left(\frac{sinTheta\_O}{sinTheta\_i \cdot v}\right)\right)\right)\right) \]

       11. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(sinTheta\_O, \color{blue}{\left(sinTheta\_i \cdot v\right)}\right)\right)\right)\right) \]

       12. *-lowering-*.f32 8.5%

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_i\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right), \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, \color{blue}{v}\right)\right)\right)\right)\right) \]

   13. Simplified 8.5%

       \[\leadsto \frac{1}{\color{blue}{\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot \left(0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_O}{v \cdot v} + \frac{sinTheta\_O}{sinTheta\_i \cdot v}\right)}} \]

   if -2.60000008e-11 < cosTheta_i

   1. Initial program 99.4%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in cosTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right) \]

      3. /-lowering-/.f32 13.0%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right) \]

   5. Simplified 13.0%

      \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

   6. Taylor expanded in cosTheta_O around 0

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right)\right) \]

      3. *-lowering-*.f32 6.3%

         \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right)\right) \]

   8. Simplified 6.3%

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   9. Taylor expanded in cosTheta_O around inf

      \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right) \]

       2. *-lowering-*.f32 42.7%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right) \]

   11. Simplified 42.7%

       \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta\_i \leq -2.6000000763470865 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot \left(0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_O}{v \cdot v} + \frac{sinTheta\_O}{v \cdot sinTheta\_i}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.4% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_O \cdot cosTheta\_i}{v} \end{array} \]

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(Float32(cosTheta_O * 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 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around inf

   \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right) \]

   3. /-lowering-/.f32 15.4%

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right) \]

5. Simplified 15.4%

   \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

6. Taylor expanded in cosTheta_O around 0

   \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right)\right) \]

   3. *-lowering-*.f32 6.3%

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right)\right) \]

8. Simplified 6.3%

   \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

9. Taylor expanded in cosTheta_O around inf

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
10. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right) \]

    2. *-lowering-*.f32 38.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right) \]

11. Simplified 38.4%

    \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
12. Add Preprocessing

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 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around inf

   \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right) \]

   3. /-lowering-/.f32 15.4%

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right) \]

5. Simplified 15.4%

   \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

6. Taylor expanded in cosTheta_O around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 6.4%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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))))))