HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 15.0s

Alternatives: 7

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. exp-sum N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in cosTheta_i around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}, 1\right), v\right)\right)\right), \mathsf{exp.f32}\left(\frac{6931}{10000}\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f32 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in cosTheta_O around 0

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

   1. sub-neg N/A

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

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

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

   3. rem-exp-log N/A

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

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

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

   5. rem-exp-log N/A

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

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

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. /-lowering-/.f32 99.8%

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

10. Simplified 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. exp-sum N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. prod-exp N/A

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

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

      \[\leadsto \mathsf{exp.f32}\left(\left(\frac{6931}{10000} + \left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}\right)\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}\right)\right)\right) \]

   4. associate--l+ N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{+.f32}\left(\log \left(\frac{\frac{1}{2}}{v}\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)\right)\right) \]

   6. rem-exp-log N/A

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

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

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

   8. rem-exp-log N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{+.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2}}{v}\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)\right)\right) \]

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

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

   10. div-sub N/A

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

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

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

   12. sub-neg N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(1\right)\right)\right), v\right)\right)\right)\right) \]

   13. metadata-eval N/A

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

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

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

   15. *-lowering-*.f32 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

   \[\leadsto e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{-1 + cosTheta\_O \cdot cosTheta\_i}{v}\right)} \]
9. Add Preprocessing

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. exp-sum N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in cosTheta_i around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}, 1\right), v\right)\right)\right), \mathsf{exp.f32}\left(\frac{6931}{10000}\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f32 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in cosTheta_O around 0

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

   1. sub-neg N/A

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

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

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

   3. rem-exp-log N/A

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

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

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

   5. rem-exp-log N/A

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

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

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. /-lowering-/.f32 99.8%

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

10. Simplified 99.8%

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

    1. prod-exp N/A

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

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

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

    3. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

    8. /-lowering-/.f32 99.8%

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

12. Applied egg-rr 99.8%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. exp-sum N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in cosTheta_i around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}, 1\right), v\right)\right)\right), \mathsf{exp.f32}\left(\frac{6931}{10000}\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f32 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in cosTheta_O around 0

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

   1. sub-neg N/A

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

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

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

   3. rem-exp-log N/A

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

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

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

   5. rem-exp-log N/A

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

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

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. /-lowering-/.f32 99.8%

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

10. Simplified 99.8%

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

    1. exp-sum N/A

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

    2. rem-exp-log N/A

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

    3. div-inv N/A

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

    4. mul-1-neg N/A

       \[\leadsto \left(\frac{\frac{1}{2}}{v} \cdot e^{\mathsf{neg}\left(\frac{1}{v}\right)}\right) \cdot e^{\frac{6931}{10000}} \]

    5. rec-exp N/A

       \[\leadsto \left(\frac{\frac{1}{2}}{v} \cdot \frac{1}{e^{\frac{1}{v}}}\right) \cdot e^{\frac{6931}{10000}} \]

    6. div-inv N/A

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

    7. associate-*l/ N/A

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

    8. associate-/l* N/A

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

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

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

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

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

    11. div-exp N/A

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

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

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

    13. --lowering--.f32 N/A

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

    14. /-lowering-/.f32 99.8%

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

12. Applied egg-rr 99.8%

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

13. Final simplification 99.8%

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

Alternative 5: 49.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{1 + sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(\frac{1}{v} + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v \cdot v}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 9.999999998199587e-24)
   (/
    1.0
    (+
     1.0
     (*
      sinTheta_O
      (*
       sinTheta_i
       (+ (/ 1.0 v) (/ (* 0.5 (* 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 <= 9.999999998199587e-24f) {
		tmp = 1.0f / (1.0f + (sinTheta_O * (sinTheta_i * ((1.0f / v) + ((0.5f * (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 <= 9.999999998199587e-24) then
        tmp = 1.0e0 / (1.0e0 + (sintheta_o * (sintheta_i * ((1.0e0 / v) + ((0.5e0 * (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(9.999999998199587e-24))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(sinTheta_O * Float32(sinTheta_i * Float32(Float32(Float32(1.0) / v) + Float32(Float32(Float32(0.5) * 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(9.999999998199587e-24))
		tmp = single(1.0) / (single(1.0) + (sinTheta_O * (sinTheta_i * ((single(1.0) / v) + ((single(0.5) * (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 < 1e-23

   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 21.1%

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

   5. Simplified 21.1%

      \[\leadsto e^{\color{blue}{0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
   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. *-lowering-*.f32 21.1%

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

   7. Applied egg-rr 21.1%

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

   8. Taylor expanded in sinTheta_O around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(1 + sinTheta\_O \cdot \left(sinTheta\_O \cdot \left(\frac{1}{6} \cdot \frac{sinTheta\_O \cdot {sinTheta\_i}^{3}}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_i}^{2}}{{v}^{2}}\right) + \frac{sinTheta\_i}{v}\right)\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \color{blue}{\left(sinTheta\_O \cdot \left(sinTheta\_O \cdot \left(\frac{1}{6} \cdot \frac{sinTheta\_O \cdot {sinTheta\_i}^{3}}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_i}^{2}}{{v}^{2}}\right) + \frac{sinTheta\_i}{v}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \color{blue}{\left(sinTheta\_O \cdot \left(\frac{1}{6} \cdot \frac{sinTheta\_O \cdot {sinTheta\_i}^{3}}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_i}^{2}}{{v}^{2}}\right) + \frac{sinTheta\_i}{v}\right)}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \left(\frac{sinTheta\_i}{v} + \color{blue}{sinTheta\_O \cdot \left(\frac{1}{6} \cdot \frac{sinTheta\_O \cdot {sinTheta\_i}^{3}}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_i}^{2}}{{v}^{2}}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\left(\frac{sinTheta\_i}{v}\right), \color{blue}{\left(sinTheta\_O \cdot \left(\frac{1}{6} \cdot \frac{sinTheta\_O \cdot {sinTheta\_i}^{3}}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_i}^{2}}{{v}^{2}}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, v\right), \left(\color{blue}{sinTheta\_O} \cdot \left(\frac{1}{6} \cdot \frac{sinTheta\_O \cdot {sinTheta\_i}^{3}}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_i}^{2}}{{v}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, v\right), \mathsf{*.f32}\left(sinTheta\_O, \color{blue}{\left(\frac{1}{6} \cdot \frac{sinTheta\_O \cdot {sinTheta\_i}^{3}}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_i}^{2}}{{v}^{2}}\right)}\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, v\right), \mathsf{*.f32}\left(sinTheta\_O, \left(\frac{1}{2} \cdot \frac{{sinTheta\_i}^{2}}{{v}^{2}} + \color{blue}{\frac{1}{6} \cdot \frac{sinTheta\_O \cdot {sinTheta\_i}^{3}}{{v}^{3}}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, v\right), \mathsf{*.f32}\left(sinTheta\_O, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{{sinTheta\_i}^{2}}{{v}^{2}}\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{sinTheta\_O \cdot {sinTheta\_i}^{3}}{{v}^{3}}\right)}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 14.3%

       \[\leadsto \frac{1}{\color{blue}{1 + sinTheta\_O \cdot \left(\frac{sinTheta\_i}{v} + sinTheta\_O \cdot \left(\frac{0.5 \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)}{v \cdot v} + \frac{0.16666666666666666 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right)\right)}{v \cdot \left(v \cdot v\right)}\right)\right)}} \]

   11. Taylor expanded in sinTheta_i around 0

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

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

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

       2. +-commutative N/A

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

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

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

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

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

       5. associate-*r/ N/A

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

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

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f32 60.7%

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

   13. Simplified 60.7%

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

   if 1e-23 < v

   1. Initial program 99.6%

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

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

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

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

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

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

      4. *-lowering-*.f32 6.6%

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

   8. Simplified 6.6%

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

   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 42.3%

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

   11. Simplified 42.3%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{1 + sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(\frac{1}{v} + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v \cdot v}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.3% 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 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 14.3%

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

5. Simplified 14.3%

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

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. +-commutative N/A

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

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

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

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

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

   4. *-lowering-*.f32 6.3%

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

8. Simplified 6.3%

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

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 37.2%

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

11. Simplified 37.2%

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

Alternative 7: 6.4% accurate, 223.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 1.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. 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 14.3%

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

5. Simplified 14.3%

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

6. Taylor expanded in cosTheta_O around 0

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

8. Simplified 6.5%

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

Reproduce

herbie shell --seed 2024161 
(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))))))