HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 15.5s

Alternatives: 8

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.0) / exp(Float32(log(Float32(v * Float32(2.0))) + 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(1.0) / exp((log((v * single(2.0))) + ((single(1.0) / v) - single(0.6931))));
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. 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. log-div N/A

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

   3. associate-+l- N/A

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

   4. exp-diff N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{e^{0}}{e^{\color{blue}{\log \left(2 \cdot v\right)} - \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   6. 1-exp N/A

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

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

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

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

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. *-lowering-*.f32 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in cosTheta_O around 0

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

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

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

   2. /-lowering-/.f32 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / ((v / single(0.5)) * (single(1.0) / exp((single(0.6931) + (((cosTheta_i * cosTheta_O) + (single(-1.0) - (sinTheta_i * sinTheta_O))) / 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. 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. log-div N/A

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

   3. associate-+l- N/A

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

   4. exp-diff N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{e^{0}}{e^{\color{blue}{\log \left(2 \cdot v\right)} - \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   6. 1-exp N/A

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

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

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

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

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

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

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

4. Applied egg-rr 99.8%

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

   1. exp-diff N/A

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

   2. div-inv N/A

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

   3. rem-exp-log N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. associate--l- N/A

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

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

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

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

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

   15. +-commutative N/A

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

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

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

   17. *-lowering-*.f32 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 24.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := cosTheta\_O \cdot \frac{cosTheta\_i}{v}\\ \mathbf{if}\;v \leq 3.000000157232057 \cdot 10^{-23}:\\ \;\;\;\;e^{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (* cosTheta_O (/ cosTheta_i v))))
   (if (<= v 3.000000157232057e-23) (exp t_0) t_0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = cosTheta_O * (cosTheta_i / v);
	float tmp;
	if (v <= 3.000000157232057e-23f) {
		tmp = expf(t_0);
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = costheta_o * (costheta_i / v)
    if (v <= 3.000000157232057e-23) then
        tmp = exp(t_0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(cosTheta_O * Float32(cosTheta_i / v))
	tmp = Float32(0.0)
	if (v <= Float32(3.000000157232057e-23))
		tmp = exp(t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cosTheta_O * (cosTheta_i / v);
	tmp = single(0.0);
	if (v <= single(3.000000157232057e-23))
		tmp = exp(t_0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 3.00000016e-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 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 22.9%

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

   5. Simplified 22.9%

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

   if 3.00000016e-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 6.5%

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

   5. Simplified 6.5%

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

   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. associate-/l* N/A

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

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

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

       3. /-lowering-/.f32 26.3%

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

   11. Simplified 26.3%

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

Alternative 4: 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 99.7%

   \[\leadsto \color{blue}{\frac{e^{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}}{v \cdot 2}} \]
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 99.7%

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

8. Taylor expanded in sinTheta_O around 0

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 5: 99.6% 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 99.7%

   \[\leadsto \color{blue}{\frac{e^{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}}{v \cdot 2}} \]
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 99.7%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. associate--l- N/A

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

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

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

   7. div-inv N/A

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

   8. *-commutative N/A

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

   9. div-inv N/A

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

   10. distribute-rgt-out N/A

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

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

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

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

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

   13. +-commutative N/A

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

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

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

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

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

   16. /-lowering-/.f32 99.3%

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

9. Applied egg-rr 99.3%

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

10. Taylor expanded in sinTheta_i around 0

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

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

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

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

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

    3. /-lowering-/.f32 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

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

Alternative 6: 97.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / exp((single(1.0) / 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. 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. log-div N/A

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

   3. associate-+l- N/A

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

   4. exp-diff N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{e^{0}}{e^{\color{blue}{\log \left(2 \cdot v\right)} - \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   6. 1-exp N/A

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

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

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

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

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. *-lowering-*.f32 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in cosTheta_O around 0

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

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

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

   2. /-lowering-/.f32 99.8%

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

10. Simplified 99.8%

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

11. Taylor expanded in v around 0

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

    1. /-lowering-/.f32 97.5%

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

13. Simplified 97.5%

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

Alternative 7: 20.4% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 13.5%

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

5. Simplified 13.5%

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

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. associate-/l* N/A

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

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

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

    3. /-lowering-/.f32 18.9%

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

11. Simplified 18.9%

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

Alternative 8: 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 13.5%

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

5. Simplified 13.5%

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

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 2024150 
(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))))))