HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.5%

Time: 14.8s

Alternatives: 6

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: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{e^{0.6931}}{v \cdot 2} \cdot e^{\frac{-1}{v}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. exp-sum N/A

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

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

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

   5. log-rec N/A

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

   6. unsub-neg N/A

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

   7. exp-diff N/A

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

   8. rem-exp-log N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

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

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. *-lowering-*.f32 99.7%

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

7. Simplified 99.7%

   \[\leadsto \frac{e^{0.6931}}{v \cdot 2} \cdot e^{\frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i} - 1}{v}} \]

8. Taylor expanded in cosTheta_O around 0

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

   1. metadata-eval N/A

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

   2. distribute-neg-frac N/A

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

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

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

   4. distribute-neg-frac N/A

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

   5. metadata-eval N/A

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

   6. /-lowering-/.f32 99.7%

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

10. Simplified 99.7%

    \[\leadsto \frac{e^{0.6931}}{v \cdot 2} \cdot \color{blue}{e^{\frac{-1}{v}}} \]
11. Add Preprocessing

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in sinTheta_i around 0

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

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

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

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

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

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

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

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

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

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

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

   8. associate-/l* N/A

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

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

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

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

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

   11. sub-neg N/A

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

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

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f32 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in cosTheta_O around 0

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

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. /-lowering-/.f32 99.7%

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

8. Simplified 99.7%

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

   1. exp-sum N/A

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

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

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

   3. rem-exp-log N/A

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

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

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

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

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

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

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

   7. /-lowering-/.f32 99.7%

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

10. Applied egg-rr 99.7%

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

Alternative 3: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in sinTheta_i around 0

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

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

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

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

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

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

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

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

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

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

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

   8. associate-/l* N/A

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

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

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

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

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

   11. sub-neg N/A

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

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

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f32 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in cosTheta_O around 0

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

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. /-lowering-/.f32 99.7%

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

8. Simplified 99.7%

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

   1. exp-sum N/A

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

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

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

   3. rem-exp-log N/A

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

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

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

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

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

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

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

   7. /-lowering-/.f32 99.7%

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

10. Applied egg-rr 99.7%

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

11. Taylor expanded in v around 0

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

    1. /-lowering-/.f32 97.5%

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

13. Simplified 97.5%

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

14. Final simplification 97.5%

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

Alternative 4: 98.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

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

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

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

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

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

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

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

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

   5. *-lowering-*.f32 97.2%

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

5. Simplified 97.2%

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

6. Taylor expanded in cosTheta_O around 0

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

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

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

   2. associate-*r/ N/A

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

   3. mul-1-neg N/A

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

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

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

   5. distribute-neg-in N/A

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

   6. metadata-eval N/A

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

   7. unsub-neg N/A

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

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

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

   9. *-lowering-*.f32 97.2%

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

8. Simplified 97.2%

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

9. Taylor expanded in sinTheta_O around 0

   \[\leadsto \color{blue}{e^{\frac{-1}{v}}} \]
10. Step-by-step derivation

    1. metadata-eval N/A

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

    2. distribute-neg-frac N/A

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

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

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

    4. distribute-neg-frac N/A

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

    5. metadata-eval N/A

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

    6. /-lowering-/.f32 97.2%

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

11. Simplified 97.2%

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

Alternative 5: 38.9% 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 99.7%

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

3. Taylor expanded in cosTheta_i around inf

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

   1. associate-/l* N/A

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

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

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

   3. /-lowering-/.f32 14.0%

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

5. Simplified 14.0%

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

6. Taylor expanded in cosTheta_O around 0

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

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

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

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

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

   3. *-lowering-*.f32 6.3%

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

8. Simplified 6.3%

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

9. Taylor expanded in cosTheta_O around inf

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

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

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

    2. *-lowering-*.f32 35.6%

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

11. Simplified 35.6%

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

Alternative 6: 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 99.7%

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

3. Taylor expanded in cosTheta_i around inf

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

   1. associate-/l* N/A

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

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

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

   3. /-lowering-/.f32 14.0%

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

5. Simplified 14.0%

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

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 2024145 
(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))))))