HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.2s

Alternatives: 11

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

Derivation

1. Initial program 99.7%

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

Alternative 2: 95.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{\log \left(u + \frac{1 - u}{1 + \frac{\frac{2 + \frac{1.3333333333333333}{v}}{v} - -2}{v}}\right)}{\frac{-1}{v}} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (-
  1.0
  (/
   (log
    (+
     u
     (/
      (- 1.0 u)
      (+ 1.0 (/ (- (/ (+ 2.0 (/ 1.3333333333333333 v)) v) -2.0) v)))))
   (/ -1.0 v))))

C

float code(float u, float v) {
	return 1.0f - (logf((u + ((1.0f - u) / (1.0f + ((((2.0f + (1.3333333333333333f / v)) / v) - -2.0f) / v))))) / (-1.0f / v));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 - (log((u + ((1.0e0 - u) / (1.0e0 + ((((2.0e0 + (1.3333333333333333e0 / v)) / v) - (-2.0e0)) / v))))) / ((-1.0e0) / v))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) - Float32(log(Float32(u + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v) - Float32(-2.0)) / v))))) / Float32(Float32(-1.0) / v)))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) - (log((u + ((single(1.0) - u) / (single(1.0) + ((((single(2.0) + (single(1.3333333333333333) / v)) / v) - single(-2.0)) / v))))) / (single(-1.0) / v));
end

Derivation

1. Initial program 99.7%

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

   1. *-commutative N/A

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

   2. remove-double-neg N/A

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

   3. distribute-rgt-neg-out N/A

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

   4. unsub-neg N/A

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

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

      \[\leadsto \mathsf{\_.f32}\left(1, \color{blue}{\left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\mathsf{neg}\left(v\right)\right)\right)}\right) \]

   6. /-rgt-identity N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\mathsf{neg}\left(v\right)\right)}{\color{blue}{1}}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(v\right)}{1}}\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(v\right)}}}\right)\right) \]

   9. un-div-inv N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}{\color{blue}{\frac{1}{\mathsf{neg}\left(v\right)}}}\right)\right) \]

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

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), \color{blue}{\left(\frac{1}{\mathsf{neg}\left(v\right)}\right)}\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around -inf

   \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \left(1 + \left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \left(1 - \frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \left(\frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} + \left(\mathsf{neg}\left(2\right)\right)\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} + -2\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v}\right), -2\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(\mathsf{neg}\left(\frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v}\right)\right), -2\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(\frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{\mathsf{neg}\left(v\right)}\right), -2\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(\frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{-1 \cdot v}\right), -2\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

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

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(2 + \frac{4}{3} \cdot \frac{1}{v}\right), \left(-1 \cdot v\right)\right), -2\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

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

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \left(\frac{4}{3} \cdot \frac{1}{v}\right)\right), \left(-1 \cdot v\right)\right), -2\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   13. associate-*r/ N/A

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \left(\frac{\frac{4}{3} \cdot 1}{v}\right)\right), \left(-1 \cdot v\right)\right), -2\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \left(\frac{\frac{4}{3}}{v}\right)\right), \left(-1 \cdot v\right)\right), -2\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

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

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{4}{3}, v\right)\right), \left(-1 \cdot v\right)\right), -2\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   16. mul-1-neg N/A

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{4}{3}, v\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), -2\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   17. neg-lowering-neg.f32 97.4%

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{4}{3}, v\right)\right), \mathsf{neg.f32}\left(v\right)\right), -2\right), v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

7. Simplified 97.4%

   \[\leadsto 1 - \frac{\log \left(u + \frac{1 - u}{\color{blue}{1 - \frac{\frac{2 + \frac{1.3333333333333333}{v}}{-v} + -2}{v}}}\right)}{\frac{-1}{v}} \]

8. Final simplification 97.4%

   \[\leadsto 1 - \frac{\log \left(u + \frac{1 - u}{1 + \frac{\frac{2 + \frac{1.3333333333333333}{v}}{v} - -2}{v}}\right)}{\frac{-1}{v}} \]
9. Add Preprocessing

Alternative 3: 93.4% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (-
  1.0
  (/
   (log (+ u (/ (- 1.0 u) (+ 1.0 (+ (/ 2.0 v) (/ 2.0 (* v v)))))))
   (/ -1.0 v))))

C

float code(float u, float v) {
	return 1.0f - (logf((u + ((1.0f - u) / (1.0f + ((2.0f / v) + (2.0f / (v * v))))))) / (-1.0f / v));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 - (log((u + ((1.0e0 - u) / (1.0e0 + ((2.0e0 / v) + (2.0e0 / (v * v))))))) / ((-1.0e0) / v))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) - Float32(log(Float32(u + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + Float32(Float32(Float32(2.0) / v) + Float32(Float32(2.0) / Float32(v * v))))))) / Float32(Float32(-1.0) / v)))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) - (log((u + ((single(1.0) - u) / (single(1.0) + ((single(2.0) / v) + (single(2.0) / (v * v))))))) / (single(-1.0) / v));
end

Derivation

1. Initial program 99.7%

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

   1. *-commutative N/A

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

   2. remove-double-neg N/A

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

   3. distribute-rgt-neg-out N/A

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

   4. unsub-neg N/A

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

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

      \[\leadsto \mathsf{\_.f32}\left(1, \color{blue}{\left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\mathsf{neg}\left(v\right)\right)\right)}\right) \]

   6. /-rgt-identity N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\mathsf{neg}\left(v\right)\right)}{\color{blue}{1}}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(v\right)}{1}}\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(v\right)}}}\right)\right) \]

   9. un-div-inv N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}{\color{blue}{\frac{1}{\mathsf{neg}\left(v\right)}}}\right)\right) \]

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

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), \color{blue}{\left(\frac{1}{\mathsf{neg}\left(v\right)}\right)}\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \color{blue}{\left(1 + \left(2 \cdot \frac{1}{v} + \frac{2}{{v}^{2}}\right)\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \left(2 \cdot \frac{1}{v} + \frac{2}{{v}^{2}}\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left(2 \cdot \frac{1}{v}\right), \left(\frac{2}{{v}^{2}}\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{2 \cdot 1}{v}\right), \left(\frac{2}{{v}^{2}}\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{2}{v}\right), \left(\frac{2}{{v}^{2}}\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \left(\frac{2}{{v}^{2}}\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(2, \left({v}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(2, \left(v \cdot v\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   8. *-lowering-*.f32 96.3%

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(2, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

7. Simplified 96.3%

   \[\leadsto 1 - \frac{\log \left(u + \frac{1 - u}{\color{blue}{1 + \left(\frac{2}{v} + \frac{2}{v \cdot v}\right)}}\right)}{\frac{-1}{v}} \]
8. Add Preprocessing

Alternative 4: 90.8% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (- 1.0 (/ (log (+ u (/ (- 1.0 u) (+ 1.0 (/ 2.0 v))))) (/ -1.0 v))))

C

float code(float u, float v) {
	return 1.0f - (logf((u + ((1.0f - u) / (1.0f + (2.0f / v))))) / (-1.0f / v));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 - (log((u + ((1.0e0 - u) / (1.0e0 + (2.0e0 / v))))) / ((-1.0e0) / v))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) - Float32(log(Float32(u + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + Float32(Float32(2.0) / v))))) / Float32(Float32(-1.0) / v)))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) - (log((u + ((single(1.0) - u) / (single(1.0) + (single(2.0) / v))))) / (single(-1.0) / v));
end

Derivation

1. Initial program 99.7%

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

   1. *-commutative N/A

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

   2. remove-double-neg N/A

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

   3. distribute-rgt-neg-out N/A

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

   4. unsub-neg N/A

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

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

      \[\leadsto \mathsf{\_.f32}\left(1, \color{blue}{\left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\mathsf{neg}\left(v\right)\right)\right)}\right) \]

   6. /-rgt-identity N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\mathsf{neg}\left(v\right)\right)}{\color{blue}{1}}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(v\right)}{1}}\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(v\right)}}}\right)\right) \]

   9. un-div-inv N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \left(\frac{\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}{\color{blue}{\frac{1}{\mathsf{neg}\left(v\right)}}}\right)\right) \]

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

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), \color{blue}{\left(\frac{1}{\mathsf{neg}\left(v\right)}\right)}\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \color{blue}{\left(1 + 2 \cdot \frac{1}{v}\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \left(2 \cdot \frac{1}{v}\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \left(\frac{2 \cdot 1}{v}\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

   3. metadata-eval N/A

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

   4. /-lowering-/.f32 94.4%

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(u, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(2, v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

7. Simplified 94.4%

   \[\leadsto 1 - \frac{\log \left(u + \frac{1 - u}{\color{blue}{1 + \frac{2}{v}}}\right)}{\frac{-1}{v}} \]
8. Add Preprocessing

Alternative 5: 91.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{u \cdot \left(0.6666666666666666 + u \cdot \left(u \cdot 8 + -4.666666666666667\right)\right) + v \cdot \left(u \cdot \left(1.3333333333333333 + u \cdot \left(u \cdot 2.6666666666666665 + -4\right)\right) + v \cdot \left(u \cdot \left(2 + u \cdot -2\right) + v \cdot \left(u \cdot 2\right)\right)\right)}{v \cdot \left(v \cdot v\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    -1.0
    (/
     (+
      (* u (+ 0.6666666666666666 (* u (+ (* u 8.0) -4.666666666666667))))
      (*
       v
       (+
        (* u (+ 1.3333333333333333 (* u (+ (* u 2.6666666666666665) -4.0))))
        (* v (+ (* u (+ 2.0 (* u -2.0))) (* v (* u 2.0)))))))
     (* v (* v v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (((u * (0.6666666666666666f + (u * ((u * 8.0f) + -4.666666666666667f)))) + (v * ((u * (1.3333333333333333f + (u * ((u * 2.6666666666666665f) + -4.0f)))) + (v * ((u * (2.0f + (u * -2.0f))) + (v * (u * 2.0f))))))) / (v * (v * v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (((u * (0.6666666666666666e0 + (u * ((u * 8.0e0) + (-4.666666666666667e0))))) + (v * ((u * (1.3333333333333333e0 + (u * ((u * 2.6666666666666665e0) + (-4.0e0))))) + (v * ((u * (2.0e0 + (u * (-2.0e0)))) + (v * (u * 2.0e0))))))) / (v * (v * v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(Float32(u * Float32(Float32(0.6666666666666666) + Float32(u * Float32(Float32(u * Float32(8.0)) + Float32(-4.666666666666667))))) + Float32(v * Float32(Float32(u * Float32(Float32(1.3333333333333333) + Float32(u * Float32(Float32(u * Float32(2.6666666666666665)) + Float32(-4.0))))) + Float32(v * Float32(Float32(u * Float32(Float32(2.0) + Float32(u * Float32(-2.0)))) + Float32(v * Float32(u * Float32(2.0)))))))) / Float32(v * Float32(v * v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (((u * (single(0.6666666666666666) + (u * ((u * single(8.0)) + single(-4.666666666666667))))) + (v * ((u * (single(1.3333333333333333) + (u * ((u * single(2.6666666666666665)) + single(-4.0))))) + (v * ((u * (single(2.0) + (u * single(-2.0)))) + (v * (u * single(2.0)))))))) / (v * (v * v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.8%

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

   if 0.0500000007 < v

   1. Initial program 92.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 82.2%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(\left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right) + \left(1 - u\right) \cdot 16\right)}{v}}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{3}} + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(u \cdot \left(\frac{8}{3} \cdot \frac{1}{{v}^{2}} + 8 \cdot \frac{1}{{v}^{3}}\right) - \left(2 \cdot \frac{1}{v} + \left(4 \cdot \frac{1}{{v}^{2}} + \frac{14}{3} \cdot \frac{1}{{v}^{3}}\right)\right)\right)\right)\right)\right)\right) - 1} \]

   8. Simplified 82.7%

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(\frac{0.6666666666666666}{v \cdot \left(v \cdot v\right)} + \left(\left(\frac{2}{v} + \frac{1.3333333333333333}{v \cdot v}\right) + u \cdot \left(u \cdot \left(\frac{2.6666666666666665}{v \cdot v} + \frac{8}{v \cdot \left(v \cdot v\right)}\right) - \left(\left(\frac{2}{v} + \frac{4}{v \cdot v}\right) + \frac{4.666666666666667}{v \cdot \left(v \cdot v\right)}\right)\right)\right)\right)\right) + -1} \]

   9. Taylor expanded in v around 0

      \[\leadsto \mathsf{+.f32}\left(\color{blue}{\left(\frac{u \cdot \left(\frac{2}{3} + u \cdot \left(8 \cdot u - \frac{14}{3}\right)\right) + v \cdot \left(u \cdot \left(\frac{4}{3} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)\right) + v \cdot \left(2 \cdot \left(u \cdot v\right) + u \cdot \left(2 + -2 \cdot u\right)\right)\right)}{{v}^{3}}\right)}, -1\right) \]

   11. Simplified 82.7%

       \[\leadsto \color{blue}{\frac{u \cdot \left(0.6666666666666666 + u \cdot \left(u \cdot 8 + -4.666666666666667\right)\right) + v \cdot \left(u \cdot \left(1.3333333333333333 + u \cdot \left(u \cdot 2.6666666666666665 + -4\right)\right) + v \cdot \left(u \cdot \left(2 + u \cdot -2\right) + \left(u \cdot 2\right) \cdot v\right)\right)}{v \cdot \left(v \cdot v\right)}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{u \cdot \left(0.6666666666666666 + u \cdot \left(u \cdot 8 + -4.666666666666667\right)\right) + v \cdot \left(u \cdot \left(1.3333333333333333 + u \cdot \left(u \cdot 2.6666666666666665 + -4\right)\right) + v \cdot \left(u \cdot \left(2 + u \cdot -2\right) + v \cdot \left(u \cdot 2\right)\right)\right)}{v \cdot \left(v \cdot v\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.3% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{\left(\frac{1.3333333333333333 + \left(u \cdot \left(u \cdot 2.6666666666666665 + -4\right) - \frac{u \cdot \left(4.666666666666667 + u \cdot -8\right) + -0.6666666666666666}{v}\right)}{v} - u \cdot 2\right) - -2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (-
        (-
         (/
          (+
           1.3333333333333333
           (-
            (* u (+ (* u 2.6666666666666665) -4.0))
            (/
             (+ (* u (+ 4.666666666666667 (* u -8.0))) -0.6666666666666666)
             v)))
          v)
         (* u 2.0))
        -2.0)
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + (((((1.3333333333333333f + ((u * ((u * 2.6666666666666665f) + -4.0f)) - (((u * (4.666666666666667f + (u * -8.0f))) + -0.6666666666666666f) / v))) / v) - (u * 2.0f)) - -2.0f) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (((((1.3333333333333333e0 + ((u * ((u * 2.6666666666666665e0) + (-4.0e0))) - (((u * (4.666666666666667e0 + (u * (-8.0e0)))) + (-0.6666666666666666e0)) / v))) / v) - (u * 2.0e0)) - (-2.0e0)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(u * Float32(Float32(u * Float32(2.6666666666666665)) + Float32(-4.0))) - Float32(Float32(Float32(u * Float32(Float32(4.666666666666667) + Float32(u * Float32(-8.0)))) + Float32(-0.6666666666666666)) / v))) / v) - Float32(u * Float32(2.0))) - Float32(-2.0)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + (((((single(1.3333333333333333) + ((u * ((u * single(2.6666666666666665)) + single(-4.0))) - (((u * (single(4.666666666666667) + (u * single(-8.0)))) + single(-0.6666666666666666)) / v))) / v) - (u * single(2.0))) - single(-2.0)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.8%

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

   if 0.0500000007 < v

   1. Initial program 92.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 82.2%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(\left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right) + \left(1 - u\right) \cdot 16\right)}{v}}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{3}} + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(u \cdot \left(\frac{8}{3} \cdot \frac{1}{{v}^{2}} + 8 \cdot \frac{1}{{v}^{3}}\right) - \left(2 \cdot \frac{1}{v} + \left(4 \cdot \frac{1}{{v}^{2}} + \frac{14}{3} \cdot \frac{1}{{v}^{3}}\right)\right)\right)\right)\right)\right)\right) - 1} \]

   8. Simplified 82.7%

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(\frac{0.6666666666666666}{v \cdot \left(v \cdot v\right)} + \left(\left(\frac{2}{v} + \frac{1.3333333333333333}{v \cdot v}\right) + u \cdot \left(u \cdot \left(\frac{2.6666666666666665}{v \cdot v} + \frac{8}{v \cdot \left(v \cdot v\right)}\right) - \left(\left(\frac{2}{v} + \frac{4}{v \cdot v}\right) + \frac{4.666666666666667}{v \cdot \left(v \cdot v\right)}\right)\right)\right)\right)\right) + -1} \]

   9. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{4}{3} + \left(-1 \cdot \frac{u \cdot \left(\frac{14}{3} + -8 \cdot u\right) - \frac{2}{3}}{v} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right)}\right), -1\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(2 + \left(\mathsf{neg}\left(\frac{\left(-1 \cdot \frac{\frac{4}{3} + \left(-1 \cdot \frac{u \cdot \left(\frac{14}{3} + -8 \cdot u\right) - \frac{2}{3}}{v} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right)\right)\right)\right), -1\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(2 - \frac{\left(-1 \cdot \frac{\frac{4}{3} + \left(-1 \cdot \frac{u \cdot \left(\frac{14}{3} + -8 \cdot u\right) - \frac{2}{3}}{v} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(2, \left(\frac{\left(-1 \cdot \frac{\frac{4}{3} + \left(-1 \cdot \frac{u \cdot \left(\frac{14}{3} + -8 \cdot u\right) - \frac{2}{3}}{v} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\left(\left(-1 \cdot \frac{\frac{4}{3} + \left(-1 \cdot \frac{u \cdot \left(\frac{14}{3} + -8 \cdot u\right) - \frac{2}{3}}{v} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)\right)}{v} + 2 \cdot u\right) - 2\right), v\right)\right)\right), -1\right) \]

   11. Simplified 82.7%

       \[\leadsto u \cdot \color{blue}{\left(2 - \frac{\left(u \cdot 2 - \frac{1.3333333333333333 + \left(u \cdot \left(u \cdot 2.6666666666666665 + -4\right) - \frac{u \cdot \left(4.666666666666667 + u \cdot -8\right) + -0.6666666666666666}{v}\right)}{v}\right) + -2}{v}\right)} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{\left(\frac{1.3333333333333333 + \left(u \cdot \left(u \cdot 2.6666666666666665 + -4\right) - \frac{u \cdot \left(4.666666666666667 + u \cdot -8\right) + -0.6666666666666666}{v}\right)}{v} - u \cdot 2\right) - -2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(\frac{0.6666666666666666}{v \cdot \left(v \cdot v\right)} + \frac{\left(2 + u \cdot -2\right) + \left(\frac{1.3333333333333333}{v} + \frac{u \cdot \left(u \cdot 2.6666666666666665 + -4\right)}{v}\right)}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (+
       (/ 0.6666666666666666 (* v (* v v)))
       (/
        (+
         (+ 2.0 (* u -2.0))
         (+
          (/ 1.3333333333333333 v)
          (/ (* u (+ (* u 2.6666666666666665) -4.0)) v)))
        v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((0.6666666666666666f / (v * (v * v))) + (((2.0f + (u * -2.0f)) + ((1.3333333333333333f / v) + ((u * ((u * 2.6666666666666665f) + -4.0f)) / v))) / v))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((0.6666666666666666e0 / (v * (v * v))) + (((2.0e0 + (u * (-2.0e0))) + ((1.3333333333333333e0 / v) + ((u * ((u * 2.6666666666666665e0) + (-4.0e0))) / v))) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(0.6666666666666666) / Float32(v * Float32(v * v))) + Float32(Float32(Float32(Float32(2.0) + Float32(u * Float32(-2.0))) + Float32(Float32(Float32(1.3333333333333333) / v) + Float32(Float32(u * Float32(Float32(u * Float32(2.6666666666666665)) + Float32(-4.0))) / v))) / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(0.6666666666666666) / (v * (v * v))) + (((single(2.0) + (u * single(-2.0))) + ((single(1.3333333333333333) / v) + ((u * ((u * single(2.6666666666666665)) + single(-4.0))) / v))) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.8%

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

   if 0.0500000007 < v

   1. Initial program 92.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \left(-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)\right) + \frac{1}{24} \cdot \frac{-96 \cdot {\left(1 - u\right)}^{4} + \left(-64 \cdot {\left(1 - u\right)}^{2} + \left(-48 \cdot {\left(1 - u\right)}^{2} + \left(16 \cdot \left(1 - u\right) + 192 \cdot {\left(1 - u\right)}^{3}\right)\right)\right)}{v}}{v} + \frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v}\right)}\right) \]

   5. Simplified 82.2%

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 - \frac{\left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right) \cdot -0.16666666666666666 + \frac{0.041666666666666664 \cdot \left(\left(-96 \cdot {\left(1 - u\right)}^{4} + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(-112 + \left(1 - u\right) \cdot 192\right)\right) + \left(1 - u\right) \cdot 16\right)}{v}}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(\frac{2}{3} \cdot \frac{1}{{v}^{3}} + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(u \cdot \left(\frac{8}{3} \cdot \frac{1}{{v}^{2}} + 8 \cdot \frac{1}{{v}^{3}}\right) - \left(2 \cdot \frac{1}{v} + \left(4 \cdot \frac{1}{{v}^{2}} + \frac{14}{3} \cdot \frac{1}{{v}^{3}}\right)\right)\right)\right)\right)\right)\right) - 1} \]

   8. Simplified 82.7%

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(\frac{0.6666666666666666}{v \cdot \left(v \cdot v\right)} + \left(\left(\frac{2}{v} + \frac{1.3333333333333333}{v \cdot v}\right) + u \cdot \left(u \cdot \left(\frac{2.6666666666666665}{v \cdot v} + \frac{8}{v \cdot \left(v \cdot v\right)}\right) - \left(\left(\frac{2}{v} + \frac{4}{v \cdot v}\right) + \frac{4.666666666666667}{v \cdot \left(v \cdot v\right)}\right)\right)\right)\right)\right) + -1} \]

   9. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{2}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right), \color{blue}{\left(\frac{2 + \left(-2 \cdot u + \left(\frac{4}{3} \cdot \frac{1}{v} + \frac{u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v}\right)\right)}{v}\right)}\right)\right)\right), -1\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{2}{3}, \mathsf{*.f32}\left(v, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\left(2 + \left(-2 \cdot u + \left(\frac{4}{3} \cdot \frac{1}{v} + \frac{u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v}\right)\right)\right), v\right)\right)\right)\right), -1\right) \]

   11. Simplified 79.6%

       \[\leadsto u \cdot \left(2 + \left(\frac{0.6666666666666666}{v \cdot \left(v \cdot v\right)} + \color{blue}{\frac{\left(2 + u \cdot -2\right) + \left(\frac{1.3333333333333333}{v} + \frac{u \cdot \left(u \cdot 2.6666666666666665 + -4\right)}{v}\right)}{v}}\right)\right) + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(\frac{0.6666666666666666}{v \cdot \left(v \cdot v\right)} + \frac{\left(2 + u \cdot -2\right) + \left(\frac{1.3333333333333333}{v} + \frac{u \cdot \left(u \cdot 2.6666666666666665 + -4\right)}{v}\right)}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{\left(\frac{1.3333333333333333 + u \cdot \left(u \cdot 2.6666666666666665 + -4\right)}{v} - u \cdot 2\right) - -2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (-
        (-
         (/ (+ 1.3333333333333333 (* u (+ (* u 2.6666666666666665) -4.0))) v)
         (* u 2.0))
        -2.0)
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + (((((1.3333333333333333f + (u * ((u * 2.6666666666666665f) + -4.0f))) / v) - (u * 2.0f)) - -2.0f) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (((((1.3333333333333333e0 + (u * ((u * 2.6666666666666665e0) + (-4.0e0)))) / v) - (u * 2.0e0)) - (-2.0e0)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(u * Float32(Float32(u * Float32(2.6666666666666665)) + Float32(-4.0)))) / v) - Float32(u * Float32(2.0))) - Float32(-2.0)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + (((((single(1.3333333333333333) + (u * ((u * single(2.6666666666666665)) + single(-4.0)))) / v) - (u * single(2.0))) - single(-2.0)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.8%

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

   if 0.0500000007 < v

   1. Initial program 92.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   5. Simplified 75.9%

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left(1 - u\right)\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 + \frac{0.16666666666666666 \cdot \left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right)}{v}}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right) - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right) + -1 \]

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

         \[\leadsto \mathsf{+.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)\right), \color{blue}{-1}\right) \]

   8. Simplified 76.5%

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(\frac{1.3333333333333333}{v \cdot v} + \left(\frac{2}{v} + u \cdot \left(\frac{u}{v \cdot v} \cdot 2.6666666666666665 - \left(\frac{2}{v} + \frac{4}{v \cdot v}\right)\right)\right)\right)\right) + -1} \]

   9. Taylor expanded in v around -inf

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{4}{3} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v} + 2 \cdot u\right) - 2}{v}\right)}\right), -1\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(2 + \left(\mathsf{neg}\left(\frac{\left(-1 \cdot \frac{\frac{4}{3} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v} + 2 \cdot u\right) - 2}{v}\right)\right)\right)\right), -1\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \left(2 - \frac{\left(-1 \cdot \frac{\frac{4}{3} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v} + 2 \cdot u\right) - 2}{v}\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(2, \left(\frac{\left(-1 \cdot \frac{\frac{4}{3} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v} + 2 \cdot u\right) - 2}{v}\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\left(\left(-1 \cdot \frac{\frac{4}{3} + u \cdot \left(\frac{8}{3} \cdot u - 4\right)}{v} + 2 \cdot u\right) - 2\right), v\right)\right)\right), -1\right) \]

   11. Simplified 76.5%

       \[\leadsto u \cdot \color{blue}{\left(2 - \frac{\left(u \cdot 2 - \frac{1.3333333333333333 + u \cdot \left(u \cdot 2.6666666666666665 + -4\right)}{v}\right) + -2}{v}\right)} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{\left(\frac{1.3333333333333333 + u \cdot \left(u \cdot 2.6666666666666665 + -4\right)}{v} - u \cdot 2\right) - -2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(\frac{1.3333333333333333}{v \cdot v} + \left(\frac{2}{v} + \frac{u \cdot -2}{v}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (+ (/ 1.3333333333333333 (* v v)) (+ (/ 2.0 v) (/ (* u -2.0) v))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((1.3333333333333333f / (v * v)) + ((2.0f / v) + ((u * -2.0f) / v)))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((1.3333333333333333e0 / (v * v)) + ((2.0e0 / v) + ((u * (-2.0e0)) / v)))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) / Float32(v * v)) + Float32(Float32(Float32(2.0) / v) + Float32(Float32(u * Float32(-2.0)) / v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(1.3333333333333333) / (v * v)) + ((single(2.0) / v) + ((u * single(-2.0)) / v)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 94.8%

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

   if 0.0500000007 < v

   1. Initial program 92.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   5. Simplified 75.9%

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left(1 - u\right)\right) - \frac{\left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) \cdot -0.5 + \frac{0.16666666666666666 \cdot \left(\left(1 - u\right) \cdot 8 + \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \left(\left(1 - u\right) \cdot 16 + -24\right)\right)}{v}}{v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right) - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right) + -1 \]

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

         \[\leadsto \mathsf{+.f32}\left(\left(u \cdot \left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)\right), \color{blue}{-1}\right) \]

   8. Simplified 76.5%

      \[\leadsto \color{blue}{u \cdot \left(2 + \left(\frac{1.3333333333333333}{v \cdot v} + \left(\frac{2}{v} + u \cdot \left(\frac{u}{v \cdot v} \cdot 2.6666666666666665 - \left(\frac{2}{v} + \frac{4}{v \cdot v}\right)\right)\right)\right)\right) + -1} \]

   9. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \color{blue}{\left(-2 \cdot \frac{u}{v}\right)}\right)\right)\right)\right), -1\right) \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \left(\frac{-2 \cdot u}{v}\right)\right)\right)\right)\right), -1\right) \]

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

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\left(-2 \cdot u\right), v\right)\right)\right)\right)\right), -1\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\left(u \cdot -2\right), v\right)\right)\right)\right)\right), -1\right) \]

       4. *-lowering-*.f32 72.3%

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(2, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{4}{3}, \mathsf{*.f32}\left(v, v\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(2, v\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(u, -2\right), v\right)\right)\right)\right)\right), -1\right) \]

   11. Simplified 72.3%

       \[\leadsto u \cdot \left(2 + \left(\frac{1.3333333333333333}{v \cdot v} + \left(\frac{2}{v} + \color{blue}{\frac{u \cdot -2}{v}}\right)\right)\right) + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(\frac{1.3333333333333333}{v \cdot v} + \left(\frac{2}{v} + \frac{u \cdot -2}{v}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.6% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

float code(float u, float v) {
	return 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0
end function

Julia

function code(u, v)
	return Float32(1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

5. Simplified 90.5%

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

Alternative 11: 6.0% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 -1.0)

C

float code(float u, float v) {
	return -1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = -1.0e0
end function

Julia

function code(u, v)
	return Float32(-1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(-1.0);
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in u around 0

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

5. Simplified 5.1%

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

Reproduce

herbie shell --seed 2024159 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0)) (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))