HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 10.2s

Alternatives: 16

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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. exp-1-e N/A

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

   6. lower-E.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 4: 94.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 95.6%

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

8. Final simplification 95.6%

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

Alternative 5: 93.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. +-commutative N/A

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

   7. lower-+.f32 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r/ N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f32 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f32 N/A

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

   20. associate-*r/ N/A

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

   21. metadata-eval N/A

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

   22. lower-/.f32 94.3

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

7. Applied rewrites 94.3%

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

8. Final simplification 94.3%

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

Alternative 6: 90.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 92.4

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

7. Applied rewrites 92.4%

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

8. Final simplification 92.4%

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

Alternative 7: 90.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{\frac{2}{v}}{u} + \frac{2}{u \cdot u}\right) + \frac{2}{v}\right) - \left(\frac{2}{u} + \frac{\frac{4}{v}}{u}\right)\right) \cdot \left(u \cdot u\right)}{-v} \cdot v + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.44999998807907104)
   1.0
   (+
    (*
     (/
      (*
       (-
        (+ (+ (/ (/ 2.0 v) u) (/ 2.0 (* u u))) (/ 2.0 v))
        (+ (/ 2.0 u) (/ (/ 4.0 v) u)))
       (* u u))
      (- v))
     v)
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.44999998807907104f) {
		tmp = 1.0f;
	} else {
		tmp = ((((((((2.0f / v) / u) + (2.0f / (u * u))) + (2.0f / v)) - ((2.0f / u) + ((4.0f / v) / u))) * (u * u)) / -v) * v) + 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.44999998807907104e0) then
        tmp = 1.0e0
    else
        tmp = ((((((((2.0e0 / v) / u) + (2.0e0 / (u * u))) + (2.0e0 / v)) - ((2.0e0 / u) + ((4.0e0 / v) / u))) * (u * u)) / -v) * v) + 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.44999998807907104))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(2.0) / v) / u) + Float32(Float32(2.0) / Float32(u * u))) + Float32(Float32(2.0) / v)) - Float32(Float32(Float32(2.0) / u) + Float32(Float32(Float32(4.0) / v) / u))) * Float32(u * u)) / Float32(-v)) * v) + Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.44999998807907104))
		tmp = single(1.0);
	else
		tmp = ((((((((single(2.0) / v) / u) + (single(2.0) / (u * u))) + (single(2.0) / v)) - ((single(2.0) / u) + ((single(4.0) / v) / u))) * (u * u)) / -v) * v) + single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.449999988

   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. Applied rewrites 95.5%

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

   if 0.449999988 < v

   1. Initial program 93.4%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 58.5%

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

   10. Applied rewrites 58.5%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{\frac{2}{v}}{u} + \frac{2}{u \cdot u}\right) + \frac{2}{v}\right) - \left(\frac{2}{u} + \frac{\frac{4}{v}}{u}\right)\right) \cdot \left(u \cdot u\right)}{-v} \cdot v + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{\frac{2}{u}}{u} - \frac{2}{u}\right) - \frac{\frac{2}{u} - 2}{v}\right) \cdot \left(u \cdot u\right)}{-v} \cdot v + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.44999998807907104)
   1.0
   (+
    (*
     (/
      (* (- (- (/ (/ 2.0 u) u) (/ 2.0 u)) (/ (- (/ 2.0 u) 2.0) v)) (* u u))
      (- v))
     v)
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.44999998807907104f) {
		tmp = 1.0f;
	} else {
		tmp = (((((((2.0f / u) / u) - (2.0f / u)) - (((2.0f / u) - 2.0f) / v)) * (u * u)) / -v) * v) + 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.44999998807907104e0) then
        tmp = 1.0e0
    else
        tmp = (((((((2.0e0 / u) / u) - (2.0e0 / u)) - (((2.0e0 / u) - 2.0e0) / v)) * (u * u)) / -v) * v) + 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.44999998807907104))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(2.0) / u) / u) - Float32(Float32(2.0) / u)) - Float32(Float32(Float32(Float32(2.0) / u) - Float32(2.0)) / v)) * Float32(u * u)) / Float32(-v)) * v) + Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.44999998807907104))
		tmp = single(1.0);
	else
		tmp = (((((((single(2.0) / u) / u) - (single(2.0) / u)) - (((single(2.0) / u) - single(2.0)) / v)) * (u * u)) / -v) * v) + single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.449999988

   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. Applied rewrites 95.5%

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

   if 0.449999988 < v

   1. Initial program 93.4%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 58.5%

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

   10. Applied rewrites 51.4%

       \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(\frac{1}{u}, \frac{2}{v}, \frac{\frac{2}{u}}{u} + \left(\frac{2}{v} - \frac{1}{u} \cdot \left(\frac{4}{v} + 2\right)\right)\right) \cdot \left(u \cdot u\right)}{-v} \]

   11. Taylor expanded in v around inf

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

   13. Applied rewrites 58.5%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{\frac{2}{u}}{u} - \frac{2}{u}\right) - \frac{\frac{2}{u} - 2}{v}\right) \cdot \left(u \cdot u\right)}{-v} \cdot v + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{2}{v} - \frac{\left(2 - \frac{-2}{v}\right) - \frac{2}{u}}{u}\right) \cdot u\right) \cdot u}{-v} \cdot v + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.44999998807907104)
   1.0
   (+
    (*
     (/ (* (* (- (/ 2.0 v) (/ (- (- 2.0 (/ -2.0 v)) (/ 2.0 u)) u)) u) u) (- v))
     v)
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.44999998807907104f) {
		tmp = 1.0f;
	} else {
		tmp = ((((((2.0f / v) - (((2.0f - (-2.0f / v)) - (2.0f / u)) / u)) * u) * u) / -v) * v) + 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.44999998807907104e0) then
        tmp = 1.0e0
    else
        tmp = ((((((2.0e0 / v) - (((2.0e0 - ((-2.0e0) / v)) - (2.0e0 / u)) / u)) * u) * u) / -v) * v) + 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.44999998807907104))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(2.0) / v) - Float32(Float32(Float32(Float32(2.0) - Float32(Float32(-2.0) / v)) - Float32(Float32(2.0) / u)) / u)) * u) * u) / Float32(-v)) * v) + Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.44999998807907104))
		tmp = single(1.0);
	else
		tmp = ((((((single(2.0) / v) - (((single(2.0) - (single(-2.0) / v)) - (single(2.0) / u)) / u)) * u) * u) / -v) * v) + single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.449999988

   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. Applied rewrites 95.5%

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

   if 0.449999988 < v

   1. Initial program 93.4%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in u around inf

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(2, 1 - u, \frac{\mathsf{fma}\left({u}^{2}, 2, \left(1 - u\right) \cdot -2\right)}{v}\right)}{-v} \]
   7. Step-by-step derivation

   8. Applied rewrites 7.3%

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(2, 1 - u, \frac{\mathsf{fma}\left(u \cdot u, 2, \left(1 - u\right) \cdot -2\right)}{v}\right)}{-v} \]

   9. Taylor expanded in u around -inf

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

   11. Applied rewrites 58.5%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{2}{v} - \frac{\left(2 - \frac{-2}{v}\right) - \frac{2}{u}}{u}\right) \cdot u\right) \cdot u}{-v} \cdot v + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u - 1\right) \cdot 2 - \frac{\left(\left(2 - \frac{2}{u}\right) \cdot u\right) \cdot u}{v}}{v} \cdot v + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.44999998807907104)
   1.0
   (+
    (* (/ (- (* (- u 1.0) 2.0) (/ (* (* (- 2.0 (/ 2.0 u)) u) u) v)) v) v)
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.44999998807907104f) {
		tmp = 1.0f;
	} else {
		tmp = (((((u - 1.0f) * 2.0f) - ((((2.0f - (2.0f / u)) * u) * u) / v)) / v) * v) + 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.44999998807907104e0) then
        tmp = 1.0e0
    else
        tmp = (((((u - 1.0e0) * 2.0e0) - ((((2.0e0 - (2.0e0 / u)) * u) * u) / v)) / v) * v) + 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.44999998807907104))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(u - Float32(1.0)) * Float32(2.0)) - Float32(Float32(Float32(Float32(Float32(2.0) - Float32(Float32(2.0) / u)) * u) * u) / v)) / v) * v) + Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.44999998807907104))
		tmp = single(1.0);
	else
		tmp = (((((u - single(1.0)) * single(2.0)) - ((((single(2.0) - (single(2.0) / u)) * u) * u) / v)) / v) * v) + single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.449999988

   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. Applied rewrites 95.5%

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

   if 0.449999988 < v

   1. Initial program 93.4%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in u around inf

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(2, 1 - u, \frac{{u}^{2} \cdot \left(2 - 2 \cdot \frac{1}{u}\right)}{v}\right)}{-v} \]
   7. Step-by-step derivation

   8. Applied rewrites 8.5%

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(2, 1 - u, \frac{\left(\left(2 - \frac{2}{u}\right) \cdot u\right) \cdot u}{v}\right)}{-v} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.1%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u - 1\right) \cdot 2 - \frac{\left(\left(2 - \frac{2}{u}\right) \cdot u\right) \cdot u}{v}}{v} \cdot v + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 - u \cdot u}{\left(\frac{v}{u} + v\right) \cdot u} \cdot -2\right) \cdot v + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.44999998807907104)
   1.0
   (+ (* (* (/ (- 1.0 (* u u)) (* (+ (/ v u) v) u)) -2.0) v) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.44999998807907104f) {
		tmp = 1.0f;
	} else {
		tmp = ((((1.0f - (u * u)) / (((v / u) + v) * u)) * -2.0f) * v) + 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.44999998807907104e0) then
        tmp = 1.0e0
    else
        tmp = ((((1.0e0 - (u * u)) / (((v / u) + v) * u)) * (-2.0e0)) * v) + 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.44999998807907104))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(1.0) - Float32(u * u)) / Float32(Float32(Float32(v / u) + v) * u)) * Float32(-2.0)) * v) + Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.44999998807907104))
		tmp = single(1.0);
	else
		tmp = ((((single(1.0) - (u * u)) / (((v / u) + v) * u)) * single(-2.0)) * v) + single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.449999988

   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. Applied rewrites 95.5%

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

   if 0.449999988 < v

   1. Initial program 93.4%

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

   3. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 50.2

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 50.3%

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

   8. Taylor expanded in u around inf

      \[\leadsto 1 + v \cdot \left(\frac{1 - u \cdot u}{u \cdot \left(v + \frac{v}{u}\right)} \cdot -2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 50.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 - u \cdot u}{\left(\frac{v}{u} + v\right) \cdot u} \cdot -2\right) \cdot v + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 - u \cdot u}{\left(u - -1\right) \cdot v} \cdot -2\right) \cdot v + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.44999998807907104)
   1.0
   (+ (* (* (/ (- 1.0 (* u u)) (* (- u -1.0) v)) -2.0) v) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.44999998807907104f) {
		tmp = 1.0f;
	} else {
		tmp = ((((1.0f - (u * u)) / ((u - -1.0f) * v)) * -2.0f) * v) + 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.44999998807907104e0) then
        tmp = 1.0e0
    else
        tmp = ((((1.0e0 - (u * u)) / ((u - (-1.0e0)) * v)) * (-2.0e0)) * v) + 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.44999998807907104))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(1.0) - Float32(u * u)) / Float32(Float32(u - Float32(-1.0)) * v)) * Float32(-2.0)) * v) + Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.44999998807907104))
		tmp = single(1.0);
	else
		tmp = ((((single(1.0) - (u * u)) / ((u - single(-1.0)) * v)) * single(-2.0)) * v) + single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.449999988

   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. Applied rewrites 95.5%

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

   if 0.449999988 < v

   1. Initial program 93.4%

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

   3. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 50.2

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 50.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 - u \cdot u}{\left(u - -1\right) \cdot v} \cdot -2\right) \cdot v + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{v - u \cdot v}{v \cdot v} \cdot -2\right) \cdot v + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.44999998807907104)
   1.0
   (+ (* (* (/ (- v (* u v)) (* v v)) -2.0) v) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.44999998807907104f) {
		tmp = 1.0f;
	} else {
		tmp = ((((v - (u * v)) / (v * v)) * -2.0f) * v) + 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.44999998807907104e0) then
        tmp = 1.0e0
    else
        tmp = ((((v - (u * v)) / (v * v)) * (-2.0e0)) * v) + 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.44999998807907104))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(v - Float32(u * v)) / Float32(v * v)) * Float32(-2.0)) * v) + Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.44999998807907104))
		tmp = single(1.0);
	else
		tmp = ((((v - (u * v)) / (v * v)) * single(-2.0)) * v) + single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.449999988

   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. Applied rewrites 95.5%

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

   if 0.449999988 < v

   1. Initial program 93.4%

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

   3. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 50.2

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 50.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{v - u \cdot v}{v \cdot v} \cdot -2\right) \cdot v + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.4% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot u + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.44999998807907104) 1.0 (+ (* 2.0 u) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.44999998807907104f) {
		tmp = 1.0f;
	} else {
		tmp = (2.0f * u) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.44999998807907104e0) then
        tmp = 1.0e0
    else
        tmp = (2.0e0 * u) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.44999998807907104))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(2.0) * u) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.44999998807907104))
		tmp = single(1.0);
	else
		tmp = (single(2.0) * u) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.449999988

   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. Applied rewrites 95.5%

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

   if 0.449999988 < v

   1. Initial program 93.4%

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

   4. Applied rewrites 4.9%

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

   5. Taylor expanded in v around inf

      \[\leadsto \color{blue}{1 + -2 \cdot \left(1 - u\right)} \]
   6. Step-by-step derivation

      1. sub-neg N/A

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

      2. neg-mul-1 N/A

         \[\leadsto 1 + -2 \cdot \left(1 + \color{blue}{-1 \cdot u}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto 1 + \color{blue}{\left(1 \cdot -2 + \left(-1 \cdot u\right) \cdot -2\right)} \]

      4. metadata-eval N/A

         \[\leadsto 1 + \left(\color{blue}{-2} + \left(-1 \cdot u\right) \cdot -2\right) \]

      5. metadata-eval N/A

         \[\leadsto 1 + \left(\color{blue}{-1 \cdot 2} + \left(-1 \cdot u\right) \cdot -2\right) \]

      6. associate-*r* N/A

         \[\leadsto 1 + \left(-1 \cdot 2 + \color{blue}{-1 \cdot \left(u \cdot -2\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto 1 + \left(-1 \cdot 2 + -1 \cdot \color{blue}{\left(-2 \cdot u\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto 1 + \color{blue}{-1 \cdot \left(2 + -2 \cdot u\right)} \]

      9. distribute-rgt-in N/A

         \[\leadsto 1 + \color{blue}{\left(2 \cdot -1 + \left(-2 \cdot u\right) \cdot -1\right)} \]

      10. metadata-eval N/A

          \[\leadsto 1 + \left(\color{blue}{-2} + \left(-2 \cdot u\right) \cdot -1\right) \]

      11. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(1 + -2\right) + \left(-2 \cdot u\right) \cdot -1} \]

      12. metadata-eval N/A

          \[\leadsto \color{blue}{-1} + \left(-2 \cdot u\right) \cdot -1 \]

      13. *-commutative N/A

          \[\leadsto -1 + \color{blue}{\left(u \cdot -2\right)} \cdot -1 \]

      14. associate-*l* N/A

          \[\leadsto -1 + \color{blue}{u \cdot \left(-2 \cdot -1\right)} \]

      15. metadata-eval N/A

          \[\leadsto -1 + u \cdot \color{blue}{2} \]

      16. *-commutative N/A

          \[\leadsto -1 + \color{blue}{2 \cdot u} \]

      17. lower-+.f32 N/A

          \[\leadsto \color{blue}{-1 + 2 \cdot u} \]

      18. *-commutative N/A

          \[\leadsto -1 + \color{blue}{u \cdot 2} \]

      19. lower-*.f32 50.3

          \[\leadsto -1 + \color{blue}{u \cdot 2} \]

   7. Applied rewrites 50.3%

      \[\leadsto \color{blue}{-1 + u \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot u + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 86.5% accurate, 231.0× 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 99.5%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 89.0%

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

Alternative 16: 5.9% accurate, 231.0× 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 99.5%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 5.8%

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

Reproduce

herbie shell --seed 2024267 
(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))))))))