HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 21.5s

Alternatives: 21

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(v, \log \left(u + e^{\frac{-2}{v} + \mathsf{log1p}\left(-u\right)}\right), 1\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (fma v (log (+ u (exp (+ (/ -2.0 v) (log1p (- u)))))) 1.0))

C

float code(float u, float v) {
	return fmaf(v, logf((u + expf(((-2.0f / v) + log1pf(-u))))), 1.0f);
}

Julia

function code(u, v)
	return fma(v, log(Float32(u + exp(Float32(Float32(Float32(-2.0) / v) + log1p(Float32(-u)))))), Float32(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. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

   1. add-exp-log 99.6%

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

   2. *-commutative 99.6%

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

   3. log-prod 99.6%

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

   4. add-log-exp 99.6%

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

   5. sub-neg 99.6%

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

   6. log1p-define 99.6%

      \[\leadsto \mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v} + \color{blue}{\mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

8. Applied egg-rr 99.6%

   \[\leadsto \mathsf{fma}\left(v, \log \left(\color{blue}{e^{\frac{-2}{v} + \mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

9. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(v, \log \left(u + e^{\frac{-2}{v} + \mathsf{log1p}\left(-u\right)}\right), 1\right) \]
10. Add Preprocessing

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(u + Float32(Float32(Float32(1.0) - u) * (Float32(exp(1)) ^ Float32(Float32(-2.0) / v))))), Float32(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. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. exp-prod 99.6%

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

8. Applied egg-rr 99.6%

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

   1. exp-1-e 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v))))), Float32(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. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 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 99.5%

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

3. Final simplification 99.5%

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

Alternative 5: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{fma}\left(v, \log u, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(-0.5 \cdot \frac{\frac{\left(u \cdot 16 - u \cdot 8\right) - \frac{u \cdot \left(\frac{8}{v} - \left(9.333333333333334 + \frac{16}{v}\right)\right)}{v}}{v} - u \cdot -4}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (fma v (log u) 1.0)
   (+
    -1.0
    (*
     u
     (+
      (*
       -0.5
       (/
        (-
         (/
          (-
           (- (* u 16.0) (* u 8.0))
           (/ (* u (- (/ 8.0 v) (+ 9.333333333333334 (/ 16.0 v)))) v))
          v)
         (* u -4.0))
        v))
      (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v))))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = fmaf(v, logf(u), 1.0f);
	} else {
		tmp = -1.0f + (u * ((-0.5f * ((((((u * 16.0f) - (u * 8.0f)) - ((u * ((8.0f / v) - (9.333333333333334f + (16.0f / v)))) / v)) / v) - (u * -4.0f)) / v)) + (v * (-1.0f + (1.0f / expf((-2.0f / v)))))));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = fma(v, log(u), Float32(1.0));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(Float32(Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0))) - Float32(Float32(u * Float32(Float32(Float32(8.0) / v) - Float32(Float32(9.333333333333334) + Float32(Float32(16.0) / v)))) / v)) / v) - Float32(u * Float32(-4.0))) / v)) + Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

         \[\leadsto \mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v} + \color{blue}{\mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

   8. Applied egg-rr 99.9%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\color{blue}{e^{\frac{-2}{v} + \mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

   9. Taylor expanded in u around inf 99.6%

      \[\leadsto \mathsf{fma}\left(v, \color{blue}{-1 \cdot \log \left(\frac{1}{u}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. mul-1-neg 99.6%

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

       2. log-rec 99.6%

          \[\leadsto \mathsf{fma}\left(v, -\color{blue}{\left(-\log u\right)}, 1\right) \]

       3. remove-double-neg 99.6%

          \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log u}, 1\right) \]

   11. Simplified 99.6%

       \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log u}, 1\right) \]

   if 0.400000006 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 83.4%

      \[\leadsto \color{blue}{u \cdot \left(-0.5 \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1} \]

   4. Taylor expanded in v around -inf 82.9%

      \[\leadsto u \cdot \left(-0.5 \cdot \color{blue}{\left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{\left(-1 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) + 8 \cdot \frac{u}{v}\right) - \left(4 \cdot \frac{9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)}{v} + \left(8 \cdot \frac{8 \cdot u - 16 \cdot u}{v} + 42.666666666666664 \cdot \frac{u}{v}\right)\right)}{v}}{v}}{v}\right)} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

   5. Taylor expanded in u around 0 82.9%

      \[\leadsto u \cdot \left(-0.5 \cdot \left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{\color{blue}{u \cdot \left(8 \cdot \frac{1}{v} - \left(9.333333333333334 + 16 \cdot \frac{1}{v}\right)\right)}}{v}}{v}}{v}\right) + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
   6. Step-by-step derivation

      1. associate-*r/ 82.9%

         \[\leadsto u \cdot \left(-0.5 \cdot \left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{u \cdot \left(\color{blue}{\frac{8 \cdot 1}{v}} - \left(9.333333333333334 + 16 \cdot \frac{1}{v}\right)\right)}{v}}{v}}{v}\right) + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      2. metadata-eval 82.9%

         \[\leadsto u \cdot \left(-0.5 \cdot \left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{u \cdot \left(\frac{\color{blue}{8}}{v} - \left(9.333333333333334 + 16 \cdot \frac{1}{v}\right)\right)}{v}}{v}}{v}\right) + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      3. associate-*r/ 82.9%

         \[\leadsto u \cdot \left(-0.5 \cdot \left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{u \cdot \left(\frac{8}{v} - \left(9.333333333333334 + \color{blue}{\frac{16 \cdot 1}{v}}\right)\right)}{v}}{v}}{v}\right) + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      4. metadata-eval 82.9%

         \[\leadsto u \cdot \left(-0.5 \cdot \left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{u \cdot \left(\frac{8}{v} - \left(9.333333333333334 + \frac{\color{blue}{16}}{v}\right)\right)}{v}}{v}}{v}\right) + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

   7. Simplified 82.9%

      \[\leadsto u \cdot \left(-0.5 \cdot \left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{\color{blue}{u \cdot \left(\frac{8}{v} - \left(9.333333333333334 + \frac{16}{v}\right)\right)}}{v}}{v}}{v}\right) + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{fma}\left(v, \log u, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(-0.5 \cdot \frac{\frac{\left(u \cdot 16 - u \cdot 8\right) - \frac{u \cdot \left(\frac{8}{v} - \left(9.333333333333334 + \frac{16}{v}\right)\right)}{v}}{v} - u \cdot -4}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(-0.5 \cdot \frac{\frac{\left(u \cdot 16 - u \cdot 8\right) - \frac{u \cdot \left(\frac{8}{v} - \left(9.333333333333334 + \frac{16}{v}\right)\right)}{v}}{v} - u \cdot -4}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (+ 1.0 (* v (log u)))
   (+
    -1.0
    (*
     u
     (+
      (*
       -0.5
       (/
        (-
         (/
          (-
           (- (* u 16.0) (* u 8.0))
           (/ (* u (- (/ 8.0 v) (+ 9.333333333333334 (/ 16.0 v)))) v))
          v)
         (* u -4.0))
        v))
      (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v))))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f + (u * ((-0.5f * ((((((u * 16.0f) - (u * 8.0f)) - ((u * ((8.0f / v) - (9.333333333333334f + (16.0f / v)))) / v)) / v) - (u * -4.0f)) / v)) + (v * (-1.0f + (1.0f / expf((-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 <= 0.4000000059604645e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = (-1.0e0) + (u * (((-0.5e0) * ((((((u * 16.0e0) - (u * 8.0e0)) - ((u * ((8.0e0 / v) - (9.333333333333334e0 + (16.0e0 / v)))) / v)) / v) - (u * (-4.0e0))) / v)) + (v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v)))))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(Float32(Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0))) - Float32(Float32(u * Float32(Float32(Float32(8.0) / v) - Float32(Float32(9.333333333333334) + Float32(Float32(16.0) / v)))) / v)) / v) - Float32(u * Float32(-4.0))) / v)) + Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(-1.0) + (u * ((single(-0.5) * ((((((u * single(16.0)) - (u * single(8.0))) - ((u * ((single(8.0) / v) - (single(9.333333333333334) + (single(16.0) / v)))) / v)) / v) - (u * single(-4.0))) / v)) + (v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

         \[\leadsto \mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v} + \color{blue}{\mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

   8. Applied egg-rr 99.9%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\color{blue}{e^{\frac{-2}{v} + \mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

   9. Taylor expanded in u around inf 99.6%

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

       1. mul-1-neg 99.6%

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

       2. distribute-rgt-neg-in 99.6%

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

       3. log-rec 99.6%

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

       4. remove-double-neg 99.6%

          \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

   11. Simplified 99.6%

       \[\leadsto \color{blue}{1 + v \cdot \log u} \]

   if 0.400000006 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 83.4%

      \[\leadsto \color{blue}{u \cdot \left(-0.5 \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1} \]

   4. Taylor expanded in v around -inf 82.9%

      \[\leadsto u \cdot \left(-0.5 \cdot \color{blue}{\left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{\left(-1 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) + 8 \cdot \frac{u}{v}\right) - \left(4 \cdot \frac{9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)}{v} + \left(8 \cdot \frac{8 \cdot u - 16 \cdot u}{v} + 42.666666666666664 \cdot \frac{u}{v}\right)\right)}{v}}{v}}{v}\right)} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

   5. Taylor expanded in u around 0 82.9%

      \[\leadsto u \cdot \left(-0.5 \cdot \left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{\color{blue}{u \cdot \left(8 \cdot \frac{1}{v} - \left(9.333333333333334 + 16 \cdot \frac{1}{v}\right)\right)}}{v}}{v}}{v}\right) + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
   6. Step-by-step derivation

      1. associate-*r/ 82.9%

         \[\leadsto u \cdot \left(-0.5 \cdot \left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{u \cdot \left(\color{blue}{\frac{8 \cdot 1}{v}} - \left(9.333333333333334 + 16 \cdot \frac{1}{v}\right)\right)}{v}}{v}}{v}\right) + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      2. metadata-eval 82.9%

         \[\leadsto u \cdot \left(-0.5 \cdot \left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{u \cdot \left(\frac{\color{blue}{8}}{v} - \left(9.333333333333334 + 16 \cdot \frac{1}{v}\right)\right)}{v}}{v}}{v}\right) + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      3. associate-*r/ 82.9%

         \[\leadsto u \cdot \left(-0.5 \cdot \left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{u \cdot \left(\frac{8}{v} - \left(9.333333333333334 + \color{blue}{\frac{16 \cdot 1}{v}}\right)\right)}{v}}{v}}{v}\right) + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      4. metadata-eval 82.9%

         \[\leadsto u \cdot \left(-0.5 \cdot \left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{u \cdot \left(\frac{8}{v} - \left(9.333333333333334 + \frac{\color{blue}{16}}{v}\right)\right)}{v}}{v}}{v}\right) + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

   7. Simplified 82.9%

      \[\leadsto u \cdot \left(-0.5 \cdot \left(-1 \cdot \frac{-4 \cdot u + -1 \cdot \frac{-1 \cdot \left(8 \cdot u - 16 \cdot u\right) + -1 \cdot \frac{\color{blue}{u \cdot \left(\frac{8}{v} - \left(9.333333333333334 + \frac{16}{v}\right)\right)}}{v}}{v}}{v}\right) + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(-0.5 \cdot \frac{\frac{\left(u \cdot 16 - u \cdot 8\right) - \frac{u \cdot \left(\frac{8}{v} - \left(9.333333333333334 + \frac{16}{v}\right)\right)}{v}}{v} - u \cdot -4}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - -0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (+ 1.0 (* v (log u)))
   (+
    -1.0
    (*
     u
     (-
      (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v)))))
      (* -0.5 (/ (+ (* u -4.0) (* -8.0 (/ u v))) v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f + (u * ((v * (-1.0f + (1.0f / expf((-2.0f / v))))) - (-0.5f * (((u * -4.0f) + (-8.0f * (u / 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 <= 0.4000000059604645e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = (-1.0e0) + (u * ((v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v))))) - ((-0.5e0) * (((u * (-4.0e0)) + ((-8.0e0) * (u / v))) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))) - Float32(Float32(-0.5) * Float32(Float32(Float32(u * Float32(-4.0)) + Float32(Float32(-8.0) * Float32(u / v))) / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(-1.0) + (u * ((v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v))))) - (single(-0.5) * (((u * single(-4.0)) + (single(-8.0) * (u / v))) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

         \[\leadsto \mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v} + \color{blue}{\mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

   8. Applied egg-rr 99.9%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\color{blue}{e^{\frac{-2}{v} + \mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

   9. Taylor expanded in u around inf 99.6%

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

       1. mul-1-neg 99.6%

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

       2. distribute-rgt-neg-in 99.6%

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

       3. log-rec 99.6%

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

       4. remove-double-neg 99.6%

          \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

   11. Simplified 99.6%

       \[\leadsto \color{blue}{1 + v \cdot \log u} \]

   if 0.400000006 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 83.4%

      \[\leadsto \color{blue}{u \cdot \left(-0.5 \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1} \]

   4. Taylor expanded in v around -inf 80.2%

      \[\leadsto u \cdot \left(-0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-4 \cdot u + 8 \cdot \frac{u}{v}\right) - 16 \cdot \frac{u}{v}}{v}\right)} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
   5. Step-by-step derivation

      1. mul-1-neg 80.2%

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

      2. distribute-neg-frac2 80.2%

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

      3. associate--l+ 80.2%

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

      4. *-commutative 80.2%

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

      5. distribute-rgt-out-- 80.2%

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

      6. metadata-eval 80.2%

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

      7. *-commutative 80.2%

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

   6. Simplified 80.2%

      \[\leadsto u \cdot \left(-0.5 \cdot \color{blue}{\frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{-v}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - -0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{\left(\frac{\left(\frac{-0.6666666666666666}{v} + \left(u \cdot 16 - u \cdot 8\right) \cdot 0.5\right) - 1.3333333333333333}{v} + u \cdot 2\right) - 2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (+ 1.0 (* v (log u)))
   (+
    -1.0
    (*
     u
     (-
      2.0
      (/
       (-
        (+
         (/
          (-
           (+ (/ -0.6666666666666666 v) (* (- (* u 16.0) (* u 8.0)) 0.5))
           1.3333333333333333)
          v)
         (* u 2.0))
        2.0)
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f + (u * (2.0f - (((((((-0.6666666666666666f / v) + (((u * 16.0f) - (u * 8.0f)) * 0.5f)) - 1.3333333333333333f) / 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 <= 0.4000000059604645e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = (-1.0e0) + (u * (2.0e0 - ((((((((-0.6666666666666666e0) / v) + (((u * 16.0e0) - (u * 8.0e0)) * 0.5e0)) - 1.3333333333333333e0) / 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(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) - Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.6666666666666666) / v) + Float32(Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0))) * Float32(0.5))) - Float32(1.3333333333333333)) / 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(0.4000000059604645))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(-1.0) + (u * (single(2.0) - (((((((single(-0.6666666666666666) / v) + (((u * single(16.0)) - (u * single(8.0))) * single(0.5))) - single(1.3333333333333333)) / v) + (u * single(2.0))) - single(2.0)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

         \[\leadsto \mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v} + \color{blue}{\mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

   8. Applied egg-rr 99.9%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\color{blue}{e^{\frac{-2}{v} + \mathsf{log1p}\left(-u\right)}} + u\right), 1\right) \]

   9. Taylor expanded in u around inf 99.6%

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

       1. mul-1-neg 99.6%

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

       2. distribute-rgt-neg-in 99.6%

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

       3. log-rec 99.6%

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

       4. remove-double-neg 99.6%

          \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

   11. Simplified 99.6%

       \[\leadsto \color{blue}{1 + v \cdot \log u} \]

   if 0.400000006 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 83.4%

      \[\leadsto \color{blue}{u \cdot \left(-0.5 \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1} \]

   4. Taylor expanded in v around -inf 76.5%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + \left(-1 \cdot \frac{0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - 0.6666666666666666}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right)} - 1 \]

   5. Taylor expanded in u around 0 77.6%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + \left(-1 \cdot \frac{\color{blue}{-0.6666666666666666}}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{\left(\frac{\left(\frac{-0.6666666666666666}{v} + \left(u \cdot 16 - u \cdot 8\right) \cdot 0.5\right) - 1.3333333333333333}{v} + u \cdot 2\right) - 2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{\left(\frac{\left(\frac{-0.6666666666666666}{v} + \left(u \cdot 16 - u \cdot 8\right) \cdot 0.5\right) - 1.3333333333333333}{v} + u \cdot 2\right) - 2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    -1.0
    (*
     u
     (-
      2.0
      (/
       (-
        (+
         (/
          (-
           (+ (/ -0.6666666666666666 v) (* (- (* u 16.0) (* u 8.0)) 0.5))
           1.3333333333333333)
          v)
         (* u 2.0))
        2.0)
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f - (((((((-0.6666666666666666f / v) + (((u * 16.0f) - (u * 8.0f)) * 0.5f)) - 1.3333333333333333f) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 - ((((((((-0.6666666666666666e0) / v) + (((u * 16.0e0) - (u * 8.0e0)) * 0.5e0)) - 1.3333333333333333e0) / 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(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) - Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.6666666666666666) / v) + Float32(Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0))) * Float32(0.5))) - Float32(1.3333333333333333)) / 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(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) - (((((((single(-0.6666666666666666) / v) + (((u * single(16.0)) - (u * single(8.0))) * single(0.5))) - single(1.3333333333333333)) / v) + (u * single(2.0))) - single(2.0)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.3%

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

   if 0.200000003 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 80.0%

      \[\leadsto \color{blue}{u \cdot \left(-0.5 \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1} \]

   4. Taylor expanded in v around -inf 73.8%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + \left(-1 \cdot \frac{0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - 0.6666666666666666}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right)} - 1 \]

   5. Taylor expanded in u around 0 74.9%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + \left(-1 \cdot \frac{\color{blue}{-0.6666666666666666}}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{\left(\frac{\left(\frac{-0.6666666666666666}{v} + \left(u \cdot 16 - u \cdot 8\right) \cdot 0.5\right) - 1.3333333333333333}{v} + u \cdot 2\right) - 2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{\left(1.3333333333333333 + u \cdot \left(-4 + \frac{-4.666666666666667}{v}\right)\right) + \frac{0.6666666666666666}{v}}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (+
        2.0
        (-
         (/
          (+
           (+ 1.3333333333333333 (* u (+ -4.0 (/ -4.666666666666667 v))))
           (/ 0.6666666666666666 v))
          v)
         (* u 2.0)))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + ((((1.3333333333333333f + (u * (-4.0f + (-4.666666666666667f / v)))) + (0.6666666666666666f / v)) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + ((((1.3333333333333333e0 + (u * ((-4.0e0) + ((-4.666666666666667e0) / v)))) + (0.6666666666666666e0 / v)) / v) - (u * 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(u * Float32(Float32(-4.0) + Float32(Float32(-4.666666666666667) / v)))) + Float32(Float32(0.6666666666666666) / v)) / v) - Float32(u * Float32(2.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + ((((single(1.3333333333333333) + (u * (single(-4.0) + (single(-4.666666666666667) / v)))) + (single(0.6666666666666666) / v)) / v) - (u * single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.3%

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

   if 0.200000003 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 80.0%

      \[\leadsto \color{blue}{u \cdot \left(-0.5 \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1} \]

   4. Taylor expanded in v around -inf 73.8%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + \left(-1 \cdot \frac{0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - 0.6666666666666666}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right)} - 1 \]

   5. Taylor expanded in u around 0 73.8%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\color{blue}{1.3333333333333333 + \left(-1 \cdot \left(u \cdot \left(4 + 4.666666666666667 \cdot \frac{1}{v}\right)\right) + 0.6666666666666666 \cdot \frac{1}{v}\right)}}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]
   6. Step-by-step derivation

      1. associate-+r+ 73.8%

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

      2. mul-1-neg 73.8%

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

      3. distribute-rgt-neg-in 73.8%

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

      4. distribute-neg-in 73.8%

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

      5. metadata-eval 73.8%

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

      6. associate-*r/ 73.8%

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

      7. metadata-eval 73.8%

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

      8. distribute-neg-frac 73.8%

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

      9. metadata-eval 73.8%

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

      10. associate-*r/ 73.8%

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

      11. metadata-eval 73.8%

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

   7. Simplified 73.8%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{\left(1.3333333333333333 + u \cdot \left(-4 + \frac{-4.666666666666667}{v}\right)\right) + \frac{0.6666666666666666}{v}}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{u \cdot -4 + 1.3333333333333333}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/ (+ 2.0 (- (/ (+ (* u -4.0) 1.3333333333333333) v) (* u 2.0))) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + ((((u * -4.0f) + 1.3333333333333333f) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + ((((u * (-4.0e0)) + 1.3333333333333333e0) / v) - (u * 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(Float32(u * Float32(-4.0)) + Float32(1.3333333333333333)) / v) - Float32(u * Float32(2.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + ((((u * single(-4.0)) + single(1.3333333333333333)) / v) - (u * single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.3%

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

   if 0.200000003 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 80.0%

      \[\leadsto \color{blue}{u \cdot \left(-0.5 \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1} \]

   4. Taylor expanded in v around -inf 73.8%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + \left(-1 \cdot \frac{0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - 0.6666666666666666}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right)} - 1 \]

   5. Taylor expanded in v around inf 68.7%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\color{blue}{1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]
   6. Step-by-step derivation

      1. distribute-rgt-out-- 68.7%

         \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + 0.5 \cdot \color{blue}{\left(u \cdot \left(8 - 16\right)\right)}}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]

      2. metadata-eval 68.7%

         \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + 0.5 \cdot \left(u \cdot \color{blue}{-8}\right)}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]

   7. Simplified 68.7%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\color{blue}{1.3333333333333333 + 0.5 \cdot \left(u \cdot -8\right)}}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]

   8. Taylor expanded in u around 0 68.7%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\color{blue}{1.3333333333333333 + -4 \cdot u}}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]
   9. Step-by-step derivation

      1. *-commutative 68.7%

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

   10. Simplified 68.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{u \cdot -4 + 1.3333333333333333}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 - \left(u \cdot 2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (-
        2.0
        (- (* u 2.0) (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) v)))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f - ((u * 2.0f) - ((1.3333333333333333f + (0.6666666666666666f / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 - ((u * 2.0e0) - ((1.3333333333333333e0 + (0.6666666666666666e0 / v)) / v))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) - Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / v))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) - ((u * single(2.0)) - ((single(1.3333333333333333) + (single(0.6666666666666666) / v)) / v))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.3%

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

   if 0.200000003 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 80.0%

      \[\leadsto \color{blue}{u \cdot \left(-0.5 \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1} \]

   4. Taylor expanded in v around -inf 73.8%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + \left(-1 \cdot \frac{0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - 0.6666666666666666}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right)} - 1 \]

   5. Taylor expanded in u around 0 69.7%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\color{blue}{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]
   6. Step-by-step derivation

      1. associate-*r/ 69.7%

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

      2. metadata-eval 69.7%

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

   7. Simplified 69.7%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 - \left(u \cdot 2 - \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{-2 + \frac{\frac{-0.6666666666666666}{v} + -1.3333333333333333}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    -1.0
    (*
     u
     (-
      2.0
      (/
       (+ -2.0 (/ (+ (/ -0.6666666666666666 v) -1.3333333333333333) v))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f - ((-2.0f + (((-0.6666666666666666f / v) + -1.3333333333333333f) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 - (((-2.0e0) + ((((-0.6666666666666666e0) / v) + (-1.3333333333333333e0)) / v)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) - Float32(Float32(Float32(-2.0) + Float32(Float32(Float32(Float32(-0.6666666666666666) / v) + Float32(-1.3333333333333333)) / v)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) - ((single(-2.0) + (((single(-0.6666666666666666) / v) + single(-1.3333333333333333)) / v)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.3%

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

   if 0.200000003 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 80.0%

      \[\leadsto \color{blue}{u \cdot \left(-0.5 \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1} \]

   4. Taylor expanded in v around -inf 73.8%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + \left(-1 \cdot \frac{0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - 0.6666666666666666}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right)} - 1 \]

   5. Taylor expanded in u around 0 64.4%

      \[\leadsto u \cdot \left(2 + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}}\right) - 1 \]
   6. Step-by-step derivation

      1. sub-neg 64.4%

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

      2. associate-*r/ 64.4%

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

      3. distribute-lft-in 64.4%

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

      4. metadata-eval 64.4%

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

      5. neg-mul-1 64.4%

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

      6. associate-*r/ 64.4%

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

      7. metadata-eval 64.4%

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

      8. distribute-neg-frac 64.4%

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

      9. metadata-eval 64.4%

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

      10. metadata-eval 64.4%

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

   7. Simplified 64.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{-2 + \frac{\frac{-0.6666666666666666}{v} + -1.3333333333333333}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    -1.0
    (* u (+ 2.0 (/ (+ 2.0 (- (/ 1.3333333333333333 v) (* u 2.0))) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + ((1.3333333333333333f / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + ((1.3333333333333333e0 / v) - (u * 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) / v) - Float32(u * Float32(2.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + ((single(1.3333333333333333) / v) - (u * single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.3%

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

   if 0.200000003 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 80.0%

      \[\leadsto \color{blue}{u \cdot \left(-0.5 \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1} \]

   4. Taylor expanded in v around -inf 73.8%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + \left(-1 \cdot \frac{0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - 0.6666666666666666}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right)} - 1 \]

   5. Taylor expanded in v around inf 68.7%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\color{blue}{1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]
   6. Step-by-step derivation

      1. distribute-rgt-out-- 68.7%

         \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + 0.5 \cdot \color{blue}{\left(u \cdot \left(8 - 16\right)\right)}}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]

      2. metadata-eval 68.7%

         \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + 0.5 \cdot \left(u \cdot \color{blue}{-8}\right)}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]

   7. Simplified 68.7%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\color{blue}{1.3333333333333333 + 0.5 \cdot \left(u \cdot -8\right)}}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]

   8. Taylor expanded in u around 0 67.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.7% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+ -1.0 (* u (- 2.0 (/ (+ -2.0 (/ -1.3333333333333333 v)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f - ((-2.0f + (-1.3333333333333333f / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 - (((-2.0e0) + ((-1.3333333333333333e0) / v)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) - Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) - ((single(-2.0) + (single(-1.3333333333333333) / v)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.3%

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

   if 0.200000003 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 80.0%

      \[\leadsto \color{blue}{u \cdot \left(-0.5 \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1} \]

   4. Taylor expanded in v around -inf 73.8%

      \[\leadsto u \cdot \color{blue}{\left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + \left(-1 \cdot \frac{0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - 0.6666666666666666}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + 2 \cdot u\right) - 2}{v}\right)} - 1 \]

   5. Taylor expanded in v around inf 68.7%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\color{blue}{1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]
   6. Step-by-step derivation

      1. distribute-rgt-out-- 68.7%

         \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + 0.5 \cdot \color{blue}{\left(u \cdot \left(8 - 16\right)\right)}}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]

      2. metadata-eval 68.7%

         \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{1.3333333333333333 + 0.5 \cdot \left(u \cdot \color{blue}{-8}\right)}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]

   7. Simplified 68.7%

      \[\leadsto u \cdot \left(2 + -1 \cdot \frac{\left(-1 \cdot \frac{\color{blue}{1.3333333333333333 + 0.5 \cdot \left(u \cdot -8\right)}}{v} + 2 \cdot u\right) - 2}{v}\right) - 1 \]

   8. Taylor expanded in u around 0 63.0%

      \[\leadsto u \cdot \left(2 + -1 \cdot \color{blue}{\left(-1 \cdot \frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v}\right)}\right) - 1 \]
   9. Step-by-step derivation

      1. associate-*r/ 63.0%

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

      2. distribute-lft-in 63.0%

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

      3. metadata-eval 63.0%

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

      4. neg-mul-1 63.0%

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

      5. associate-*r/ 63.0%

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

      6. metadata-eval 63.0%

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

      7. distribute-neg-frac 63.0%

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

      8. metadata-eval 63.0%

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

   10. Simplified 63.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 89.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \left(\frac{-2}{v} \cdot \left(1 - u\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ 1.0 (* v (* (/ -2.0 v) (- 1.0 u))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (v * ((-2.0f / v) * (1.0f - u)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (v * (((-2.0e0) / v) * (1.0e0 - u)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(v * Float32(Float32(Float32(-2.0) / v) * Float32(Float32(1.0) - u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (v * ((single(-2.0) / v) * (single(1.0) - u)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.3%

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

   if 0.200000003 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in v around inf 50.8%

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

      1. associate-*r/ 50.8%

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

      2. *-commutative 50.8%

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

      3. associate-/l* 50.8%

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

   5. Simplified 50.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \left(\frac{-2}{v} \cdot \left(1 - u\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 89.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \frac{-2 \cdot \left(1 - u\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ 1.0 (* v (/ (* -2.0 (- 1.0 u)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (v * ((-2.0f * (1.0f - u)) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (v * (((-2.0e0) * (1.0e0 - u)) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(v * Float32(Float32(Float32(-2.0) * Float32(Float32(1.0) - u)) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (v * ((single(-2.0) * (single(1.0) - u)) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.3%

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

   if 0.200000003 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in v around inf 50.8%

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

      1. associate-*r/ 50.8%

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

   5. Simplified 50.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \frac{-2 \cdot \left(1 - u\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 89.8% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ 1.0 (* -2.0 (- 1.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (-2.0f * (1.0f - u));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((-2.0e0) * (1.0e0 - u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(1.0) - u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (single(-2.0) * (single(1.0) - u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.3%

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

   if 0.200000003 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in v around inf 50.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 89.8% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.3%

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

   if 0.200000003 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 80.0%

      \[\leadsto \color{blue}{u \cdot \left(-0.5 \cdot \frac{u \cdot \left(v \cdot {\left(1 + -1 \cdot e^{\frac{-2}{v}}\right)}^{2}\right)}{{\left(e^{\frac{-2}{v}}\right)}^{2}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1} \]

   4. Taylor expanded in v around inf 50.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 6.1% 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 99.5%

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

3. Taylor expanded in u around 0 5.8%

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

4. Final simplification 5.8%

   \[\leadsto -1 \]
5. Add Preprocessing

Alternative 21: 86.5% 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 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in v around 0 86.1%

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

8. Final simplification 86.1%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(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))))))))