HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 16.9s

Alternatives: 20

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. *-commutative 99.5%

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

   2. add-log-exp 99.5%

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

   3. exp-to-pow 99.5%

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

   4. +-commutative 99.5%

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

   5. fma-undefine 99.5%

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

4. Applied egg-rr 99.5%

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

   1. log-pow 99.5%

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

   2. expm1-log1p-u 12.1%

      \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

   3. expm1-define 12.1%

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

   4. sub-neg 12.1%

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

   5. metadata-eval 12.1%

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

   6. distribute-rgt-in 12.1%

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

   7. log1p-undefine 12.1%

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

   8. rem-exp-log 99.5%

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

   9. neg-mul-1 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + ((v * (single(1.0) + log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))))) - 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. Step-by-step derivation

   1. *-commutative 99.5%

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

   2. add-log-exp 99.5%

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

   3. exp-to-pow 99.5%

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

   4. +-commutative 99.5%

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

   5. fma-undefine 99.5%

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

4. Applied egg-rr 99.5%

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

   1. log-pow 99.5%

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

   2. expm1-log1p-u 12.1%

      \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

   3. expm1-define 12.1%

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

   4. sub-neg 12.1%

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

   5. metadata-eval 12.1%

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

   6. distribute-rgt-in 12.1%

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

   7. log1p-undefine 12.1%

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

   8. rem-exp-log 99.5%

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

   9. neg-mul-1 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in v around 0 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + v \cdot \log \left(u \cdot \left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(e^{\frac{2}{v}} + -1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (* v (log (* u (- (expm1 (/ -2.0 v)))))))
   (+ (* u (* v (+ (exp (/ 2.0 v)) -1.0))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v * logf((u * -expm1f((-2.0f / v)))));
	} else {
		tmp = (u * (v * (expf((2.0f / v)) + -1.0f))) + -1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(u * Float32(-expm1(Float32(Float32(-2.0) / v)))))));
	else
		tmp = Float32(Float32(u * Float32(v * Float32(exp(Float32(Float32(2.0) / v)) + Float32(-1.0)))) + Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. div-inv 100.0%

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

      2. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in u around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. expm1-define 100.0%

         \[\leadsto 1 + v \cdot \log \left(-u \cdot \color{blue}{\mathsf{expm1}\left(\frac{-2}{v}\right)}\right) \]

      3. distribute-rgt-neg-in 100.0%

         \[\leadsto 1 + v \cdot \log \color{blue}{\left(u \cdot \left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)\right)} \]

   7. Simplified 100.0%

      \[\leadsto 1 + v \cdot \log \color{blue}{\left(u \cdot \left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)\right)} \]

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in u around 0 73.9%

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

   4. Taylor expanded in u around -inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. *-commutative 73.9%

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

      3. distribute-rgt-neg-in 73.9%

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

      4. +-commutative 73.9%

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

      5. mul-1-neg 73.9%

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

      6. unsub-neg 73.9%

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

      7. associate-*r/ 73.9%

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

      8. metadata-eval 73.9%

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

      9. rec-exp 73.9%

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

      10. expm1-define 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right) \cdot \left(-u\right) \]

      11. distribute-neg-frac 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right)\right) \cdot \left(-u\right) \]

      12. metadata-eval 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right)\right) \cdot \left(-u\right) \]

   6. Simplified 73.9%

      \[\leadsto 1 + \color{blue}{\left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) \cdot \left(-u\right)} \]

   7. Taylor expanded in u around 0 75.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + v \cdot \log \left(u \cdot \left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(e^{\frac{2}{v}} + -1\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * (Float32(exp(1)) ^ Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * (single(2.71828182845904523536) ^ (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. Step-by-step derivation

   1. *-un-lft-identity 99.5%

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

   2. exp-prod 99.5%

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

4. Applied egg-rr 99.5%

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

   1. exp-1-e 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 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 6: 91.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \frac{-1}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (- v v))
   (* u (+ (* v (expm1 (/ 2.0 v))) (/ -1.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v - v);
	} else {
		tmp = u * ((v * expm1f((2.0f / v))) + (-1.0f / u));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v - v));
	else
		tmp = Float32(u * Float32(Float32(v * expm1(Float32(Float32(2.0) / v))) + Float32(Float32(-1.0) / u)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-log-exp 99.9%

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

      3. exp-to-pow 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. log-pow 100.0%

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

      2. expm1-log1p-u 8.8%

         \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

      3. expm1-define 8.8%

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

      4. sub-neg 8.8%

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

      5. metadata-eval 8.8%

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

      6. distribute-rgt-in 8.8%

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

      7. log1p-undefine 8.8%

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

      8. rem-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around inf 93.6%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in u around 0 73.9%

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

   4. Taylor expanded in u around -inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. *-commutative 73.9%

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

      3. distribute-rgt-neg-in 73.9%

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

      4. +-commutative 73.9%

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

      5. mul-1-neg 73.9%

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

      6. unsub-neg 73.9%

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

      7. associate-*r/ 73.9%

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

      8. metadata-eval 73.9%

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

      9. rec-exp 73.9%

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

      10. expm1-define 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right) \cdot \left(-u\right) \]

      11. distribute-neg-frac 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right)\right) \cdot \left(-u\right) \]

      12. metadata-eval 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right)\right) \cdot \left(-u\right) \]

   6. Simplified 73.9%

      \[\leadsto 1 + \color{blue}{\left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) \cdot \left(-u\right)} \]

   7. Taylor expanded in u around inf 74.5%

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

      1. expm1-define 74.5%

         \[\leadsto u \cdot \left(v \cdot \color{blue}{\mathsf{expm1}\left(\frac{2}{v}\right)} - \frac{1}{u}\right) \]

   9. Simplified 74.5%

      \[\leadsto \color{blue}{u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) - \frac{1}{u}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \frac{-1}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(e^{\frac{2}{v}} + -1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (- v v))
   (+ (* u (* v (+ (exp (/ 2.0 v)) -1.0))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v - v);
	} else {
		tmp = (u * (v * (expf((2.0f / v)) + -1.0f))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0 + (v - v)
    else
        tmp = (u * (v * (exp((2.0e0 / v)) + (-1.0e0)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v - v));
	else
		tmp = Float32(Float32(u * Float32(v * Float32(exp(Float32(Float32(2.0) / v)) + Float32(-1.0)))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0) + (v - v);
	else
		tmp = (u * (v * (exp((single(2.0) / v)) + single(-1.0)))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-log-exp 99.9%

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

      3. exp-to-pow 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. log-pow 100.0%

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

      2. expm1-log1p-u 8.8%

         \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

      3. expm1-define 8.8%

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

      4. sub-neg 8.8%

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

      5. metadata-eval 8.8%

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

      6. distribute-rgt-in 8.8%

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

      7. log1p-undefine 8.8%

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

      8. rem-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around inf 93.6%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in u around 0 73.9%

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

   4. Taylor expanded in u around -inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. *-commutative 73.9%

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

      3. distribute-rgt-neg-in 73.9%

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

      4. +-commutative 73.9%

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

      5. mul-1-neg 73.9%

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

      6. unsub-neg 73.9%

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

      7. associate-*r/ 73.9%

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

      8. metadata-eval 73.9%

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

      9. rec-exp 73.9%

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

      10. expm1-define 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right) \cdot \left(-u\right) \]

      11. distribute-neg-frac 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right)\right) \cdot \left(-u\right) \]

      12. metadata-eval 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right)\right) \cdot \left(-u\right) \]

   6. Simplified 73.9%

      \[\leadsto 1 + \color{blue}{\left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) \cdot \left(-u\right)} \]

   7. Taylor expanded in u around 0 75.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(e^{\frac{2}{v}} + -1\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (- v v))
   (+ (* u (* v (expm1 (/ 2.0 v)))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v - v);
	} else {
		tmp = (u * (v * expm1f((2.0f / v)))) + -1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v - v));
	else
		tmp = Float32(Float32(u * Float32(v * expm1(Float32(Float32(2.0) / v)))) + Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-log-exp 99.9%

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

      3. exp-to-pow 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. log-pow 100.0%

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

      2. expm1-log1p-u 8.8%

         \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

      3. expm1-define 8.8%

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

      4. sub-neg 8.8%

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

      5. metadata-eval 8.8%

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

      6. distribute-rgt-in 8.8%

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

      7. log1p-undefine 8.8%

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

      8. rem-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around inf 93.6%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in u around 0 73.9%

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

   4. Taylor expanded in u around -inf 74.5%

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

      1. mul-1-neg 74.5%

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

      2. distribute-lft-in 75.0%

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

      3. rgt-mult-inverse 75.0%

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

      4. distribute-neg-in 75.0%

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

      5. mul-1-neg 75.0%

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

      6. distribute-rgt-neg-in 75.0%

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

      7. rec-exp 75.0%

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

      8. expm1-define 75.0%

         \[\leadsto \left(-u \cdot \left(v \cdot \left(-\color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right)\right)\right) + \left(-1\right) \]

      9. distribute-neg-frac 75.0%

         \[\leadsto \left(-u \cdot \left(v \cdot \left(-\mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right)\right)\right)\right) + \left(-1\right) \]

      10. metadata-eval 75.0%

          \[\leadsto \left(-u \cdot \left(v \cdot \left(-\mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right)\right)\right)\right) + \left(-1\right) \]

      11. metadata-eval 75.0%

          \[\leadsto \left(-u \cdot \left(v \cdot \left(-\mathsf{expm1}\left(\frac{2}{v}\right)\right)\right)\right) + \color{blue}{-1} \]

   6. Simplified 75.0%

      \[\leadsto \color{blue}{\left(-u \cdot \left(v \cdot \left(-\mathsf{expm1}\left(\frac{2}{v}\right)\right)\right)\right) + -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + 2 \cdot \frac{-1}{u}\right) + \frac{u \cdot \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v} - u \cdot -2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (- v v))
   (+
    1.0
    (+
     (* u (+ 2.0 (* 2.0 (/ -1.0 u))))
     (/
      (-
       (* u (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) v))
       (* u -2.0))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v - v);
	} else {
		tmp = 1.0f + ((u * (2.0f + (2.0f * (-1.0f / u)))) + (((u * ((1.3333333333333333f + (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0 + (v - v)
    else
        tmp = 1.0e0 + ((u * (2.0e0 + (2.0e0 * ((-1.0e0) / u)))) + (((u * ((1.3333333333333333e0 + (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(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v - v));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(2.0) * Float32(Float32(-1.0) / u)))) + Float32(Float32(Float32(u * Float32(Float32(Float32(1.3333333333333333) + 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(0.10000000149011612))
		tmp = single(1.0) + (v - v);
	else
		tmp = single(1.0) + ((u * (single(2.0) + (single(2.0) * (single(-1.0) / u)))) + (((u * ((single(1.3333333333333333) + (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 < 0.100000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-log-exp 99.9%

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

      3. exp-to-pow 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. log-pow 100.0%

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

      2. expm1-log1p-u 8.8%

         \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

      3. expm1-define 8.8%

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

      4. sub-neg 8.8%

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

      5. metadata-eval 8.8%

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

      6. distribute-rgt-in 8.8%

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

      7. log1p-undefine 8.8%

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

      8. rem-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around inf 93.6%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in u around 0 73.9%

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

   4. Taylor expanded in u around -inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. *-commutative 73.9%

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

      3. distribute-rgt-neg-in 73.9%

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

      4. +-commutative 73.9%

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

      5. mul-1-neg 73.9%

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

      6. unsub-neg 73.9%

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

      7. associate-*r/ 73.9%

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

      8. metadata-eval 73.9%

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

      9. rec-exp 73.9%

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

      10. expm1-define 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right) \cdot \left(-u\right) \]

      11. distribute-neg-frac 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right)\right) \cdot \left(-u\right) \]

      12. metadata-eval 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right)\right) \cdot \left(-u\right) \]

   6. Simplified 73.9%

      \[\leadsto 1 + \color{blue}{\left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) \cdot \left(-u\right)} \]

   7. Taylor expanded in v around -inf 70.8%

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

   8. Taylor expanded in u around 0 70.8%

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

      1. associate-/l* 70.8%

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

      2. associate-*r/ 70.8%

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

      3. metadata-eval 70.8%

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

   10. Simplified 70.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + 2 \cdot \frac{-1}{u}\right) + \frac{u \cdot \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v} - u \cdot -2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} - \left(2 + u \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (- v v))
   (+
    1.0
    (-
     (/
      (-
       (/ (+ (* 0.6666666666666666 (/ u v)) (* u 1.3333333333333333)) v)
       (* u -2.0))
      v)
     (+ 2.0 (* u -2.0))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v - v);
	} else {
		tmp = 1.0f + ((((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / v) - (2.0f + (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0 + (v - v)
    else
        tmp = 1.0e0 + ((((((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v) - (2.0e0 + (u * (-2.0e0))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v - v));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u / v)) + Float32(u * Float32(1.3333333333333333))) / v) - Float32(u * Float32(-2.0))) / v) - Float32(Float32(2.0) + Float32(u * Float32(-2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0) + (v - v);
	else
		tmp = single(1.0) + ((((((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) - (single(2.0) + (u * single(-2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-log-exp 99.9%

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

      3. exp-to-pow 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. log-pow 100.0%

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

      2. expm1-log1p-u 8.8%

         \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

      3. expm1-define 8.8%

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

      4. sub-neg 8.8%

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

      5. metadata-eval 8.8%

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

      6. distribute-rgt-in 8.8%

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

      7. log1p-undefine 8.8%

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

      8. rem-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around inf 93.6%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in u around 0 73.9%

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

   4. Taylor expanded in v around -inf 70.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} - \left(2 + u \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + 2 \cdot \frac{-1}{u}\right) - \frac{u \cdot -2 + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (- v v))
   (+
    1.0
    (-
     (* u (+ 2.0 (* 2.0 (/ -1.0 u))))
     (/ (+ (* u -2.0) (* (/ u v) -1.3333333333333333)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v - v);
	} else {
		tmp = 1.0f + ((u * (2.0f + (2.0f * (-1.0f / u)))) - (((u * -2.0f) + ((u / v) * -1.3333333333333333f)) / 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 <= 0.10000000149011612e0) then
        tmp = 1.0e0 + (v - v)
    else
        tmp = 1.0e0 + ((u * (2.0e0 + (2.0e0 * ((-1.0e0) / u)))) - (((u * (-2.0e0)) + ((u / v) * (-1.3333333333333333e0))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v - v));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(2.0) * Float32(Float32(-1.0) / u)))) - Float32(Float32(Float32(u * Float32(-2.0)) + Float32(Float32(u / v) * Float32(-1.3333333333333333))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0) + (v - v);
	else
		tmp = single(1.0) + ((u * (single(2.0) + (single(2.0) * (single(-1.0) / u)))) - (((u * single(-2.0)) + ((u / v) * single(-1.3333333333333333))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-log-exp 99.9%

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

      3. exp-to-pow 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. log-pow 100.0%

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

      2. expm1-log1p-u 8.8%

         \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

      3. expm1-define 8.8%

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

      4. sub-neg 8.8%

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

      5. metadata-eval 8.8%

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

      6. distribute-rgt-in 8.8%

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

      7. log1p-undefine 8.8%

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

      8. rem-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around inf 93.6%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in u around 0 73.9%

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

   4. Taylor expanded in u around -inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. *-commutative 73.9%

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

      3. distribute-rgt-neg-in 73.9%

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

      4. +-commutative 73.9%

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

      5. mul-1-neg 73.9%

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

      6. unsub-neg 73.9%

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

      7. associate-*r/ 73.9%

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

      8. metadata-eval 73.9%

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

      9. rec-exp 73.9%

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

      10. expm1-define 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right) \cdot \left(-u\right) \]

      11. distribute-neg-frac 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right)\right) \cdot \left(-u\right) \]

      12. metadata-eval 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right)\right) \cdot \left(-u\right) \]

   6. Simplified 73.9%

      \[\leadsto 1 + \color{blue}{\left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) \cdot \left(-u\right)} \]

   7. Taylor expanded in v around -inf 68.6%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + 2 \cdot \frac{-1}{u}\right) - \frac{u \cdot -2 + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(2 + u \cdot -2\right) + \frac{u \cdot -2 + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (- v v))
   (-
    1.0
    (+
     (+ 2.0 (* u -2.0))
     (/ (+ (* u -2.0) (* (/ u v) -1.3333333333333333)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v - v);
	} else {
		tmp = 1.0f - ((2.0f + (u * -2.0f)) + (((u * -2.0f) + ((u / v) * -1.3333333333333333f)) / 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 <= 0.10000000149011612e0) then
        tmp = 1.0e0 + (v - v)
    else
        tmp = 1.0e0 - ((2.0e0 + (u * (-2.0e0))) + (((u * (-2.0e0)) + ((u / v) * (-1.3333333333333333e0))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v - v));
	else
		tmp = Float32(Float32(1.0) - Float32(Float32(Float32(2.0) + Float32(u * Float32(-2.0))) + Float32(Float32(Float32(u * Float32(-2.0)) + Float32(Float32(u / v) * Float32(-1.3333333333333333))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0) + (v - v);
	else
		tmp = single(1.0) - ((single(2.0) + (u * single(-2.0))) + (((u * single(-2.0)) + ((u / v) * single(-1.3333333333333333))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-log-exp 99.9%

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

      3. exp-to-pow 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. log-pow 100.0%

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

      2. expm1-log1p-u 8.8%

         \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

      3. expm1-define 8.8%

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

      4. sub-neg 8.8%

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

      5. metadata-eval 8.8%

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

      6. distribute-rgt-in 8.8%

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

      7. log1p-undefine 8.8%

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

      8. rem-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around inf 93.6%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in u around 0 73.9%

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

   4. Taylor expanded in v around -inf 68.5%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(2 + u \cdot -2\right) + \frac{u \cdot -2 + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;1 - u \cdot \left(-2 + \left(\frac{2}{u} - \frac{2 + \frac{1.3333333333333333}{v}}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (- v v))
   (-
    1.0
    (* u (+ -2.0 (- (/ 2.0 u) (/ (+ 2.0 (/ 1.3333333333333333 v)) v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v - v);
	} else {
		tmp = 1.0f - (u * (-2.0f + ((2.0f / u) - ((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 <= 0.10000000149011612e0) then
        tmp = 1.0e0 + (v - v)
    else
        tmp = 1.0e0 - (u * ((-2.0e0) + ((2.0e0 / u) - ((2.0e0 + (1.3333333333333333e0 / v)) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v - v));
	else
		tmp = Float32(Float32(1.0) - Float32(u * Float32(Float32(-2.0) + Float32(Float32(Float32(2.0) / u) - 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(0.10000000149011612))
		tmp = single(1.0) + (v - v);
	else
		tmp = single(1.0) - (u * (single(-2.0) + ((single(2.0) / u) - ((single(2.0) + (single(1.3333333333333333) / v)) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-log-exp 99.9%

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

      3. exp-to-pow 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. log-pow 100.0%

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

      2. expm1-log1p-u 8.8%

         \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

      3. expm1-define 8.8%

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

      4. sub-neg 8.8%

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

      5. metadata-eval 8.8%

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

      6. distribute-rgt-in 8.8%

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

      7. log1p-undefine 8.8%

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

      8. rem-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around inf 93.6%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in u around 0 73.9%

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

   4. Taylor expanded in u around -inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. *-commutative 73.9%

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

      3. distribute-rgt-neg-in 73.9%

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

      4. +-commutative 73.9%

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

      5. mul-1-neg 73.9%

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

      6. unsub-neg 73.9%

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

      7. associate-*r/ 73.9%

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

      8. metadata-eval 73.9%

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

      9. rec-exp 73.9%

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

      10. expm1-define 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right) \cdot \left(-u\right) \]

      11. distribute-neg-frac 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right)\right) \cdot \left(-u\right) \]

      12. metadata-eval 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right)\right) \cdot \left(-u\right) \]

   6. Simplified 73.9%

      \[\leadsto 1 + \color{blue}{\left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) \cdot \left(-u\right)} \]

   7. Taylor expanded in v around inf 68.2%

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

      1. sub-neg 68.2%

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

      2. +-commutative 68.2%

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

      3. mul-1-neg 68.2%

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

      4. unsub-neg 68.2%

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

      5. associate-*r/ 68.2%

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

      6. metadata-eval 68.2%

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

      7. associate-*r/ 68.2%

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

      8. metadata-eval 68.2%

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

      9. metadata-eval 68.2%

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

   9. Simplified 68.2%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;1 - u \cdot \left(-2 + \left(\frac{2}{u} - \frac{2 + \frac{1.3333333333333333}{v}}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.6% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(\frac{2}{v} + \frac{-1}{u}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (- v v))
   (* u (+ 2.0 (+ (/ 2.0 v) (/ -1.0 u))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v - v);
	} else {
		tmp = u * (2.0f + ((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 <= 0.10000000149011612e0) then
        tmp = 1.0e0 + (v - v)
    else
        tmp = u * (2.0e0 + ((2.0e0 / v) + ((-1.0e0) / u)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v - v));
	else
		tmp = Float32(u * Float32(Float32(2.0) + 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(0.10000000149011612))
		tmp = single(1.0) + (v - v);
	else
		tmp = u * (single(2.0) + ((single(2.0) / v) + (single(-1.0) / u)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-log-exp 99.9%

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

      3. exp-to-pow 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. log-pow 100.0%

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

      2. expm1-log1p-u 8.8%

         \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

      3. expm1-define 8.8%

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

      4. sub-neg 8.8%

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

      5. metadata-eval 8.8%

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

      6. distribute-rgt-in 8.8%

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

      7. log1p-undefine 8.8%

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

      8. rem-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around inf 93.6%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in u around 0 73.9%

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

   4. Taylor expanded in u around -inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. *-commutative 73.9%

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

      3. distribute-rgt-neg-in 73.9%

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

      4. +-commutative 73.9%

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

      5. mul-1-neg 73.9%

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

      6. unsub-neg 73.9%

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

      7. associate-*r/ 73.9%

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

      8. metadata-eval 73.9%

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

      9. rec-exp 73.9%

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

      10. expm1-define 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right) \cdot \left(-u\right) \]

      11. distribute-neg-frac 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right)\right) \cdot \left(-u\right) \]

      12. metadata-eval 73.9%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right)\right) \cdot \left(-u\right) \]

   6. Simplified 73.9%

      \[\leadsto 1 + \color{blue}{\left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) \cdot \left(-u\right)} \]

   7. Taylor expanded in v around inf 64.0%

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

      1. +-commutative 64.0%

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

      2. mul-1-neg 64.0%

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

      3. unsub-neg 64.0%

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

      4. sub-neg 64.0%

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

      5. associate-*r/ 64.0%

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

      6. metadata-eval 64.0%

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

      7. metadata-eval 64.0%

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

   9. Simplified 64.0%

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

   10. Taylor expanded in u around inf 64.3%

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

       1. associate--l+ 64.3%

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

       2. associate-*r/ 64.3%

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

       3. metadata-eval 64.3%

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

   12. Simplified 64.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(\frac{2}{v} + \frac{-1}{u}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (- v v))
   (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v - v);
	} else {
		tmp = -1.0f + (2.0f * (u + (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0 + (v - v)
    else
        tmp = (-1.0e0) + (2.0e0 * (u + (u / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v - v));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(2.0) * Float32(u + Float32(u / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0) + (v - v);
	else
		tmp = single(-1.0) + (single(2.0) * (u + (u / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-log-exp 99.9%

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

      3. exp-to-pow 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. log-pow 100.0%

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

      2. expm1-log1p-u 8.8%

         \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

      3. expm1-define 8.8%

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

      4. sub-neg 8.8%

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

      5. metadata-eval 8.8%

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

      6. distribute-rgt-in 8.8%

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

      7. log1p-undefine 8.8%

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

      8. rem-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around inf 93.6%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in u around 0 73.9%

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

   4. Taylor expanded in v around inf 64.2%

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

      1. sub-neg 64.2%

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

      2. distribute-lft-out 64.2%

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

      3. metadata-eval 64.2%

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

   6. Simplified 64.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 89.9% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) (+ 1.0 (- v v)) (+ 1.0 (* (- 1.0 u) -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v - v);
	} else {
		tmp = 1.0f + ((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 <= 0.10000000149011612e0) then
        tmp = 1.0e0 + (v - v)
    else
        tmp = 1.0e0 + ((1.0e0 - u) * (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v - v));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0) + (v - v);
	else
		tmp = single(1.0) + ((single(1.0) - u) * single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-log-exp 99.9%

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

      3. exp-to-pow 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. log-pow 100.0%

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

      2. expm1-log1p-u 8.8%

         \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

      3. expm1-define 8.8%

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

      4. sub-neg 8.8%

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

      5. metadata-eval 8.8%

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

      6. distribute-rgt-in 8.8%

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

      7. log1p-undefine 8.8%

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

      8. rem-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around inf 93.6%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in v around inf 56.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 14.3% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;2 \cdot \frac{u}{v}\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) (* 2.0 (/ u v)) (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 2.0f * (u / v);
	} 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 <= 0.10000000149011612e0) then
        tmp = 2.0e0 * (u / v)
    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(0.10000000149011612))
		tmp = Float32(Float32(2.0) * Float32(u / v));
	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(0.10000000149011612))
		tmp = single(2.0) * (u / v);
	else
		tmp = single(-1.0) + (u * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   3. Taylor expanded in u around 0 6.3%

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

   4. Taylor expanded in u around -inf 6.3%

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

      1. mul-1-neg 6.3%

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

      2. *-commutative 6.3%

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

      3. distribute-rgt-neg-in 6.3%

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

      4. +-commutative 6.3%

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

      5. mul-1-neg 6.3%

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

      6. unsub-neg 6.3%

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

      7. associate-*r/ 6.3%

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

      8. metadata-eval 6.3%

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

      9. rec-exp 6.3%

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

      10. expm1-define 6.3%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right) \cdot \left(-u\right) \]

      11. distribute-neg-frac 6.3%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right)\right) \cdot \left(-u\right) \]

      12. metadata-eval 6.3%

          \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right)\right) \cdot \left(-u\right) \]

   6. Simplified 6.3%

      \[\leadsto 1 + \color{blue}{\left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) \cdot \left(-u\right)} \]

   7. Taylor expanded in v around inf 10.7%

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

      1. +-commutative 10.7%

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

      2. mul-1-neg 10.7%

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

      3. unsub-neg 10.7%

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

      4. sub-neg 10.7%

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

      5. associate-*r/ 10.7%

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

      6. metadata-eval 10.7%

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

      7. metadata-eval 10.7%

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

   9. Simplified 10.7%

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

   10. Taylor expanded in v around 0 11.4%

       \[\leadsto \color{blue}{2 \cdot \frac{u}{v}} \]

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in u around 0 73.9%

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

   4. Taylor expanded in v around inf 56.1%

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

4. Final simplification 14.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;2 \cdot \frac{u}{v}\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 89.9% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) (+ 1.0 (- v v)) (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v - v);
	} 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 <= 0.10000000149011612e0) then
        tmp = 1.0e0 + (v - v)
    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(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v - v));
	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(0.10000000149011612))
		tmp = single(1.0) + (v - v);
	else
		tmp = single(-1.0) + (u * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-log-exp 99.9%

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

      3. exp-to-pow 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. log-pow 100.0%

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

      2. expm1-log1p-u 8.8%

         \[\leadsto 1 + v \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)} \]

      3. expm1-define 8.8%

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

      4. sub-neg 8.8%

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

      5. metadata-eval 8.8%

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

      6. distribute-rgt-in 8.8%

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

      7. log1p-undefine 8.8%

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

      8. rem-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around inf 93.6%

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

   if 0.100000001 < v

   1. Initial program 92.6%

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

   3. Taylor expanded in u around 0 73.9%

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

   4. Taylor expanded in v around inf 56.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + \left(v - v\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 11.1% accurate, 42.6× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{u}{v} \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (* 2.0 (/ u v)))

C

float code(float u, float v) {
	return 2.0f * (u / v);
}

Fortran

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

Julia

function code(u, v)
	return Float32(Float32(2.0) * Float32(u / v))
end

MATLAB

function tmp = code(u, v)
	tmp = single(2.0) * (u / 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. Taylor expanded in u around 0 10.8%

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

4. Taylor expanded in u around -inf 10.8%

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

   1. mul-1-neg 10.8%

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

   2. *-commutative 10.8%

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

   3. distribute-rgt-neg-in 10.8%

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

   4. +-commutative 10.8%

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

   5. mul-1-neg 10.8%

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

   6. unsub-neg 10.8%

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

   7. associate-*r/ 10.8%

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

   8. metadata-eval 10.8%

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

   9. rec-exp 10.8%

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

   10. expm1-define 10.8%

       \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \color{blue}{\mathsf{expm1}\left(-\frac{-2}{v}\right)}\right) \cdot \left(-u\right) \]

   11. distribute-neg-frac 10.8%

       \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\color{blue}{\frac{--2}{v}}\right)\right) \cdot \left(-u\right) \]

   12. metadata-eval 10.8%

       \[\leadsto 1 + \left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{\color{blue}{2}}{v}\right)\right) \cdot \left(-u\right) \]

6. Simplified 10.8%

   \[\leadsto 1 + \color{blue}{\left(\frac{2}{u} - v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) \cdot \left(-u\right)} \]

7. Taylor expanded in v around inf 14.2%

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

   1. +-commutative 14.2%

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

   2. mul-1-neg 14.2%

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

   3. unsub-neg 14.2%

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

   4. sub-neg 14.2%

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

   5. associate-*r/ 14.2%

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

   6. metadata-eval 14.2%

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

   7. metadata-eval 14.2%

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

9. Simplified 14.2%

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

10. Taylor expanded in v around 0 10.9%

    \[\leadsto \color{blue}{2 \cdot \frac{u}{v}} \]

11. Final simplification 10.9%

    \[\leadsto 2 \cdot \frac{u}{v} \]
12. Add Preprocessing

Alternative 20: 5.9% 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 6.1%

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

4. Final simplification 6.1%

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

Reproduce

herbie shell --seed 2024077 
(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))))))))