HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 13.3s

Alternatives: 21

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-define 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-define 99.4%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. exp-prod 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + exp(Float32(Float32(Float32(-2.0) / v) + log1p(Float32(-u))))))))
end

Derivation

1. Initial program 99.4%

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

   1. add-exp-log 99.4%

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

   2. *-commutative 99.4%

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

   3. log-prod 99.4%

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

   4. add-log-exp 99.5%

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

   5. sub-neg 99.5%

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

   6. log1p-define 99.5%

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

4. Applied egg-rr 99.5%

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

Alternative 3: 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 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. exp-prod 99.5%

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

4. Applied egg-rr 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. exp-1-e 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-lft-identity 99.4%

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

8. Simplified 99.4%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

Alternative 5: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. add-exp-log 99.4%

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

   2. *-commutative 99.4%

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

   3. log-prod 99.4%

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

   4. add-log-exp 99.5%

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

   5. sub-neg 99.5%

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

   6. log1p-define 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in u around 0 97.5%

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

   1. neg-mul-1 97.5%

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

   2. associate-*r/ 97.5%

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

   3. metadata-eval 97.5%

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

7. Simplified 97.5%

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

8. Final simplification 97.5%

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

Alternative 6: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return 1.0f + (v * logf((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 + exp(((-2.0e0) / v)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u around 0 95.8%

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

Alternative 7: 97.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot {\left(1 - u\right)}^{2}\right) + 0.16666666666666666 \cdot \frac{u \cdot \left(8 - u \cdot \left(24 + u \cdot -16\right)\right)}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (- 1.0 (* v (log (/ 1.0 u))))
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (+
       (* -0.5 (- (* 4.0 (+ u -1.0)) (* -4.0 (pow (- 1.0 u) 2.0))))
       (* 0.16666666666666666 (/ (* u (- 8.0 (* u (+ 24.0 (* u -16.0))))) v)))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f - (v * logf((1.0f / u)));
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((-0.5f * ((4.0f * (u + -1.0f)) - (-4.0f * powf((1.0f - u), 2.0f)))) + (0.16666666666666666f * ((u * (8.0f - (u * (24.0f + (u * -16.0f))))) / v))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.4000000059604645e0) then
        tmp = 1.0e0 - (v * log((1.0e0 / u)))
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((((-0.5e0) * ((4.0e0 * (u + (-1.0e0))) - ((-4.0e0) * ((1.0e0 - u) ** 2.0e0)))) + (0.16666666666666666e0 * ((u * (8.0e0 - (u * (24.0e0 + (u * (-16.0e0)))))) / v))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) - Float32(v * log(Float32(Float32(1.0) / u))));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(4.0) * Float32(u + Float32(-1.0))) - Float32(Float32(-4.0) * (Float32(Float32(1.0) - u) ^ Float32(2.0))))) + Float32(Float32(0.16666666666666666) * Float32(Float32(u * Float32(Float32(8.0) - Float32(u * Float32(Float32(24.0) + Float32(u * Float32(-16.0)))))) / v))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0) - (v * log((single(1.0) / u)));
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (((single(-0.5) * ((single(4.0) * (u + single(-1.0))) - (single(-4.0) * ((single(1.0) - u) ^ single(2.0))))) + (single(0.16666666666666666) * ((u * (single(8.0) - (u * (single(24.0) + (u * single(-16.0)))))) / v))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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. add-exp-log 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-log-exp 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in u around inf 99.6%

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

   if 0.400000006 < v

   1. Initial program 91.8%

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

      1. +-commutative 91.8%

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

      2. fma-define 91.8%

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

      3. +-commutative 91.8%

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

      4. fma-define 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in v around -inf 77.7%

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0 77.7%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{\color{blue}{u \cdot \left(u \cdot \left(24 + -16 \cdot u\right) - 8\right)}}{v}}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot {\left(1 - u\right)}^{2}\right) + 0.16666666666666666 \cdot \frac{u \cdot \left(8 - u \cdot \left(24 + u \cdot -16\right)\right)}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{0.16666666666666666 \cdot \frac{u \cdot \left(8 - u \cdot 24\right)}{v} - -0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + \left(1 - u\right) \cdot 4\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (- 1.0 (* v (log (/ 1.0 u))))
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (-
       (* 0.16666666666666666 (/ (* u (- 8.0 (* u 24.0))) v))
       (* -0.5 (+ (* -4.0 (pow (- 1.0 u) 2.0)) (* (- 1.0 u) 4.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f - (v * logf((1.0f / u)));
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((0.16666666666666666f * ((u * (8.0f - (u * 24.0f))) / v)) - (-0.5f * ((-4.0f * powf((1.0f - u), 2.0f)) + ((1.0f - u) * 4.0f)))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.4000000059604645e0) then
        tmp = 1.0e0 - (v * log((1.0e0 / u)))
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + (((0.16666666666666666e0 * ((u * (8.0e0 - (u * 24.0e0))) / v)) - ((-0.5e0) * (((-4.0e0) * ((1.0e0 - u) ** 2.0e0)) + ((1.0e0 - u) * 4.0e0)))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) - Float32(v * log(Float32(Float32(1.0) / u))));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(0.16666666666666666) * Float32(Float32(u * Float32(Float32(8.0) - Float32(u * Float32(24.0)))) / v)) - Float32(Float32(-0.5) * Float32(Float32(Float32(-4.0) * (Float32(Float32(1.0) - u) ^ Float32(2.0))) + Float32(Float32(Float32(1.0) - u) * Float32(4.0))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0) - (v * log((single(1.0) / u)));
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (((single(0.16666666666666666) * ((u * (single(8.0) - (u * single(24.0)))) / v)) - (single(-0.5) * ((single(-4.0) * ((single(1.0) - u) ^ single(2.0))) + ((single(1.0) - u) * single(4.0))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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. add-exp-log 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-log-exp 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in u around inf 99.6%

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

   if 0.400000006 < v

   1. Initial program 91.8%

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

      1. +-commutative 91.8%

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

      2. fma-define 91.8%

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

      3. +-commutative 91.8%

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

      4. fma-define 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in v around -inf 77.7%

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0 75.4%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{\color{blue}{u \cdot \left(24 \cdot u - 8\right)}}{v}}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{0.16666666666666666 \cdot \frac{u \cdot \left(8 - u \cdot 24\right)}{v} - -0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + \left(1 - u\right) \cdot 4\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot {\left(1 - u\right)}^{2}\right) - 0.16666666666666666 \cdot \frac{u \cdot -8}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (- 1.0 (* v (log (/ 1.0 u))))
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (-
       (* -0.5 (- (* 4.0 (+ u -1.0)) (* -4.0 (pow (- 1.0 u) 2.0))))
       (* 0.16666666666666666 (/ (* u -8.0) v)))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f - (v * logf((1.0f / u)));
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((-0.5f * ((4.0f * (u + -1.0f)) - (-4.0f * powf((1.0f - u), 2.0f)))) - (0.16666666666666666f * ((u * -8.0f) / v))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.4000000059604645e0) then
        tmp = 1.0e0 - (v * log((1.0e0 / u)))
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((((-0.5e0) * ((4.0e0 * (u + (-1.0e0))) - ((-4.0e0) * ((1.0e0 - u) ** 2.0e0)))) - (0.16666666666666666e0 * ((u * (-8.0e0)) / v))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) - Float32(v * log(Float32(Float32(1.0) / u))));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(4.0) * Float32(u + Float32(-1.0))) - Float32(Float32(-4.0) * (Float32(Float32(1.0) - u) ^ Float32(2.0))))) - Float32(Float32(0.16666666666666666) * Float32(Float32(u * Float32(-8.0)) / v))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0) - (v * log((single(1.0) / u)));
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (((single(-0.5) * ((single(4.0) * (u + single(-1.0))) - (single(-4.0) * ((single(1.0) - u) ^ single(2.0))))) - (single(0.16666666666666666) * ((u * single(-8.0)) / v))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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. add-exp-log 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-log-exp 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in u around inf 99.6%

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

   if 0.400000006 < v

   1. Initial program 91.8%

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

      1. +-commutative 91.8%

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

      2. fma-define 91.8%

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

      3. +-commutative 91.8%

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

      4. fma-define 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in v around -inf 77.7%

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0 74.3%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{\color{blue}{-8 \cdot u}}{v}}{v}\right) \]
   7. Step-by-step derivation

      1. *-commutative 74.3%

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

   8. Simplified 74.3%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{\color{blue}{u \cdot -8}}{v}}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot {\left(1 - u\right)}^{2}\right) - 0.16666666666666666 \cdot \frac{u \cdot -8}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.4000000059604645e0) then
        tmp = 1.0e0 - (v * log((1.0e0 / u)))
    else
        tmp = (-1.0e0) + (u * (v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v))))))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 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. add-exp-log 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-log-exp 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in u around inf 99.6%

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

   if 0.400000006 < v

   1. Initial program 91.8%

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

      1. +-commutative 91.8%

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

      2. fma-define 91.8%

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

      3. +-commutative 91.8%

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

      4. fma-define 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in u around 0 72.9%

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

4. Final simplification 97.9%

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

Alternative 11: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (- 1.0 (* v (log (/ 1.0 u))))
   (+ 1.0 (- (* u (* v (expm1 (/ 2.0 v)))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f - (v * logf((1.0f / u)));
	} else {
		tmp = 1.0f + ((u * (v * expm1f((2.0f / v)))) - 2.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) - Float32(v * log(Float32(Float32(1.0) / u))));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(v * expm1(Float32(Float32(2.0) / v)))) - Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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. add-exp-log 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-log-exp 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in u around inf 99.6%

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

   if 0.400000006 < v

   1. Initial program 91.8%

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

   3. Taylor expanded in u around 0 72.6%

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

      1. pow1 72.6%

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

      2. rec-exp 72.6%

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

      3. expm1-define 72.6%

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

   5. Applied egg-rr 72.6%

      \[\leadsto 1 + \left(u \cdot \color{blue}{{\left(v \cdot \mathsf{expm1}\left(-\frac{-2}{v}\right)\right)}^{1}} - 2\right) \]
   6. Step-by-step derivation

      1. unpow1 72.6%

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

      2. expm1-undefine 72.6%

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

      3. sub-neg 72.6%

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

      4. metadata-eval 72.6%

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

      5. metadata-eval 72.6%

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

      6. sub-neg 72.6%

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

      7. expm1-undefine 72.6%

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

      8. distribute-neg-frac 72.6%

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

      9. metadata-eval 72.6%

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

      10. metadata-eval 72.6%

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

      11. associate-*r/ 72.6%

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

      12. remove-double-neg 72.6%

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

      13. expm1-define 72.6%

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

      14. rec-exp 72.6%

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

      15. rec-exp 72.6%

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

      16. expm1-define 72.6%

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

      17. remove-double-neg 72.6%

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

      18. associate-*r/ 72.6%

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

      19. metadata-eval 72.6%

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

   7. Simplified 72.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) - Float32(v * log(Float32(Float32(1.0) / u))));
	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(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0) - (v * log((single(1.0) / u)));
	else
		tmp = single(-1.0) + ((((((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) + (u * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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. add-exp-log 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-log-exp 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in u around inf 99.6%

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

   if 0.400000006 < v

   1. Initial program 91.8%

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

   3. Taylor expanded in u around 0 72.6%

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

   4. Taylor expanded in v around -inf 71.4%

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

4. Final simplification 97.8%

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

Alternative 13: 97.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u / v)) + Float32(u * Float32(1.3333333333333333))) / v) - Float32(u * Float32(-2.0))) / v) + Float32(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(-1.0) + ((((((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) + (u * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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. add-exp-log 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-log-exp 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in u around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. distribute-rgt-neg-in 99.6%

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

      3. log-rec 99.6%

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

      4. remove-double-neg 99.6%

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

   7. Simplified 99.6%

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

   if 0.400000006 < v

   1. Initial program 91.8%

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

   3. Taylor expanded in u around 0 72.6%

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

   4. Taylor expanded in v around -inf 71.4%

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

4. Final simplification 97.8%

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

Alternative 14: 90.8% accurate, 7.6× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / v) + (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 <= 0.4000000059604645e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((((((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v) + (u * 2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	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(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((((((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) + (u * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.8%

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

   if 0.400000006 < v

   1. Initial program 91.8%

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

   3. Taylor expanded in u around 0 72.6%

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

   4. Taylor expanded in v around -inf 71.4%

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

4. Final simplification 91.4%

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

Alternative 15: 90.8% accurate, 8.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f + ((2.0f + ((1.3333333333333333f + (0.6666666666666666f * (1.0f / v))) / v)) / v))) - 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 <= 0.4000000059604645e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 + ((2.0e0 + ((1.3333333333333333e0 + (0.6666666666666666e0 * (1.0e0 / v))) / v)) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) * Float32(Float32(1.0) / v))) / v)) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) + ((single(2.0) + ((single(1.3333333333333333) + (single(0.6666666666666666) * (single(1.0) / v))) / v)) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.8%

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

   if 0.400000006 < v

   1. Initial program 91.8%

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

   3. Taylor expanded in u around 0 72.6%

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

   4. Taylor expanded in v around -inf 71.0%

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

4. Final simplification 91.4%

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

Alternative 16: 90.8% accurate, 9.7× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   1.0
   (+ 1.0 (- (+ (* u 2.0) (/ (* u (+ 2.0 (/ 1.3333333333333333 v))) v)) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((u * 2.0f) + ((u * (2.0f + (1.3333333333333333f / v))) / v)) - 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 <= 0.4000000059604645e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((u * 2.0e0) + ((u * (2.0e0 + (1.3333333333333333e0 / v))) / v)) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(u * Float32(2.0)) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v))) / v)) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((u * single(2.0)) + ((u * (single(2.0) + (single(1.3333333333333333) / v))) / v)) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.8%

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

   if 0.400000006 < v

   1. Initial program 91.8%

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

   3. Taylor expanded in u around 0 72.6%

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

   4. Taylor expanded in v around -inf 70.2%

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

   5. Taylor expanded in u around 0 70.2%

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

      1. associate-*r/ 70.2%

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

      2. metadata-eval 70.2%

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

   7. Simplified 70.2%

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

4. Final simplification 91.3%

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

Alternative 17: 90.8% accurate, 10.6× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   1.0
   (* u (- (+ (/ (+ 2.0 (/ 1.3333333333333333 v)) v) (/ -1.0 u)) -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = u * ((((2.0f + (1.3333333333333333f / v)) / v) + (-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 <= 0.4000000059604645e0) then
        tmp = 1.0e0
    else
        tmp = u * ((((2.0e0 + (1.3333333333333333e0 / v)) / v) + ((-1.0e0) / u)) - (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(u * Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v) + 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(0.4000000059604645))
		tmp = single(1.0);
	else
		tmp = u * ((((single(2.0) + (single(1.3333333333333333) / v)) / v) + (single(-1.0) / u)) - single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.8%

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

   if 0.400000006 < v

   1. Initial program 91.8%

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

   3. Taylor expanded in u around 0 72.6%

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

   4. Taylor expanded in v around -inf 70.2%

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

   5. Taylor expanded in u around -inf 69.9%

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

      1. mul-1-neg 69.9%

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

      2. *-commutative 69.9%

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

      3. distribute-rgt-neg-in 69.9%

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

      4. sub-neg 69.9%

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

      5. +-commutative 69.9%

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

      6. mul-1-neg 69.9%

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

      7. unsub-neg 69.9%

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

      8. associate-*r/ 69.9%

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

      9. metadata-eval 69.9%

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

      10. metadata-eval 69.9%

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

   7. Simplified 69.9%

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

4. Final simplification 91.3%

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

Alternative 18: 90.6% accurate, 15.2× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645) 1.0 (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} 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 <= 0.4000000059604645e0) then
        tmp = 1.0e0
    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(0.4000000059604645))
		tmp = Float32(1.0);
	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(0.4000000059604645))
		tmp = single(1.0);
	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 < 0.400000006

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.8%

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

   if 0.400000006 < v

   1. Initial program 91.8%

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

   3. Taylor expanded in u around 0 72.6%

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

   4. Taylor expanded in v around inf 64.0%

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

      1. sub-neg 64.0%

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

      2. distribute-lft-out 64.0%

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

      3. metadata-eval 64.0%

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

   6. Simplified 64.0%

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

4. Final simplification 90.9%

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

Alternative 19: 90.0% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.4000000059604645e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.8%

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

   if 0.400000006 < v

   1. Initial program 91.8%

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

   3. Taylor expanded in u around 0 72.6%

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

   4. Taylor expanded in v around inf 53.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 87.2% 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-define 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-define 99.4%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. exp-prod 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in v around 0 87.0%

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

Alternative 21: 5.7% 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-define 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-define 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u around 0 5.8%

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

Reproduce

herbie shell --seed 2024136 
(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))))))))