HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 17.9s

Alternatives: 13

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. Step-by-step derivation

   1. exp-1-e 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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. fma-undefine 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. exp-prod 99.5%

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

8. Applied egg-rr 99.5%

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

   1. exp-1-e 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 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. exp-1-e 99.5%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 5: 95.9% 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. 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. exp-1-e 99.5%

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

6. Simplified 99.4%

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

7. Taylor expanded in u around 0 95.8%

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

   1. log-E 95.8%

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

   2. associate-*r/ 95.8%

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

   3. metadata-eval 95.8%

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

9. Simplified 95.8%

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

10. Final simplification 95.8%

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

Alternative 6: 90.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (v * (u * ((((2.0f + (1.3333333333333333f / v)) / powf(v, 2.0f)) - ((2.0f / u) / v)) - (-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
    else
        tmp = 1.0e0 + (v * (u * ((((2.0e0 + (1.3333333333333333e0 / v)) / (v ** 2.0e0)) - ((2.0e0 / u) / v)) - ((-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(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(v * Float32(u * Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / (v ^ Float32(2.0))) - Float32(Float32(Float32(2.0) / u) / v)) - Float32(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);
	else
		tmp = single(1.0) + (v * (u * ((((single(2.0) + (single(1.3333333333333333) / v)) / (v ^ single(2.0))) - ((single(2.0) / u) / v)) - (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. Taylor expanded in v around -inf -0.0%

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

      1. mul-1-neg -0.0%

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

      2. unsub-neg -0.0%

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

      3. mul-1-neg -0.0%

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

      4. unsub-neg -0.0%

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

      5. sub-neg -0.0%

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

      6. associate-*r/ -0.0%

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

      7. metadata-eval -0.0%

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

      8. distribute-neg-frac -0.0%

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

      9. metadata-eval -0.0%

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

   5. Simplified -0.0%

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

   6. Taylor expanded in v around 0 93.2%

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

   if 0.100000001 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 56.6%

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

   4. Taylor expanded in v around -inf 58.0%

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

   5. Taylor expanded in u around -inf 57.9%

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

      1. mul-1-neg 57.9%

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

      2. *-commutative 57.9%

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

      3. distribute-rgt-neg-in 57.9%

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

   7. Simplified 58.2%

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

4. Final simplification 90.6%

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

Alternative 7: 90.6% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) - Float32(v * Float32(Float32(Float32(2.0) + Float32(Float32(u * Float32(-2.0)) - Float32(Float32(Float32(Float32(1.3333333333333333) * Float32(u / v)) + Float32(u * Float32(2.0))) / 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);
	else
		tmp = single(1.0) - (v * ((single(2.0) + ((u * single(-2.0)) - (((single(1.3333333333333333) * (u / v)) + (u * single(2.0))) / 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. Taylor expanded in v around -inf -0.0%

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

      1. mul-1-neg -0.0%

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

      2. unsub-neg -0.0%

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

      3. mul-1-neg -0.0%

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

      4. unsub-neg -0.0%

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

      5. sub-neg -0.0%

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

      6. associate-*r/ -0.0%

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

      7. metadata-eval -0.0%

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

      8. distribute-neg-frac -0.0%

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

      9. metadata-eval -0.0%

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

   5. Simplified -0.0%

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

   6. Taylor expanded in v around 0 93.2%

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

   if 0.100000001 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 56.6%

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

   4. Taylor expanded in v around -inf 58.0%

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

   5. Taylor expanded in v around -inf 58.2%

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

4. Final simplification 90.6%

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

Alternative 8: 90.6% accurate, 8.9× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (-
    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;
	} 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
    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(1.0);
	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);
	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. Taylor expanded in v around -inf -0.0%

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

      1. mul-1-neg -0.0%

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

      2. unsub-neg -0.0%

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

      3. mul-1-neg -0.0%

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

      4. unsub-neg -0.0%

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

      5. sub-neg -0.0%

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

      6. associate-*r/ -0.0%

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

      7. metadata-eval -0.0%

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

      8. distribute-neg-frac -0.0%

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

      9. metadata-eval -0.0%

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

   5. Simplified -0.0%

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

   6. Taylor expanded in v around 0 93.2%

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

   if 0.100000001 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 56.6%

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

   4. Taylor expanded in v around -inf 58.1%

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

4. Final simplification 90.6%

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

Alternative 9: 90.7% accurate, 10.6× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f - (((1.3333333333333333f * (-1.0f / v)) - 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
    else
        tmp = (-1.0e0) + (u * (2.0e0 - (((1.3333333333333333e0 * ((-1.0e0) / v)) - 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(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) - Float32(Float32(Float32(Float32(1.3333333333333333) * Float32(Float32(-1.0) / v)) - 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);
	else
		tmp = single(-1.0) + (u * (single(2.0) - (((single(1.3333333333333333) * (single(-1.0) / v)) - 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. Taylor expanded in v around -inf -0.0%

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

      1. mul-1-neg -0.0%

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

      2. unsub-neg -0.0%

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

      3. mul-1-neg -0.0%

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

      4. unsub-neg -0.0%

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

      5. sub-neg -0.0%

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

      6. associate-*r/ -0.0%

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

      7. metadata-eval -0.0%

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

      8. distribute-neg-frac -0.0%

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

      9. metadata-eval -0.0%

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

   5. Simplified -0.0%

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

   6. Taylor expanded in v around 0 93.2%

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

   if 0.100000001 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 56.6%

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

   4. Taylor expanded in v around -inf 58.0%

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

   5. Taylor expanded in u around 0 58.1%

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

4. Final simplification 90.6%

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

Alternative 10: 90.7% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (1.3333333333333333e0 / v)) / v)))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   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 v around -inf -0.0%

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

      1. mul-1-neg -0.0%

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

      2. unsub-neg -0.0%

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

      3. mul-1-neg -0.0%

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

      4. unsub-neg -0.0%

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

      5. sub-neg -0.0%

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

      6. associate-*r/ -0.0%

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

      7. metadata-eval -0.0%

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

      8. distribute-neg-frac -0.0%

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

      9. metadata-eval -0.0%

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

   5. Simplified -0.0%

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

   6. Taylor expanded in v around 0 93.2%

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

   if 0.100000001 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 56.6%

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

   4. Taylor expanded in v around -inf 58.0%

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

   5. Taylor expanded in u around 0 58.1%

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

      1. *-un-lft-identity 58.1%

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

      2. un-div-inv 58.1%

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

   7. Applied egg-rr 58.1%

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

      1. *-lft-identity 58.1%

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

   9. Simplified 58.1%

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

4. Final simplification 90.6%

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

Alternative 11: 90.4% accurate, 15.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		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 <= 0.10000000149011612e0) 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(0.10000000149011612))
		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(0.10000000149011612))
		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 < 0.100000001

   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 v around -inf -0.0%

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

      1. mul-1-neg -0.0%

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

      2. unsub-neg -0.0%

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

      3. mul-1-neg -0.0%

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

      4. unsub-neg -0.0%

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

      5. sub-neg -0.0%

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

      6. associate-*r/ -0.0%

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

      7. metadata-eval -0.0%

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

      8. distribute-neg-frac -0.0%

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

      9. metadata-eval -0.0%

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

   5. Simplified -0.0%

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

   6. Taylor expanded in v around 0 93.2%

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

   if 0.100000001 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 56.6%

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

   4. Taylor expanded in v around inf 53.8%

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

      1. sub-neg 53.8%

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

      2. distribute-lft-out 53.8%

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

      3. metadata-eval 53.8%

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

   6. Simplified 53.8%

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

4. Final simplification 90.3%

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

Alternative 12: 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 99.4%

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

3. Taylor expanded in u around 0 5.5%

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

4. Final simplification 5.5%

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

Alternative 13: 86.7% 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. Add Preprocessing

3. Taylor expanded in v around -inf 3.2%

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

   1. mul-1-neg 3.2%

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

   2. unsub-neg 3.2%

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

   3. mul-1-neg 3.2%

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

   4. unsub-neg 3.2%

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

   5. sub-neg 3.2%

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

   6. associate-*r/ 3.2%

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

   7. metadata-eval 3.2%

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

   8. distribute-neg-frac 3.2%

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

   9. metadata-eval 3.2%

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

5. Simplified 3.2%

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

6. Taylor expanded in v around 0 86.9%

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

7. Final simplification 86.9%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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))))))))