HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 16.6s

Alternatives: 10

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right) \cdot \sqrt[3]{v \cdot v}, \sqrt[3]{v}, 1\right) \end{array} \]

FPCore

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

C

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

Julia

function code(u, v)
	return fma(Float32(log(fma(Float32(Float32(1.0) - u), exp(Float32(Float32(-2.0) / v)), u)) * cbrt(Float32(v * v))), cbrt(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. *-commutative 99.4%

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

   3. add-cube-cbrt 99.3%

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

   4. associate-*r* 99.4%

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

   5. fma-def 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\sqrt[3]{v} \cdot \sqrt[3]{v}\right), \sqrt[3]{v}, 1\right)} \]

   6. +-commutative 99.4%

      \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}} + u\right)} \cdot \left(\sqrt[3]{v} \cdot \sqrt[3]{v}\right), \sqrt[3]{v}, 1\right) \]

   7. fma-udef 99.3%

      \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)} \cdot \left(\sqrt[3]{v} \cdot \sqrt[3]{v}\right), \sqrt[3]{v}, 1\right) \]

   8. cbrt-unprod 99.5%

      \[\leadsto \mathsf{fma}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right) \cdot \color{blue}{\sqrt[3]{v \cdot v}}, \sqrt[3]{v}, 1\right) \]

3. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right) \cdot \sqrt[3]{v \cdot v}, \sqrt[3]{v}, 1\right)} \]

4. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(\log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right) \cdot \sqrt[3]{v \cdot v}, \sqrt[3]{v}, 1\right) \]

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(fma(Float32(Float32(1.0) - u), exp(Float32(Float32(-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. Taylor expanded in v around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. fma-def 99.4%

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

4. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 3: 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. Final simplification 99.4%

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

Alternative 4: 91.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot \left(\frac{2}{v} + \left(2 + \frac{1.3333333333333333}{v \cdot v}\right)\right) - \left(u \cdot u\right) \cdot \left(\frac{2}{v} + \frac{4}{v \cdot v}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    (-
     (* u (+ (/ 2.0 v) (+ 2.0 (/ 1.3333333333333333 (* v v)))))
     (* (* u u) (+ (/ 2.0 v) (/ 4.0 (* v v)))))
    -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = ((u * ((2.0f / v) + (2.0f + (1.3333333333333333f / (v * v))))) - ((u * u) * ((2.0f / v) + (4.0f / (v * v))))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = ((u * ((2.0e0 / v) + (2.0e0 + (1.3333333333333333e0 / (v * v))))) - ((u * u) * ((2.0e0 / v) + (4.0e0 / (v * v))))) + (-1.0e0)
    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(Float32(u * Float32(Float32(Float32(2.0) / v) + Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / Float32(v * v))))) - Float32(Float32(u * u) * Float32(Float32(Float32(2.0) / v) + Float32(Float32(4.0) / Float32(v * v))))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = ((u * ((single(2.0) / v) + (single(2.0) + (single(1.3333333333333333) / (v * v))))) - ((u * u) * ((single(2.0) / v) + (single(4.0) / (v * v))))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-udef 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-udef 99.9%

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

   3. Applied egg-rr 99.9%

      \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left({\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{v} - 1\right)} \]
   4. Step-by-step derivation

      1. pow-to-exp 99.9%

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

      2. expm1-def 100.0%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.0%

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

   if 0.100000001 < v

   1. Initial program 93.7%

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

   2. Taylor expanded in u around 0 79.5%

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

      1. sub-neg 79.5%

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

   4. Simplified 79.5%

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

   5. Taylor expanded in v around -inf 70.5%

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

      1. fma-def 70.5%

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

      2. distribute-rgt-out-- 70.5%

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

      3. metadata-eval 70.5%

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

      4. unpow2 70.5%

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

      5. times-frac 70.5%

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

      6. unpow2 70.5%

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

      7. +-commutative 70.5%

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

      8. mul-1-neg 70.5%

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

      9. unsub-neg 70.5%

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

   7. Simplified 70.5%

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

   8. Taylor expanded in u around 0 70.5%

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

      1. +-commutative 70.5%

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

      2. mul-1-neg 70.5%

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

      3. unsub-neg 70.5%

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

      4. *-commutative 70.5%

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

      5. associate-+r+ 70.5%

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

      6. +-commutative 70.5%

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

      7. associate-*r/ 70.5%

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

      8. metadata-eval 70.5%

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

      9. associate-*r/ 70.5%

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

      10. metadata-eval 70.5%

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

      11. unpow2 70.5%

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

   10. Simplified 70.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot \left(\frac{2}{v} + \left(2 + \frac{1.3333333333333333}{v \cdot v}\right)\right) - \left(u \cdot u\right) \cdot \left(\frac{2}{v} + \frac{4}{v \cdot v}\right)\right) + -1\\ \end{array} \]

Alternative 5: 90.7% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (u * ((2.0e0 / v) + (2.0e0 + (1.3333333333333333e0 / (v * v))))) + (-1.0e0)
    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(u * Float32(Float32(Float32(2.0) / v) + Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / Float32(v * v))))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-udef 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-udef 99.9%

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

   3. Applied egg-rr 99.9%

      \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left({\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{v} - 1\right)} \]
   4. Step-by-step derivation

      1. pow-to-exp 99.9%

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

      2. expm1-def 100.0%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.0%

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

   if 0.100000001 < v

   1. Initial program 93.7%

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

   2. Taylor expanded in u around 0 79.5%

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

      1. sub-neg 79.5%

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

   4. Simplified 79.5%

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

   5. Taylor expanded in v around -inf 70.5%

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

      1. fma-def 70.5%

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

      2. distribute-rgt-out-- 70.5%

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

      3. metadata-eval 70.5%

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

      4. unpow2 70.5%

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

      5. times-frac 70.5%

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

      6. unpow2 70.5%

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

      7. +-commutative 70.5%

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

      8. mul-1-neg 70.5%

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

      9. unsub-neg 70.5%

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

   7. Simplified 70.5%

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

   8. Taylor expanded in u around 0 65.6%

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

      1. *-commutative 65.6%

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

      2. associate-+r+ 65.6%

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

      3. +-commutative 65.6%

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

      4. associate-*r/ 65.6%

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

      5. metadata-eval 65.6%

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

      6. associate-*r/ 65.6%

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

      7. metadata-eval 65.6%

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

      8. unpow2 65.6%

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

   10. Simplified 65.6%

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

4. Final simplification 91.4%

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

Alternative 6: 90.5% accurate, 14.1× speedup

Math

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

FPCore

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

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 * (1.0f / 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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 + (2.0e0 * (1.0e0 / v)))) - 2.0e0)
    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(u * Float32(Float32(2.0) + Float32(Float32(2.0) * Float32(Float32(1.0) / v)))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-udef 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-udef 99.9%

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

   3. Applied egg-rr 99.9%

      \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left({\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{v} - 1\right)} \]
   4. Step-by-step derivation

      1. pow-to-exp 99.9%

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

      2. expm1-def 100.0%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.0%

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

   if 0.100000001 < v

   1. Initial program 93.7%

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

   2. Taylor expanded in v around inf 62.6%

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

   3. Taylor expanded in u around 0 63.5%

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

4. Final simplification 91.2%

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

Alternative 7: 90.5% accurate, 16.2× speedup

Math

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

FPCore

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

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))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((2.0e0 * (u + (u / v))) - 2.0e0)
    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(u / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-udef 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-udef 99.9%

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

   3. Applied egg-rr 99.9%

      \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left({\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{v} - 1\right)} \]
   4. Step-by-step derivation

      1. pow-to-exp 99.9%

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

      2. expm1-def 100.0%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 94.0%

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

   if 0.100000001 < v

   1. Initial program 93.7%

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

   2. Taylor expanded in u around 0 70.1%

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

   3. Taylor expanded in v around inf 63.5%

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

      1. distribute-lft-out 63.5%

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

   5. Simplified 63.5%

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

4. Final simplification 91.2%

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

Alternative 8: 89.9% accurate, 29.9× 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. log1p-expm1-u 99.9%

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

      2. expm1-udef 99.8%

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

      3. *-commutative 99.8%

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

      4. exp-to-pow 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-udef 99.8%

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

   3. Applied egg-rr 99.8%

      \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left({\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{v} - 1\right)} \]
   4. Step-by-step derivation

      1. pow-to-exp 99.8%

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

      2. expm1-def 99.9%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 93.0%

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

   if 0.400000006 < v

   1. Initial program 93.2%

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

   2. Taylor expanded in v around inf 58.6%

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

      1. *-commutative 58.6%

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

   4. Simplified 58.6%

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

   5. Taylor expanded in u around 0 58.6%

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

4. Final simplification 90.3%

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

Alternative 9: 89.3% accurate, 68.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.4000000059604645e0) then
        tmp = 1.0e0
    else
        tmp = -1.0e0
    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(-1.0);
	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);
	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. log1p-expm1-u 99.9%

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

      2. expm1-udef 99.8%

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

      3. *-commutative 99.8%

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

      4. exp-to-pow 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-udef 99.8%

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

   3. Applied egg-rr 99.8%

      \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left({\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}^{v} - 1\right)} \]
   4. Step-by-step derivation

      1. pow-to-exp 99.8%

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

      2. expm1-def 99.9%

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

      3. log1p-expm1-u 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in v around 0 93.0%

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

   if 0.400000006 < v

   1. Initial program 93.2%

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

   2. Taylor expanded in u around 0 49.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10: 5.6% 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. Taylor expanded in u around 0 6.9%

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

3. Final simplification 6.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023207 
(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))))))))