HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 10.5s

Alternatives: 12

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, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. exp-1-e N/A

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

   6. lower-E.f32 99.3

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

4. Applied rewrites 99.3%

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

5. Final simplification 99.3%

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

Alternative 2: 90.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 91.5%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 5.6%

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

   6. Taylor expanded in u around -inf

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

   8. Applied rewrites 74.0%

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

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.0%

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

4. Final simplification 90.7%

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

Alternative 3: 90.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 91.5%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 5.6%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 3.2%

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

   10. Applied rewrites 73.6%

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

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.0%

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

4. Final simplification 90.6%

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

Alternative 4: 89.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \cdot v \leq -1.2000000476837158:\\ \;\;\;\;\left(2 \cdot u - \frac{-2 \cdot u - \frac{\mathsf{fma}\left(0.6666666666666666, \frac{u}{v}, 1.3333333333333333 \cdot u\right)}{v}}{v}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (- u (* (+ -1.0 u) (exp (/ -2.0 v))))) v) -1.2000000476837158)
   (-
    (-
     (* 2.0 u)
     (/
      (-
       (* -2.0 u)
       (/ (fma 0.6666666666666666 (/ u v) (* 1.3333333333333333 u)) v))
      v))
    1.0)
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((logf((u - ((-1.0f + u) * expf((-2.0f / v))))) * v) <= -1.2000000476837158f) {
		tmp = ((2.0f * u) - (((-2.0f * u) - (fmaf(0.6666666666666666f, (u / v), (1.3333333333333333f * u)) / v)) / v)) - 1.0f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(log(Float32(u - Float32(Float32(Float32(-1.0) + u) * exp(Float32(Float32(-2.0) / v))))) * v) <= Float32(-1.2000000476837158))
		tmp = Float32(Float32(Float32(Float32(2.0) * u) - Float32(Float32(Float32(Float32(-2.0) * u) - Float32(fma(Float32(0.6666666666666666), Float32(u / v), Float32(Float32(1.3333333333333333) * u)) / v)) / v)) - Float32(1.0));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1.20000005

   1. Initial program 91.7%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 52.8

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

   5. Applied rewrites 49.2%

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

   7. Applied rewrites 49.3%

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

   8. Taylor expanded in v around -inf

      \[\leadsto \left(-1 \cdot \frac{-2 \cdot u + -1 \cdot \frac{\frac{2}{3} \cdot \frac{u}{v} + \frac{4}{3} \cdot u}{v}}{v} + 2 \cdot u\right) - \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.9%

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

   if -1.20000005 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 91.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \cdot v \leq -1.2000000476837158:\\ \;\;\;\;\left(2 \cdot u - \frac{-2 \cdot u - \frac{\mathsf{fma}\left(0.6666666666666666, \frac{u}{v}, 1.3333333333333333 \cdot u\right)}{v}}{v}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 91.5%

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

   4. Applied rewrites 7.7%

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

   5. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 62.0

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

   7. Applied rewrites 62.0%

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

   9. Applied rewrites 62.0%

      \[\leadsto 1 + \frac{1}{\frac{1 + u}{1 - u \cdot u}} \cdot -2 \]

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.0%

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

4. Final simplification 89.8%

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

Alternative 6: 90.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 91.5%

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

   4. Applied rewrites 8.7%

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

   5. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 62.0

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

   7. Applied rewrites 62.0%

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

   9. Applied rewrites 62.0%

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

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.0%

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

4. Final simplification 89.8%

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

Alternative 7: 90.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 91.5%

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

   3. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 62.0

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

   5. Applied rewrites 62.0%

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

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.0%

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

4. Final simplification 89.8%

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

Alternative 8: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-neg.f32 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in u around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f32 N/A

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

   7. Applied rewrites 99.1%

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

   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 v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 7.1%

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

   6. Taylor expanded in u around -inf

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

   8. Applied rewrites 66.4%

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

4. Final simplification 96.3%

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

Alternative 9: 89.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 91.5%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 51.3%

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

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.0%

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

4. Final simplification 89.0%

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

Alternative 10: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Final simplification 99.3%

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

Alternative 11: 47.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5)
   (+ (* (log (fma 1.0 (- 1.0 u) u)) v) 1.0)
   (+
    (*
     (/ (* (* (- (/ 2.0 v) (/ (- (- 2.0 (/ -2.0 v)) (/ 2.0 u)) u)) u) u) (- v))
     v)
    1.0)))

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 99.9%

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in v around inf

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

   7. Applied rewrites 49.7%

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

   if 0.5 < v

   1. Initial program 91.2%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 5.7%

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

   6. Taylor expanded in u around -inf

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

   8. Applied rewrites 76.4%

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

4. Final simplification 52.3%

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

Alternative 12: 6.0% accurate, 231.0× 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 99.3%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Applied rewrites 6.7%

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

Reproduce

herbie shell --seed 2024270 
(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))))))))