HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 8.9s

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-2}{v}}\\ 1 + v \cdot \log \left(t\_0 - \left(u \cdot t\_0 - u\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (exp (/ -2.0 v)))) (+ 1.0 (* v (log (- t_0 (- (* u t_0) u)))))))

C

float code(float u, float v) {
	float t_0 = expf((-2.0f / v));
	return 1.0f + (v * logf((t_0 - ((u * t_0) - u))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    t_0 = exp(((-2.0e0) / v))
    code = 1.0e0 + (v * log((t_0 - ((u * t_0) - u))))
end function

Julia

function code(u, v)
	t_0 = exp(Float32(Float32(-2.0) / v))
	return Float32(Float32(1.0) + Float32(v * log(Float32(t_0 - Float32(Float32(u * t_0) - u)))))
end

MATLAB

function tmp = code(u, v)
	t_0 = exp((single(-2.0) / v));
	tmp = single(1.0) + (v * log((t_0 - ((u * t_0) - u))));
end

Derivation

1. Initial program 99.5%

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

   1. 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. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 99.5

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 99.4

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

6. Applied rewrites 99.4%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. div-sub N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. associate-+l- N/A

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

   8. lower--.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lift-/.f32 N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f32 N/A

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

   15. lower-exp.f32 N/A

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

   16. lower--.f32 N/A

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

8. Applied rewrites 99.5%

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

Alternative 2: 90.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v)))))) <= single(-1.0))
		tmp = single(1.0) + (v * ((((single(2.0) * u) - ((single(2.0) / v) * v)) / (v * u)) * u));
	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 92.9%

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

   3. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 55.4

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 55.4%

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

   10. Applied rewrites 55.4%

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

   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 94.0%

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

Alternative 3: 90.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v)))))) <= single(-1.0))
		tmp = single(1.0) + (v * (((single(1.0) - u) / v) * single(-2.0)));
	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 92.9%

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

   3. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 55.4

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

   5. Applied rewrites 55.4%

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

   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 94.0%

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

Alternative 4: 90.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v)))))) <= single(-1.0))
		tmp = single(1.0) + ((single(1.0) - u) * single(-2.0));
	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 92.9%

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

   3. Taylor expanded in v around 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 55.4

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

   5. Applied rewrites 55.4%

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

   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 94.0%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

Alternative 6: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{1 - \frac{\frac{\frac{-1.3333333333333333}{v} + -2}{v} - 2}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+
  1.0
  (*
   v
   (log
    (+
     u
     (*
      (- 1.0 u)
      (/
       1.0
       (- 1.0 (/ (- (/ (+ (/ -1.3333333333333333 v) -2.0) v) 2.0) v)))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * (1.0f / (1.0f - (((((-1.3333333333333333f / v) + -2.0f) / v) - 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) * (1.0e0 / (1.0e0 - ((((((-1.3333333333333333e0) / v) + (-2.0e0)) / v) - 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) * Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(Float32(-1.3333333333333333) / v) + Float32(-2.0)) / v) - Float32(2.0)) / v))))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * (single(1.0) / (single(1.0) - (((((single(-1.3333333333333333) / v) + single(-2.0)) / v) - single(2.0)) / v)))))));
end

Derivation

1. Initial program 99.5%

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

   1. 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. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}\right)\right)}}\right) \]

   2. unsub-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}}}\right) \]

   3. lower--.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}}}\right) \]

   4. lower-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{1 - \color{blue}{\frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}}}\right) \]

7. Applied rewrites 96.0%

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

Alternative 7: 93.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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) * (1.0e0 / ((((2.0e0 / v) / v) + (2.0e0 / v)) + 1.0e0))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing
3. 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. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. associate-*r/ N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f32 94.4

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

7. Applied rewrites 94.4%

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

Alternative 8: 93.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. 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. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 99.5

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 99.4

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. +-commutative N/A

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

   6. lower-+.f32 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 94.4

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

9. Applied rewrites 94.4%

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

Alternative 9: 91.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

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) * (1.0e0 / ((2.0e0 / v) + 1.0e0))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing
3. 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. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 92.1

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

7. Applied rewrites 92.1%

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

Alternative 10: 91.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. 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. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 99.5

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 99.4

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 92.1

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

9. Applied rewrites 92.1%

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

Alternative 11: 87.2% 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 99.5%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 88.0%

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

Alternative 12: 5.8% 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 99.5%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 6.0%

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

Reproduce

herbie shell --seed 2024313 
(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))))))))