HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 8.8s

Alternatives: 8

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 2: 89.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (+ (* (exp (/ -2.0 v)) (- 1.0 u)) u)) v) -1.0)
   (/
    (- (pow (* -2.0 (- 1.0 u)) 2.0) 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(((expf((-2.0f / v)) * (1.0f - u)) + u)) * v) <= -1.0f) {
		tmp = (powf((-2.0f * (1.0f - u)), 2.0f) - 1.0f) / ((((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(((exp(((-2.0e0) / v)) * (1.0e0 - u)) + u)) * v) <= (-1.0e0)) then
        tmp = ((((-2.0e0) * (1.0e0 - u)) ** 2.0e0) - 1.0e0) / ((((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(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)) * v) <= Float32(-1.0))
		tmp = Float32(Float32((Float32(Float32(-2.0) * Float32(Float32(1.0) - u)) ^ Float32(2.0)) - Float32(1.0)) / 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(((exp((single(-2.0) / v)) * (single(1.0) - u)) + u)) * v) <= single(-1.0))
		tmp = (((single(-2.0) * (single(1.0) - u)) ^ single(2.0)) - single(1.0)) / ((((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 92.5%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 60.9

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

   5. Applied rewrites 60.9%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 60.9%

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

   9. Applied rewrites 60.9%

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

   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 91.7%

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

4. Final simplification 89.3%

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

Alternative 3: 89.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float t_0 = -2.0f * (1.0f - u);
	float tmp;
	if ((logf(((expf((-2.0f / v)) * (1.0f - u)) + u)) * v) <= -1.0f) {
		tmp = (1.0f / (1.0f - t_0)) * (1.0f - powf(t_0, 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) :: t_0
    real(4) :: tmp
    t_0 = (-2.0e0) * (1.0e0 - u)
    if ((log(((exp(((-2.0e0) / v)) * (1.0e0 - u)) + u)) * v) <= (-1.0e0)) then
        tmp = (1.0e0 / (1.0e0 - t_0)) * (1.0e0 - (t_0 ** 2.0e0))
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(Float32(-2.0) * Float32(Float32(1.0) - u))
	tmp = Float32(0.0)
	if (Float32(log(Float32(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)) * v) <= Float32(-1.0))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(1.0) - t_0)) * Float32(Float32(1.0) - (t_0 ^ Float32(2.0))));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = single(-2.0) * (single(1.0) - u);
	tmp = single(0.0);
	if ((log(((exp((single(-2.0) / v)) * (single(1.0) - u)) + u)) * v) <= single(-1.0))
		tmp = (single(1.0) / (single(1.0) - t_0)) * (single(1.0) - (t_0 ^ 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 92.5%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 60.9

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

   5. Applied rewrites 60.9%

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

      1. lift-+.f32 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f32 N/A

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

   7. Applied rewrites 60.9%

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

   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 91.7%

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

4. Final simplification 89.3%

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

Alternative 4: 89.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if ((logf(((expf((-2.0f / v)) * (1.0f - u)) + u)) * v) <= -1.0f) {
		tmp = 1.0f / (1.0f / ((-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(((exp(((-2.0e0) / v)) * (1.0e0 - u)) + u)) * v) <= (-1.0e0)) then
        tmp = 1.0e0 / (1.0e0 / (((-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(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)) * v) <= Float32(-1.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) / 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(((exp((single(-2.0) / v)) * (single(1.0) - u)) + u)) * v) <= single(-1.0))
		tmp = single(1.0) / (single(1.0) / ((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 92.5%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 60.9

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

   5. Applied rewrites 60.9%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 60.9%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. clear-num N/A

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

   9. Applied rewrites 60.9%

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

   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 91.7%

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

4. Final simplification 89.3%

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

Alternative 5: 89.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if ((logf(((expf((-2.0f / v)) * (1.0f - u)) + u)) * 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(((exp(((-2.0e0) / v)) * (1.0e0 - u)) + u)) * 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(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)) * 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(((exp((single(-2.0) / v)) * (single(1.0) - u)) + u)) * 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 92.5%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 60.9

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

   5. Applied rewrites 60.9%

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

   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 91.7%

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

4. Final simplification 89.3%

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

Alternative 6: 38.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ (* (log (fma (- u) (exp (/ -2.0 v)) u)) v) 1.0)
   (/
    (- (pow (* -2.0 (- 1.0 u)) 2.0) 1.0)
    (- (/ (* (- 1.0 (* u u)) -2.0) (+ u 1.0)) 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 = (powf((-2.0f * (1.0f - u)), 2.0f) - 1.0f) / ((((1.0f - (u * u)) * -2.0f) / (u + 1.0f)) - 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(-2.0) * Float32(Float32(1.0) - u)) ^ Float32(2.0)) - Float32(1.0)) / Float32(Float32(Float32(Float32(Float32(1.0) - Float32(u * u)) * Float32(-2.0)) / Float32(u + Float32(1.0))) - 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. mul-1-neg N/A

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

      8. lower-neg.f32 N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. lower-exp.f32 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f32 99.8

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

   7. Applied rewrites 99.8%

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

   if 0.100000001 < v

   1. Initial program 92.8%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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 55.6

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

   5. Applied rewrites 55.6%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 55.6%

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

   9. Applied rewrites 55.7%

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

4. Final simplification 96.0%

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

Alternative 7: 89.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if ((logf(((expf((-2.0f / v)) * (1.0f - u)) + u)) * 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(((exp(((-2.0e0) / v)) * (1.0e0 - u)) + u)) * 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(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)) * 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(((exp((single(-2.0) / v)) * (single(1.0) - u)) + u)) * 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 92.5%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 50.4%

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

   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 91.7%

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

4. Final simplification 88.5%

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

Alternative 8: 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 99.4%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 6.8%

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

Reproduce

herbie shell --seed 2024248 
(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))))))))