HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 10.1s

Alternatives: 11

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

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

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 99.2%

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

Alternative 2: 91.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (- (* (* u v) (/ (+ (/ (+ (/ 1.3333333333333333 v) 2.0) v) 2.0) v)) 1.0)
   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 = ((u * v) * (((((1.3333333333333333f / v) + 2.0f) / v) + 2.0f) / 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 ((v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v)))))) <= (-1.0e0)) then
        tmp = ((u * v) * (((((1.3333333333333333e0 / v) + 2.0e0) / v) + 2.0e0) / v)) - 1.0e0
    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(Float32(u * v) * Float32(Float32(Float32(Float32(Float32(Float32(1.3333333333333333) / v) + Float32(2.0)) / v) + Float32(2.0)) / 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 ((v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v)))))) <= single(-1.0))
		tmp = ((u * v) * (((((single(1.3333333333333333) / v) + single(2.0)) / v) + single(2.0)) / 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 91.6%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-expm1.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 51.9

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

   5. Applied rewrites 55.1%

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

   7. Applied rewrites 66.2%

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

   8. Taylor expanded in v around -inf

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

   10. Applied rewrites 67.4%

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

   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 99.8%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

4. Final simplification 90.6%

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

Alternative 3: 91.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (- (* (* u v) (- (+ (+ (/ 2.0 (* v v)) (/ 2.0 v)) 1.0) 1.0)) 1.0)
   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 = ((u * v) * ((((2.0f / (v * v)) + (2.0f / v)) + 1.0f) - 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 ((v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v)))))) <= (-1.0e0)) then
        tmp = ((u * v) * ((((2.0e0 / (v * v)) + (2.0e0 / v)) + 1.0e0) - 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(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))) <= Float32(-1.0))
		tmp = Float32(Float32(Float32(u * v) * Float32(Float32(Float32(Float32(Float32(2.0) / Float32(v * v)) + Float32(Float32(2.0) / v)) + Float32(1.0)) - 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 ((v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v)))))) <= single(-1.0))
		tmp = ((u * v) * ((((single(2.0) / (v * v)) + (single(2.0) / v)) + single(1.0)) - 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 91.6%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-expm1.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 51.9

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

   5. Applied rewrites 51.9%

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

   7. Applied rewrites 66.2%

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

   8. Taylor expanded in v around inf

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

   10. Applied rewrites 64.6%

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

   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 99.8%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

Alternative 4: 91.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (- (* 2.0 (+ u (/ u v))) 1.0)
   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 = (2.0f * (u + (u / 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 ((v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v)))))) <= (-1.0e0)) then
        tmp = (2.0e0 * (u + (u / v))) - 1.0e0
    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(Float32(2.0) * Float32(u + Float32(u / 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 ((v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v)))))) <= single(-1.0))
		tmp = (single(2.0) * (u + (u / 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 91.6%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-expm1.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 64.5%

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

   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 99.8%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

Alternative 5: 90.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))) -1.0)
   (- (+ u u) 1.0)
   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 = (u + 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 ((v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v)))))) <= (-1.0e0)) then
        tmp = (u + u) - 1.0e0
    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(u + 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 ((v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v)))))) <= single(-1.0))
		tmp = (u + 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 91.6%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-expm1.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in v around inf

      \[\leadsto 2 \cdot u - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 55.1%

      \[\leadsto 2 \cdot u - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 55.1%

       \[\leadsto \left(u + u\right) - 1 \]

   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 99.8%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

Alternative 6: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot \log \left(\left(-u\right) \cdot \left(e^{\frac{-2}{v}} - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot v\right) \cdot \frac{\frac{\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}}{v} + 2}{v} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5)
   (+ 1.0 (* v (log (* (- u) (- (exp (/ -2.0 v)) 1.0)))))
   (-
    (*
     (* u v)
     (/
      (+
       (/
        (+ (+ 2.0 (/ 0.6666666666666666 (* v v))) (/ 1.3333333333333333 v))
        v)
       2.0)
      v))
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f + (v * logf((-u * (expf((-2.0f / v)) - 1.0f))));
	} else {
		tmp = ((u * v) * (((((2.0f + (0.6666666666666666f / (v * v))) + (1.3333333333333333f / v)) / v) + 2.0f) / v)) - 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.5e0) then
        tmp = 1.0e0 + (v * log((-u * (exp(((-2.0e0) / v)) - 1.0e0))))
    else
        tmp = ((u * v) * (((((2.0e0 + (0.6666666666666666e0 / (v * v))) + (1.3333333333333333e0 / v)) / v) + 2.0e0) / v)) - 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(Float32(-u) * Float32(exp(Float32(Float32(-2.0) / v)) - Float32(1.0))))));
	else
		tmp = Float32(Float32(Float32(u * v) * Float32(Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(0.6666666666666666) / Float32(v * v))) + Float32(Float32(1.3333333333333333) / v)) / v) + Float32(2.0)) / v)) - Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.5))
		tmp = single(1.0) + (v * log((-u * (exp((single(-2.0) / v)) - single(1.0)))));
	else
		tmp = ((u * v) * (((((single(2.0) + (single(0.6666666666666666) / (v * v))) + (single(1.3333333333333333) / v)) / v) + single(2.0)) / v)) - single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 99.8%

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

   3. Taylor expanded in u around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f32 N/A

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

      4. lower-neg.f32 N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-expm1.f32 N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 47.4

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

   5. Applied rewrites 46.6%

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

   7. Applied rewrites 98.6%

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

   if 0.5 < v

   1. Initial program 91.1%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-expm1.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 54.0

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 71.5%

      \[\leadsto \left(u \cdot v\right) \cdot \frac{\frac{\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}}{-v} - 2}{-v} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot \log \left(\left(-u\right) \cdot \left(e^{\frac{-2}{v}} - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot v\right) \cdot \frac{\frac{\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}}{v} + 2}{v} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot v\right) \cdot \frac{\frac{\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}}{v} + 2}{v} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   (+ 1.0 (* v (log (fma (- 1.0 u) (exp (/ -2.0 v)) u))))
   (-
    (*
     (* u v)
     (/
      (+
       (/
        (+ (+ 2.0 (/ 0.6666666666666666 (* v v))) (/ 1.3333333333333333 v))
        v)
       2.0)
      v))
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f + (v * logf(fmaf((1.0f - u), expf((-2.0f / v)), u)));
	} else {
		tmp = ((u * v) * (((((2.0f + (0.6666666666666666f / (v * v))) + (1.3333333333333333f / v)) / v) + 2.0f) / v)) - 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		tmp = Float32(Float32(1.0) + Float32(v * log(fma(Float32(Float32(1.0) - u), exp(Float32(Float32(-2.0) / v)), u))));
	else
		tmp = Float32(Float32(Float32(u * v) * Float32(Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(0.6666666666666666) / Float32(v * v))) + Float32(Float32(1.3333333333333333) / v)) / v) + Float32(2.0)) / v)) - Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   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. 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. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lower-fma.f32 99.2

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

   4. Applied rewrites 99.2%

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

   if 0.119999997 < v

   1. Initial program 92.1%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-expm1.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 45.1

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 60.6%

      \[\leadsto \left(u \cdot v\right) \cdot \frac{\frac{\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}}{-v} - 2}{-v} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot v\right) \cdot \frac{\frac{\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}}{v} + 2}{v} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.0% accurate, 1.0× speedup

Math

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

FPCore

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

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;
	} 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
    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(-1.0);
	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);
	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 91.6%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 44.3%

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

   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 99.8%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.6%

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

Alternative 9: 49.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.07000000029802322:\\ \;\;\;\;1 + \log \left(\mathsf{fma}\left(1 - \frac{2}{v}, 1 - u, u\right)\right) \cdot v\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot v\right) \cdot \frac{\frac{\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}}{v} + 2}{v} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.07000000029802322)
   (+ 1.0 (* (log (fma (- 1.0 (/ 2.0 v)) (- 1.0 u) u)) v))
   (-
    (*
     (* u v)
     (/
      (+
       (/
        (+ (+ 2.0 (/ 0.6666666666666666 (* v v))) (/ 1.3333333333333333 v))
        v)
       2.0)
      v))
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.07000000029802322f) {
		tmp = 1.0f + (logf(fmaf((1.0f - (2.0f / v)), (1.0f - u), u)) * v);
	} else {
		tmp = ((u * v) * (((((2.0f + (0.6666666666666666f / (v * v))) + (1.3333333333333333f / v)) / v) + 2.0f) / v)) - 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.07000000029802322))
		tmp = Float32(Float32(1.0) + Float32(log(fma(Float32(Float32(1.0) - Float32(Float32(2.0) / v)), Float32(Float32(1.0) - u), u)) * v));
	else
		tmp = Float32(Float32(Float32(u * v) * Float32(Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(0.6666666666666666) / Float32(v * v))) + Float32(Float32(1.3333333333333333) / v)) / v) + Float32(2.0)) / v)) - Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0700000003

   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

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

      2. *-commutative N/A

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

      3. lower-*.f32 100.0

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in v around inf

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

      1. lower--.f32 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f32 45.4

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

   7. Applied rewrites 47.2%

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

   if 0.0700000003 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0

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

      1. lower--.f32 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-expm1.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 41.1

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

   5. Applied rewrites 41.1%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 56.9%

      \[\leadsto \left(u \cdot v\right) \cdot \frac{\frac{\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}}{-v} - 2}{-v} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.07000000029802322:\\ \;\;\;\;1 + \log \left(\mathsf{fma}\left(1 - \frac{2}{v}, 1 - u, u\right)\right) \cdot v\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot v\right) \cdot \frac{\frac{\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}}{v} + 2}{v} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot v\right) \cdot \frac{\frac{\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}}{v} + 2}{v} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (-
    (*
     (* u v)
     (/
      (+
       (/
        (+ (+ 2.0 (/ 0.6666666666666666 (* v v))) (/ 1.3333333333333333 v))
        v)
       2.0)
      v))
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = ((u * v) * (((((2.0f + (0.6666666666666666f / (v * v))) + (1.3333333333333333f / v)) / v) + 2.0f) / v)) - 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = ((u * v) * (((((2.0e0 + (0.6666666666666666e0 / (v * v))) + (1.3333333333333333e0 / v)) / v) + 2.0e0) / v)) - 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(u * v) * Float32(Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(0.6666666666666666) / Float32(v * v))) + Float32(Float32(1.3333333333333333) / v)) / v) + Float32(2.0)) / v)) - Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = ((u * v) * (((((single(2.0) + (single(0.6666666666666666) / (v * v))) + (single(1.3333333333333333) / v)) / v) + single(2.0)) / v)) - single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   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%

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

   if 0.119999997 < v

   1. Initial program 92.1%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-expm1.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 45.1

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 60.6%

      \[\leadsto \left(u \cdot v\right) \cdot \frac{\frac{\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}}{-v} - 2}{-v} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot v\right) \cdot \frac{\frac{\left(2 + \frac{0.6666666666666666}{v \cdot v}\right) + \frac{1.3333333333333333}{v}}{v} + 2}{v} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 99.2%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 6.4%

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

Reproduce

herbie shell --seed 2024326 
(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))))))))