HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.6% → 99.6%

Time: 12.6s

Alternatives: 15

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: 99.6% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. log-lowering-log.f32 N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. exp-lowering-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f32 N/A

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

   16. --lowering--.f32 99.4

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

5. Simplified 99.4%

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

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -1.600000023841858:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{expm1}\left(\frac{2}{v}\right), v \cdot u, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -1.600000023841858)
   (fma (expm1 (/ 2.0 v)) (* v u) -1.0)
   (fma v (log (* (expm1 (/ -2.0 v)) (- u))) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -1.600000023841858f) {
		tmp = fmaf(expm1f((2.0f / v)), (v * u), -1.0f);
	} else {
		tmp = fmaf(v, logf((expm1f((-2.0f / v)) * -u)), 1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-1.600000023841858))
		tmp = fma(expm1(Float32(Float32(2.0) / v)), Float32(v * u), Float32(-1.0));
	else
		tmp = fma(v, log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u))), Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1.60000002

   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 u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. accelerator-lowering-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f32 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f32 79.9

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

   5. Simplified 79.9%

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

   if -1.60000002 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   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. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. log-lowering-log.f32 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f32 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. exp-lowering-exp.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f32 N/A

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

      16. --lowering--.f32 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in u around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. neg-mul-1 N/A

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

      11. *-lowering-*.f32 N/A

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

   8. Simplified 98.8%

      \[\leadsto \mathsf{fma}\left(v, \log \color{blue}{\left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right)}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -1.600000023841858:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{expm1}\left(\frac{2}{v}\right), v \cdot u, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5:\\ \;\;\;\;1 + \mathsf{fma}\left(u, \mathsf{fma}\left(u, \frac{-2}{v} + \frac{-4}{v \cdot v}, 2 + \frac{2 + \frac{1.3333333333333333}{v}}{v}\right), -2\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* v (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u))))) -0.5)
   (+
    1.0
    (fma
     u
     (fma
      u
      (+ (/ -2.0 v) (/ -4.0 (* v v)))
      (+ 2.0 (/ (+ 2.0 (/ 1.3333333333333333 v)) v)))
     -2.0))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -0.5f) {
		tmp = 1.0f + fmaf(u, fmaf(u, ((-2.0f / v) + (-4.0f / (v * v))), (2.0f + ((2.0f + (1.3333333333333333f / v)) / v))), -2.0f);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= Float32(-0.5))
		tmp = Float32(Float32(1.0) + fma(u, fma(u, Float32(Float32(Float32(-2.0) / v) + Float32(Float32(-4.0) / Float32(v * v))), Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v))), Float32(-2.0)));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.5

   1. Initial program 93.4%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v} + \frac{1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v} + 2 \cdot \left(1 - u\right)}{v}\right)} \]

   5. Simplified 70.7%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v}} \]

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

   8. Simplified 71.5%

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

   if -0.5 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   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. Taylor expanded in v around 0

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

   5. Simplified 93.7%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \leq -0.5:\\ \;\;\;\;1 + \mathsf{fma}\left(u, \mathsf{fma}\left(u, \frac{-2}{v} + \frac{-4}{v \cdot v}, 2 + \frac{2 + \frac{1.3333333333333333}{v}}{v}\right), -2\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.3%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v} + \frac{1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v} + 2 \cdot \left(1 - u\right)}{v}\right)} \]

   5. Simplified 72.4%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(\frac{1}{2}, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \color{blue}{\left(-8 \cdot u\right)} \cdot \frac{\frac{-1}{6}}{v}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(\frac{1}{2}, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \color{blue}{\left(u \cdot -8\right)} \cdot \frac{\frac{-1}{6}}{v}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)} \]

      2. *-lowering-*.f32 72.2

         \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \color{blue}{\left(u \cdot -8\right)} \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v} \]

   8. Simplified 72.2%

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \color{blue}{\left(u \cdot -8\right)} \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v} \]

   9. Taylor expanded in v around 0

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

       1. /-lowering-/.f32 N/A

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

   11. Simplified 72.5%

       \[\leadsto 1 + \color{blue}{\frac{\mathsf{fma}\left(v, \left(1 - u\right) \cdot \mathsf{fma}\left(-2, v, 0.5 \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right), u \cdot 1.3333333333333333\right)}{v \cdot v}} \]

   12. Taylor expanded in u around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. accelerator-lowering-fma.f32 N/A

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

   14. Simplified 73.0%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.4%

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

4. Final simplification 91.7%

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

Alternative 5: 91.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.3%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v} + \frac{1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v} + 2 \cdot \left(1 - u\right)}{v}\right)} \]

   5. Simplified 72.4%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(\frac{1}{2}, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \color{blue}{\left(-8 \cdot u\right)} \cdot \frac{\frac{-1}{6}}{v}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(\frac{1}{2}, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \color{blue}{\left(u \cdot -8\right)} \cdot \frac{\frac{-1}{6}}{v}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)} \]

      2. *-lowering-*.f32 72.2

         \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \color{blue}{\left(u \cdot -8\right)} \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v} \]

   8. Simplified 72.2%

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \color{blue}{\left(u \cdot -8\right)} \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v} \]

   9. Taylor expanded in v around 0

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

       1. /-lowering-/.f32 N/A

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

   11. Simplified 72.5%

       \[\leadsto 1 + \color{blue}{\frac{\mathsf{fma}\left(v, \left(1 - u\right) \cdot \mathsf{fma}\left(-2, v, 0.5 \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right), u \cdot 1.3333333333333333\right)}{v \cdot v}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-lowering-+.f32 N/A

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

   13. Applied egg-rr 72.6%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.4%

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

4. Final simplification 91.6%

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

Alternative 6: 91.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.3%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v} + \frac{1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v} + 2 \cdot \left(1 - u\right)}{v}\right)} \]

   5. Simplified 72.4%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v}} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(\frac{1}{2}, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \color{blue}{\left(-8 \cdot u\right)} \cdot \frac{\frac{-1}{6}}{v}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(\frac{1}{2}, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \color{blue}{\left(u \cdot -8\right)} \cdot \frac{\frac{-1}{6}}{v}\right)}{\mathsf{neg}\left(v\right)}\right)}{\mathsf{neg}\left(v\right)} \]

      2. *-lowering-*.f32 72.2

         \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \color{blue}{\left(u \cdot -8\right)} \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v} \]

   8. Simplified 72.2%

      \[\leadsto 1 + v \cdot \frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \color{blue}{\left(u \cdot -8\right)} \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v} \]

   9. Taylor expanded in v around 0

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

       1. /-lowering-/.f32 N/A

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

   11. Simplified 72.5%

       \[\leadsto 1 + \color{blue}{\frac{\mathsf{fma}\left(v, \left(1 - u\right) \cdot \mathsf{fma}\left(-2, v, 0.5 \cdot \mathsf{fma}\left(-4, 1 - u, 4\right)\right), u \cdot 1.3333333333333333\right)}{v \cdot v}} \]

   12. Taylor expanded in u around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f32 72.5

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

   14. Simplified 72.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.4%

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

4. Final simplification 91.6%

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

Alternative 7: 91.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.3%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v} + \frac{1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v} + 2 \cdot \left(1 - u\right)}{v}\right)} \]

   5. Simplified 72.4%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v}} \]

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

   8. Simplified 70.6%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. +-lowering-+.f32 N/A

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

      8. /-lowering-/.f32 N/A

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

      9. +-lowering-+.f32 N/A

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

      10. /-lowering-/.f32 N/A

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

      11. metadata-eval 70.8

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

   10. Applied egg-rr 70.8%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.4%

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

4. Final simplification 91.5%

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

Alternative 8: 91.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.3%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v} + \frac{1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v} + 2 \cdot \left(1 - u\right)}{v}\right)} \]

   5. Simplified 72.4%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v}} \]

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

   8. Simplified 70.6%

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

   9. Taylor expanded in v around 0

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

       1. /-lowering-/.f32 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. accelerator-lowering-fma.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(u, 2 + \frac{\color{blue}{\mathsf{fma}\left(v, 2, \frac{4}{3}\right)}}{{v}^{2}}, -2\right) \]

       5. unpow2 N/A

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

       6. *-lowering-*.f32 70.6

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

   11. Simplified 70.6%

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

       5. accelerator-lowering-fma.f32 N/A

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

       6. +-lowering-+.f32 N/A

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

       7. /-lowering-/.f32 N/A

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

       8. accelerator-lowering-fma.f32 N/A

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

       9. *-lowering-*.f32 70.8

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

   13. Applied egg-rr 70.8%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.4%

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

4. Final simplification 91.5%

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

Alternative 9: 91.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.5

   1. Initial program 93.4%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. accelerator-lowering-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

      11. *-lowering-*.f32 N/A

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

      12. --lowering--.f32 N/A

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

      13. accelerator-lowering-fma.f32 N/A

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

      14. --lowering--.f32 N/A

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

      15. /-lowering-/.f32 N/A

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

      16. sub-neg N/A

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

      17. neg-mul-1 N/A

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

   5. Simplified 63.6%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-lowering-+.f32 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f32 64.4

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

   8. Simplified 64.4%

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

   if -0.5 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   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. Taylor expanded in v around 0

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

   5. Simplified 93.7%

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

4. Final simplification 91.1%

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

Alternative 10: 91.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -0.5

   1. Initial program 93.4%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. accelerator-lowering-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

      11. *-lowering-*.f32 N/A

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

      12. --lowering--.f32 N/A

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

      13. accelerator-lowering-fma.f32 N/A

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

      14. --lowering--.f32 N/A

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

      15. /-lowering-/.f32 N/A

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

      16. sub-neg N/A

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

      17. neg-mul-1 N/A

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

   5. Simplified 63.6%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. +-lowering-+.f32 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f32 64.3

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

   8. Simplified 64.3%

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

   if -0.5 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   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. Taylor expanded in v around 0

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

   5. Simplified 93.7%

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

4. Final simplification 91.1%

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

Alternative 11: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.3%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. accelerator-lowering-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

      11. *-lowering-*.f32 N/A

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

      12. --lowering--.f32 N/A

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

      13. accelerator-lowering-fma.f32 N/A

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

      14. --lowering--.f32 N/A

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

      15. /-lowering-/.f32 N/A

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

      16. sub-neg N/A

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

      17. neg-mul-1 N/A

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

   5. Simplified 64.8%

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

   6. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot u - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f32 55.2

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

   8. Simplified 55.2%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.4%

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

4. Final simplification 90.2%

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

Alternative 12: 90.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if ((v * logf((u + (expf((-2.0f / v)) * (1.0f - u))))) <= -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 + (exp(((-2.0e0) / v)) * (1.0e0 - u))))) <= (-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(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u))))) <= 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 + (exp((single(-2.0) / v)) * (single(1.0) - u))))) <= 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 93.3%

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

   3. Taylor expanded in u around 0

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

   5. Simplified 46.4%

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

   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 100.0%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 93.4%

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

4. Final simplification 89.4%

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

Alternative 13: 91.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(8 + \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right)\right), \frac{0.16666666666666666}{v}, \left(1 - u\right) \cdot \left(-0.5 \cdot \mathsf{fma}\left(1 - u, -4, 4\right)\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.1599999964237213)
   1.0
   (-
    (fma -2.0 (- 1.0 u) 1.0)
    (/
     (fma
      (* (- 1.0 u) (+ 8.0 (* (- 1.0 u) (fma (- 1.0 u) 16.0 -24.0))))
      (/ 0.16666666666666666 v)
      (* (- 1.0 u) (* -0.5 (fma (- 1.0 u) -4.0 4.0))))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.1599999964237213f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f) - (fmaf(((1.0f - u) * (8.0f + ((1.0f - u) * fmaf((1.0f - u), 16.0f, -24.0f)))), (0.16666666666666666f / v), ((1.0f - u) * (-0.5f * fmaf((1.0f - u), -4.0f, 4.0f)))) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.1599999964237213))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0)) - Float32(fma(Float32(Float32(Float32(1.0) - u) * Float32(Float32(8.0) + Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0))))), Float32(Float32(0.16666666666666666) / v), Float32(Float32(Float32(1.0) - u) * Float32(Float32(-0.5) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.159999996

   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. Taylor expanded in v around 0

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

   5. Simplified 94.0%

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

   if 0.159999996 < v

   1. Initial program 93.7%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. log-lowering-log.f32 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f32 N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. exp-lowering-exp.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f32 N/A

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

      16. --lowering--.f32 94.0

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

   5. Simplified 94.0%

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

   6. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   8. Simplified 70.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right) - \frac{\mathsf{fma}\left(\left(1 - u\right) \cdot \left(8 + \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right)\right), \frac{0.16666666666666666}{v}, \left(1 - u\right) \cdot \left(-0.5 \cdot \mathsf{fma}\left(1 - u, -4, 4\right)\right)\right)}{v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 91.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\mathsf{fma}\left(v, \left(1 - u\right) \cdot \mathsf{fma}\left(v, -2, \mathsf{fma}\left(1 - u, -4, 4\right) \cdot 0.5\right), \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(u, -8, \mathsf{fma}\left(1 - u, 16, -24\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right), -1.3333333333333333\right)\right)}{v \cdot v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.1599999964237213)
   1.0
   (+
    1.0
    (/
     (fma
      v
      (* (- 1.0 u) (fma v -2.0 (* (fma (- 1.0 u) -4.0 4.0) 0.5)))
      (fma
       -0.16666666666666666
       (fma u -8.0 (* (fma (- 1.0 u) 16.0 -24.0) (* (- 1.0 u) (- 1.0 u))))
       -1.3333333333333333))
     (* v v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.1599999964237213f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (fmaf(v, ((1.0f - u) * fmaf(v, -2.0f, (fmaf((1.0f - u), -4.0f, 4.0f) * 0.5f))), fmaf(-0.16666666666666666f, fmaf(u, -8.0f, (fmaf((1.0f - u), 16.0f, -24.0f) * ((1.0f - u) * (1.0f - u)))), -1.3333333333333333f)) / (v * v));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.1599999964237213))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(fma(v, Float32(Float32(Float32(1.0) - u) * fma(v, Float32(-2.0), Float32(fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0)) * Float32(0.5)))), fma(Float32(-0.16666666666666666), fma(u, Float32(-8.0), Float32(fma(Float32(Float32(1.0) - u), Float32(16.0), Float32(-24.0)) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)))), Float32(-1.3333333333333333))) / Float32(v * v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.159999996

   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. Taylor expanded in v around 0

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

   5. Simplified 94.0%

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

   if 0.159999996 < v

   1. Initial program 93.7%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v} + \frac{1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right)}{v} + 2 \cdot \left(1 - u\right)}{v}\right)} \]

   5. Simplified 70.0%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{\mathsf{fma}\left(1 - u, 2, \frac{\mathsf{fma}\left(0.5, \left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), \mathsf{fma}\left(1 - u, 16, -24\right), \mathsf{fma}\left(8, -u, 8\right)\right) \cdot \frac{-0.16666666666666666}{v}\right)}{-v}\right)}{-v}} \]

   6. Taylor expanded in v around 0

      \[\leadsto 1 + \color{blue}{\frac{\frac{-1}{6} \cdot \left(8 + \left(-8 \cdot u + {\left(1 - u\right)}^{2} \cdot \left(16 \cdot \left(1 - u\right) - 24\right)\right)\right) + v \cdot \left(-2 \cdot \left(v \cdot \left(1 - u\right)\right) + \frac{1}{2} \cdot \left(\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)\right)\right)}{{v}^{2}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto 1 + \color{blue}{\frac{\frac{-1}{6} \cdot \left(8 + \left(-8 \cdot u + {\left(1 - u\right)}^{2} \cdot \left(16 \cdot \left(1 - u\right) - 24\right)\right)\right) + v \cdot \left(-2 \cdot \left(v \cdot \left(1 - u\right)\right) + \frac{1}{2} \cdot \left(\left(4 + -4 \cdot \left(1 - u\right)\right) \cdot \left(1 - u\right)\right)\right)}{{v}^{2}}} \]

   8. Simplified 69.9%

      \[\leadsto 1 + \color{blue}{\frac{\mathsf{fma}\left(v, \left(1 - u\right) \cdot \mathsf{fma}\left(v, -2, 0.5 \cdot \mathsf{fma}\left(1 - u, -4, 4\right)\right), \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(u, -8, \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) \cdot \mathsf{fma}\left(1 - u, 16, -24\right)\right), -1.3333333333333333\right)\right)}{v \cdot v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\mathsf{fma}\left(v, \left(1 - u\right) \cdot \mathsf{fma}\left(v, -2, \mathsf{fma}\left(1 - u, -4, 4\right) \cdot 0.5\right), \mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(u, -8, \mathsf{fma}\left(1 - u, 16, -24\right) \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right), -1.3333333333333333\right)\right)}{v \cdot v}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 5.6% 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. Simplified 6.9%

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

Reproduce

herbie shell --seed 2024201 
(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))))))))