HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 11.4s

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: 99.5% 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

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

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. lower-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. lower-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. lower-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. lower-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. lower-/.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. lower--.f32 99.5

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

5. Simplified 99.5%

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

Alternative 2: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{fma}\left(v, \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \mathsf{fma}\left(-2, 1 - u, 1\right)\right) + \frac{0.16666666666666666 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, \mathsf{fma}\left(1 - u, -16, 24\right), -8\right)\right)}{v \cdot v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (fma v (log (* (expm1 (/ -2.0 v)) (- u))) 1.0)
   (+
    (fma
     0.5
     (/ (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0)) v)
     (fma -2.0 (- 1.0 u) 1.0))
    (/
     (*
      0.16666666666666666
      (* (- 1.0 u) (fma (- 1.0 u) (fma (- 1.0 u) -16.0 24.0) -8.0)))
     (* v v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = fmaf(v, logf((expm1f((-2.0f / v)) * -u)), 1.0f);
	} else {
		tmp = fmaf(0.5f, (((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)) / v), fmaf(-2.0f, (1.0f - u), 1.0f)) + ((0.16666666666666666f * ((1.0f - u) * fmaf((1.0f - u), fmaf((1.0f - u), -16.0f, 24.0f), -8.0f))) / (v * v));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = fma(v, log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u))), Float32(1.0));
	else
		tmp = Float32(fma(Float32(0.5), Float32(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))) / v), fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0))) + Float32(Float32(Float32(0.16666666666666666) * Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), fma(Float32(Float32(1.0) - u), Float32(-16.0), Float32(24.0)), Float32(-8.0)))) / Float32(v * v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. lower-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. lower-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. lower-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. lower-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. lower-/.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. lower--.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. lower-*.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 99.9%

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

   if 0.200000003 < v

   1. Initial program 90.6%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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. lower-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. lower-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. lower-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. lower-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. lower-/.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. lower--.f32 91.8

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

   5. Simplified 91.8%

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

   6. Taylor expanded in v around inf

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

   8. Simplified 81.8%

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

Alternative 3: 91.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \mathsf{fma}\left(-2, 1 - u, 1\right)\right) + \frac{0.16666666666666666 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, \mathsf{fma}\left(1 - u, -16, 24\right), -8\right)\right)}{v \cdot v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    (fma
     0.5
     (/ (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0)) v)
     (fma -2.0 (- 1.0 u) 1.0))
    (/
     (*
      0.16666666666666666
      (* (- 1.0 u) (fma (- 1.0 u) (fma (- 1.0 u) -16.0 24.0) -8.0)))
     (* v v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(0.5f, (((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)) / v), fmaf(-2.0f, (1.0f - u), 1.0f)) + ((0.16666666666666666f * ((1.0f - u) * fmaf((1.0f - u), fmaf((1.0f - u), -16.0f, 24.0f), -8.0f))) / (v * v));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(0.5), Float32(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))) / v), fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0))) + Float32(Float32(Float32(0.16666666666666666) * Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), fma(Float32(Float32(1.0) - u), Float32(-16.0), Float32(24.0)), Float32(-8.0)))) / Float32(v * v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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. lower-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. lower-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. lower-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. lower-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. lower-/.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. lower--.f32 91.8

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

   5. Simplified 91.8%

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

   6. Taylor expanded in v around inf

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

   8. Simplified 81.8%

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

Alternative 4: 91.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \frac{0.16666666666666666 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, \mathsf{fma}\left(1 - u, -16, 24\right), -8\right)\right)}{v \cdot v}\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    (fma
     -2.0
     (- 1.0 u)
     (fma
      0.5
      (/ (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0)) v)
      (/
       (*
        0.16666666666666666
        (* (- 1.0 u) (fma (- 1.0 u) (fma (- 1.0 u) -16.0 24.0) -8.0)))
       (* v v))))
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), fmaf(0.5f, (((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)) / v), ((0.16666666666666666f * ((1.0f - u) * fmaf((1.0f - u), fmaf((1.0f - u), -16.0f, 24.0f), -8.0f))) / (v * v)))) + 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(1.0) - u), fma(Float32(0.5), Float32(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))) / v), Float32(Float32(Float32(0.16666666666666666) * Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), fma(Float32(Float32(1.0) - u), Float32(-16.0), Float32(24.0)), Float32(-8.0)))) / Float32(v * v)))) + Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around 0

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

      1. lower-*.f32 N/A

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

      2. lower-log.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower-exp.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. lower--.f32 91.5

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

   5. Simplified 91.5%

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

   6. Taylor expanded in v around inf

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

   8. Simplified 81.6%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \mathsf{fma}\left(0.5, \frac{\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right)}{v}, \frac{0.16666666666666666 \cdot \left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, \mathsf{fma}\left(1 - u, -16, 24\right), -8\right)\right)}{v \cdot v}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \mathsf{fma}\left(0.16666666666666666, \frac{u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 16, -24\right), 8\right)}{v \cdot v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{0.5}{v}, 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (fma
    -2.0
    (- 1.0 u)
    (fma
     0.16666666666666666
     (/ (* u (fma u (fma u 16.0 -24.0) 8.0)) (* v v))
     (fma (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0)) (/ 0.5 v) 1.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), fmaf(0.16666666666666666f, ((u * fmaf(u, fmaf(u, 16.0f, -24.0f), 8.0f)) / (v * v)), fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), (0.5f / v), 1.0f)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = fma(Float32(-2.0), Float32(Float32(1.0) - u), fma(Float32(0.16666666666666666), Float32(Float32(u * fma(u, fma(u, Float32(16.0), Float32(-24.0)), Float32(8.0))) / Float32(v * v)), fma(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))), Float32(Float32(0.5) / v), Float32(1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 81.1%

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

   6. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f32 81.1

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

   8. Simplified 81.1%

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

Alternative 6: 90.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(u, 2, \frac{\mathsf{fma}\left(u, -1.3333333333333333, \frac{u}{v} \cdot -0.6666666666666666\right)}{-v}\right)}{v} - \mathsf{fma}\left(-2, u, 2\right)}{v}, v, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (fma
    (/
     (-
      (/
       (fma
        u
        2.0
        (/ (fma u -1.3333333333333333 (* (/ u v) -0.6666666666666666)) (- v)))
       v)
      (fma -2.0 u 2.0))
     v)
    v
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf((((fmaf(u, 2.0f, (fmaf(u, -1.3333333333333333f, ((u / v) * -0.6666666666666666f)) / -v)) / v) - fmaf(-2.0f, u, 2.0f)) / v), v, 1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = fma(Float32(Float32(Float32(fma(u, Float32(2.0), Float32(fma(u, Float32(-1.3333333333333333), Float32(Float32(u / v) * Float32(-0.6666666666666666))) / Float32(-v))) / v) - fma(Float32(-2.0), u, Float32(2.0))) / v), v, Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f32 N/A

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

      3. rec-exp N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-expm1.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f32 78.0

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

   5. Simplified 78.0%

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

      1. lift-/.f32 N/A

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

      2. lift-expm1.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f32 78.5

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

   7. Applied egg-rr 78.5%

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

   8. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f32 N/A

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

   10. Simplified 75.3%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(u, 2, \frac{\mathsf{fma}\left(u, -1.3333333333333333, \frac{u}{v} \cdot -0.6666666666666666\right)}{-v}\right)}{v} - \mathsf{fma}\left(-2, u, 2\right)}{v}, v, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, \mathsf{fma}\left(0.16666666666666666, \frac{u \cdot \mathsf{fma}\left(u, -24, 8\right)}{v \cdot v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \mathsf{fma}\left(1 - u, -4, 4\right), \frac{0.5}{v}, 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (fma
    -2.0
    (- 1.0 u)
    (fma
     0.16666666666666666
     (/ (* u (fma u -24.0 8.0)) (* v v))
     (fma (* (- 1.0 u) (fma (- 1.0 u) -4.0 4.0)) (/ 0.5 v) 1.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), fmaf(0.16666666666666666f, ((u * fmaf(u, -24.0f, 8.0f)) / (v * v)), fmaf(((1.0f - u) * fmaf((1.0f - u), -4.0f, 4.0f)), (0.5f / v), 1.0f)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = fma(Float32(-2.0), Float32(Float32(1.0) - u), fma(Float32(0.16666666666666666), Float32(Float32(u * fma(u, Float32(-24.0), Float32(8.0))) / Float32(v * v)), fma(Float32(Float32(Float32(1.0) - u) * fma(Float32(Float32(1.0) - u), Float32(-4.0), Float32(4.0))), Float32(Float32(0.5) / v), Float32(1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 81.1%

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

   6. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 75.0

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

   8. Simplified 75.0%

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

Alternative 8: 90.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, u, -1\right) - \frac{\mathsf{fma}\left(u, -2, \frac{\mathsf{fma}\left(u, 1.3333333333333333, \frac{u \cdot 0.6666666666666666}{v}\right)}{-v}\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (-
    (fma 2.0 u -1.0)
    (/
     (fma
      u
      -2.0
      (/ (fma u 1.3333333333333333 (/ (* u 0.6666666666666666) v)) (- v)))
     v))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(2.0f, u, -1.0f) - (fmaf(u, -2.0f, (fmaf(u, 1.3333333333333333f, ((u * 0.6666666666666666f) / v)) / -v)) / v);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(2.0), u, Float32(-1.0)) - Float32(fma(u, Float32(-2.0), Float32(fma(u, Float32(1.3333333333333333), Float32(Float32(u * Float32(0.6666666666666666)) / v)) / Float32(-v))) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f32 N/A

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

      3. rec-exp N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-expm1.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f32 78.0

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

   5. Simplified 78.0%

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

   6. Taylor expanded in v around -inf

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

      1. associate-+r+ N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. associate-+r+ N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-/.f32 N/A

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

   8. Simplified 74.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, u, -1\right) - \frac{\mathsf{fma}\left(u, -2, \frac{\mathsf{fma}\left(u, 1.3333333333333333, \frac{u \cdot 0.6666666666666666}{v}\right)}{-v}\right)}{v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 90.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(2, u, -2\right) + \frac{u \cdot \left(\frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v} - -2\right)}{v}\right) + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    (+
     (fma 2.0 u -2.0)
     (/
      (* u (- (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) v) -2.0))
      v))
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = (fmaf(2.0f, u, -2.0f) + ((u * (((1.3333333333333333f + (0.6666666666666666f / v)) / v) - -2.0f)) / v)) + 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(fma(Float32(2.0), u, Float32(-2.0)) + Float32(Float32(u * Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / v) - Float32(-2.0))) / v)) + Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f32 N/A

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

      3. rec-exp N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-expm1.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f32 78.0

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

   5. Simplified 78.0%

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

   6. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-/.f32 N/A

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

   8. Simplified 74.7%

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

   9. Taylor expanded in u around 0

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

       1. lower-/.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. lower-+.f32 N/A

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

       6. mul-1-neg N/A

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

       7. distribute-neg-frac2 N/A

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

       8. mul-1-neg N/A

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

       9. lower-/.f32 N/A

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

       10. lower-+.f32 N/A

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

       11. associate-*r/ N/A

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

       12. metadata-eval N/A

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

       13. lower-/.f32 N/A

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

       14. mul-1-neg N/A

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

       15. lower-neg.f32 74.7

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

   11. Simplified 74.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(2, u, -2\right) + \frac{u \cdot \left(\frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v} - -2\right)}{v}\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-2}{v}, u + -1, 2\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (fma u (fma (/ -2.0 v) (+ u -1.0) 2.0) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(u, fmaf((-2.0f / v), (u + -1.0f), 2.0f), -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = fma(u, fma(Float32(Float32(-2.0) / v), Float32(u + Float32(-1.0)), Float32(2.0)), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-fma.f32 N/A

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

      4. lower--.f32 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f32 N/A

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

   5. Simplified 72.5%

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

   6. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f32 N/A

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

   8. Simplified 73.3%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, \mathsf{fma}\left(\frac{-2}{v}, u + -1, 2\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u, 2 + \frac{2}{v}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (fma u (+ 2.0 (/ 2.0 v)) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(u, (2.0f + (2.0f / v)), -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = fma(u, Float32(Float32(2.0) + Float32(Float32(2.0) / v)), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. lower-fma.f32 N/A

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

      4. lower--.f32 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f32 N/A

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

   5. Simplified 72.5%

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

   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. lower-fma.f32 N/A

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

      4. lower-+.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. lower-/.f32 68.6

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

   8. Simplified 68.6%

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

Alternative 12: 89.9% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, 1 - u, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (fma -2.0 (- 1.0 u) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(-2.0f, (1.0f - u), 1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower--.f32 61.5

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

   5. Simplified 61.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, 1 - u, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 89.9% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, u, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (fma 2.0 u -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(2.0f, u, -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = fma(Float32(2.0), u, Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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 92.6%

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

   if 0.200000003 < v

   1. Initial program 90.6%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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. lower-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. lower-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. lower-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. lower-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. lower-/.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. lower--.f32 91.8

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

   5. Simplified 91.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. exp-prod N/A

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

      4. lower-pow.f32 N/A

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

      5. lower-exp.f32 N/A

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

      6. lower-/.f32 90.8

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

   7. Applied egg-rr 90.8%

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

   8. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f32 61.5

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

   10. Simplified 61.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(2, u, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 86.6% 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 v around 0

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

5. Simplified 87.3%

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

Alternative 15: 6.1% 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 5.9%

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

Reproduce

herbie shell --seed 2024219 
(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))))))))