HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 16.6s

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: 99.5% 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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. add-sqr-sqrt 99.3%

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

   2. log-prod 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-undefine 99.4%

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

   5. +-commutative 99.4%

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

   6. fma-undefine 99.4%

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

4. Applied egg-rr 99.4%

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

   1. count-2 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in v around 0 99.4%

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

8. Final simplification 99.4%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. fma-define 99.2%

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

   3. +-commutative 99.2%

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

   4. fma-define 99.3%

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

3. Simplified 99.3%

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

   1. fma-undefine 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. fma-define 99.2%

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

   3. +-commutative 99.2%

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

   4. fma-define 99.3%

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

3. Simplified 99.3%

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

   1. fma-undefine 99.3%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in u around inf 99.2%

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

   1. *-commutative 99.2%

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

   2. *-rgt-identity 99.2%

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

   3. associate-/l* 99.3%

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

   4. distribute-lft-out 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (+ 1.0 (* v (log (* (expm1 (/ -2.0 v)) (- u)))))
   (+ -1.0 (* (* v u) (expm1 (/ 2.0 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f + (v * logf((expm1f((-2.0f / v)) * -u)));
	} else {
		tmp = -1.0f + ((v * u) * expm1f((2.0f / v)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u)))));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(v * u) * expm1(Float32(Float32(2.0) / v))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in u around inf 99.3%

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

      1. *-commutative 99.3%

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

      2. +-commutative 99.3%

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

      3. mul-1-neg 99.3%

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

      4. metadata-eval 99.3%

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

      5. distribute-neg-in 99.3%

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

      6. metadata-eval 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-lft-neg-in 99.3%

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

      9. distribute-rgt-neg-in 99.3%

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

      10. expm1-define 99.3%

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

   9. Simplified 99.3%

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

   if 0.400000006 < v

   1. Initial program 91.5%

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

      1. +-commutative 91.5%

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

      2. fma-define 91.1%

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

      3. +-commutative 91.1%

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

      4. fma-define 91.7%

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

   3. Simplified 91.7%

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

      1. fma-undefine 91.1%

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

   6. Applied egg-rr 91.1%

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

      1. *-un-lft-identity 91.1%

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

      2. exp-prod 92.2%

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

   8. Applied egg-rr 92.2%

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

      1. exp-1-e 92.2%

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

   10. Simplified 92.2%

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

   11. Taylor expanded in u around 0 66.9%

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

       1. associate-+r+ 66.9%

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

       2. log-E 67.9%

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

       3. metadata-eval 67.9%

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

       4. metadata-eval 67.9%

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

       5. associate-*r* 67.9%

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

       6. *-commutative 67.9%

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

       7. rec-exp 67.9%

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

       8. log-E 67.9%

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

       9. metadata-eval 67.9%

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

       10. log-E 67.9%

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

       11. associate-*r/ 67.9%

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

       12. log-E 67.9%

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

       13. metadata-eval 67.9%

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

       14. metadata-eval 67.9%

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

   13. Simplified 67.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Final simplification 99.3%

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

Alternative 6: 91.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896) 1.0 (+ -1.0 (* (* v u) (expm1 (/ 2.0 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((v * u) * expm1f((2.0f / v)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(v * u) * expm1(Float32(Float32(2.0) / v))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 94.6%

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

   if 0.300000012 < v

   1. Initial program 91.7%

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

      1. +-commutative 91.7%

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

      2. fma-define 91.3%

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

      3. +-commutative 91.3%

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

      4. fma-define 91.9%

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

   3. Simplified 91.9%

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

      1. fma-undefine 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. *-un-lft-identity 91.3%

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

      2. exp-prod 92.4%

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

   8. Applied egg-rr 92.4%

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

      1. exp-1-e 92.4%

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

   10. Simplified 92.4%

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

   11. Taylor expanded in u around 0 63.9%

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

       1. associate-+r+ 63.8%

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

       2. log-E 64.8%

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

       3. metadata-eval 64.8%

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

       4. metadata-eval 64.8%

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

       5. associate-*r* 64.8%

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

       6. *-commutative 64.8%

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

       7. rec-exp 64.8%

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

       8. log-E 64.8%

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

       9. metadata-eval 64.8%

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

       10. log-E 64.8%

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

       11. associate-*r/ 64.8%

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

       12. log-E 64.8%

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

       13. metadata-eval 64.8%

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

       14. metadata-eval 64.8%

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

   13. Simplified 64.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (((-2.0e0) * (u / v)) + (2.0e0 * (1.0e0 / v)))))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(-2.0) * (u / v)) + (single(2.0) * (single(1.0) / v)))));
	end
	tmp_2 = 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 95.0%

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

   if 0.200000003 < v

   1. Initial program 92.1%

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

   3. Taylor expanded in v around inf 57.3%

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

      1. fma-define 57.3%

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

      2. associate-*r/ 57.3%

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

      3. *-commutative 57.3%

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

      4. associate-/l* 57.3%

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

      5. *-commutative 57.3%

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

      6. unpow2 57.3%

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

      7. associate-*l* 57.3%

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

      8. *-commutative 57.3%

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

      9. distribute-lft-out 57.3%

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

   5. Simplified 57.3%

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

   6. Taylor expanded in u around 0 57.3%

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

4. Final simplification 91.7%

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

Alternative 8: 90.9% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (u * ((2.0e0 / v) + (2.0e0 - (2.0e0 / u))))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (u * ((single(2.0) / v) + (single(2.0) - (single(2.0) / u))));
	end
	tmp_2 = 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 95.0%

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

   if 0.200000003 < v

   1. Initial program 92.1%

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

   3. Taylor expanded in v around inf 57.3%

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

      1. fma-define 57.3%

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

      2. associate-*r/ 57.3%

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

      3. *-commutative 57.3%

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

      4. associate-/l* 57.3%

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

      5. *-commutative 57.3%

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

      6. unpow2 57.3%

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

      7. associate-*l* 57.3%

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

      8. *-commutative 57.3%

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

      9. distribute-lft-out 57.3%

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

   5. Simplified 57.3%

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

   6. Taylor expanded in u around 0 56.1%

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

   7. Taylor expanded in u around inf 56.3%

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

      1. +-commutative 56.3%

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

      2. associate--l+ 56.3%

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

      3. associate-*r/ 56.3%

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

      4. metadata-eval 56.3%

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

      5. associate-*r/ 56.3%

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

      6. metadata-eval 56.3%

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

   9. Simplified 56.3%

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

4. Final simplification 91.7%

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

Alternative 9: 91.0% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = u * (2.0e0 + ((2.0e0 / v) + ((-1.0e0) / u)))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = u * (single(2.0) + ((single(2.0) / v) + (single(-1.0) / u)));
	end
	tmp_2 = 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 95.0%

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

   if 0.200000003 < v

   1. Initial program 92.1%

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

   3. Taylor expanded in v around inf 57.3%

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

      1. fma-define 57.3%

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

      2. associate-*r/ 57.3%

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

      3. *-commutative 57.3%

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

      4. associate-/l* 57.3%

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

      5. *-commutative 57.3%

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

      6. unpow2 57.3%

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

      7. associate-*l* 57.3%

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

      8. *-commutative 57.3%

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

      9. distribute-lft-out 57.3%

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

   5. Simplified 57.3%

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

   6. Taylor expanded in u around 0 56.1%

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

   7. Taylor expanded in u around inf 56.2%

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

      1. associate--l+ 56.2%

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

      2. associate-*r/ 56.2%

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

      3. metadata-eval 56.2%

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

   9. Simplified 56.2%

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

4. Final simplification 91.6%

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

Alternative 10: 91.0% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (2.0e0 * (1.0e0 / v))))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + (single(2.0) * (single(1.0) / v))));
	end
	tmp_2 = 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 95.0%

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

   if 0.200000003 < v

   1. Initial program 92.1%

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

   3. Taylor expanded in v around inf 57.3%

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

      1. fma-define 57.3%

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

      2. associate-*r/ 57.3%

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

      3. *-commutative 57.3%

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

      4. associate-/l* 57.3%

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

      5. *-commutative 57.3%

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

      6. unpow2 57.3%

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

      7. associate-*l* 57.3%

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

      8. *-commutative 57.3%

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

      9. distribute-lft-out 57.3%

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

   5. Simplified 57.3%

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

   6. Taylor expanded in u around 0 56.2%

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

4. Final simplification 91.6%

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

Alternative 11: 91.0% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = u * (((-1.0e0) / u) - ((-2.0e0) + ((-2.0e0) / v)))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = u * ((single(-1.0) / u) - (single(-2.0) + (single(-2.0) / v)));
	end
	tmp_2 = 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 95.0%

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

   if 0.200000003 < v

   1. Initial program 92.1%

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

   3. Taylor expanded in v around inf 57.3%

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

      1. fma-define 57.3%

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

      2. associate-*r/ 57.3%

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

      3. *-commutative 57.3%

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

      4. associate-/l* 57.3%

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

      5. *-commutative 57.3%

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

      6. unpow2 57.3%

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

      7. associate-*l* 57.3%

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

      8. *-commutative 57.3%

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

      9. distribute-lft-out 57.3%

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

   5. Simplified 57.3%

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

   6. Taylor expanded in u around 0 56.1%

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

   7. Taylor expanded in u around -inf 56.2%

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

      1. mul-1-neg 56.2%

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

      2. *-commutative 56.2%

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

      3. distribute-rgt-neg-in 56.2%

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

      4. sub-neg 56.2%

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

      5. distribute-neg-in 56.2%

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

      6. metadata-eval 56.2%

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

      7. associate-*r/ 56.2%

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

      8. metadata-eval 56.2%

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

      9. distribute-neg-frac 56.2%

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

      10. metadata-eval 56.2%

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

   9. Simplified 56.2%

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

4. Final simplification 91.6%

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

Alternative 12: 90.3% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = u * (2.0e0 + ((-1.0e0) / u))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = u * (single(2.0) + (single(-1.0) / u));
	end
	tmp_2 = 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 95.0%

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

   if 0.200000003 < v

   1. Initial program 92.1%

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

   3. Taylor expanded in v around inf 48.3%

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

   4. Taylor expanded in u around inf 48.3%

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

4. Final simplification 91.0%

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

Alternative 13: 90.3% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot u\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (2.0e0 * u)
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (single(2.0) * u);
	end
	tmp_2 = 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. log-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-undefine 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 95.0%

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

   if 0.200000003 < v

   1. Initial program 92.1%

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

   3. Taylor expanded in v around inf 48.3%

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

   4. Taylor expanded in u around 0 48.3%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 6.0% accurate, 213.0× 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 99.3%

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

3. Taylor expanded in u around 0 6.4%

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

4. Final simplification 6.4%

   \[\leadsto -1 \]
5. Add Preprocessing

Alternative 15: 87.0% accurate, 213.0× 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 99.3%

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

   1. add-sqr-sqrt 99.3%

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

   2. log-prod 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-undefine 99.4%

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

   5. +-commutative 99.4%

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

   6. fma-undefine 99.4%

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

4. Applied egg-rr 99.4%

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

   1. count-2 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in v around 0 87.2%

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

8. Final simplification 87.2%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(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))))))))