HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 13.3s

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.5%

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

   2. log-prod 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-undefine 99.5%

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

   5. +-commutative 99.5%

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

   6. fma-undefine 99.5%

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

4. Applied egg-rr 99.5%

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

   1. count-2 99.5%

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

6. Simplified 99.5%

   \[\leadsto 1 + v \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right)\right)} \]
7. 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 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

Alternative 4: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(v, u \cdot \mathsf{expm1}\left(\frac{2}{v}\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5) 1.0 (fma v (* u (expm1 (/ 2.0 v))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(v, (u * expm1f((2.0f / v))), -1.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = Float32(1.0);
	else
		tmp = fma(v, Float32(u * expm1(Float32(Float32(2.0) / v))), Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   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 92.6%

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

   if 0.5 < v

   1. Initial program 91.4%

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

   3. Taylor expanded in u around 0 81.6%

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

      1. *-commutative 81.6%

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

      2. associate-*l* 81.6%

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

      3. *-commutative 81.6%

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

      4. fma-neg 81.6%

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

      5. rec-exp 81.6%

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

      6. expm1-define 81.6%

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

      7. distribute-neg-frac 81.6%

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

      8. metadata-eval 81.6%

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

      9. metadata-eval 81.6%

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

   5. Simplified 81.6%

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

Alternative 5: 90.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (v * (-1.0f + expf((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 <= 0.5e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (v * ((-1.0e0) + exp((2.0e0 / v)))))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   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 92.6%

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

   if 0.5 < v

   1. Initial program 91.4%

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

   3. Taylor expanded in u around 0 81.6%

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

      1. *-commutative 81.6%

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

      2. associate-*l* 81.6%

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

      3. *-commutative 81.6%

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

      4. fma-neg 81.6%

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

      5. rec-exp 81.6%

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

      6. expm1-define 81.6%

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

      7. distribute-neg-frac 81.6%

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

      8. metadata-eval 81.6%

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

      9. metadata-eval 81.6%

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

   5. Simplified 81.6%

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

   6. Taylor expanded in v around 0 81.6%

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

4. Final simplification 92.0%

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

Alternative 6: 90.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + 2 \cdot u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    -1.0
    (+
     (/
      (-
       (/ (+ (* 0.6666666666666666 (/ u v)) (* u 1.3333333333333333)) v)
       (* u -2.0))
      v)
     (* 2.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / v) + (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 <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((((((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v) + (2.0e0 * u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u / v)) + Float32(u * Float32(1.3333333333333333))) / v) - Float32(u * Float32(-2.0))) / v) + Float32(Float32(2.0) * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((((((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) + (single(2.0) * u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   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 93.3%

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

   if 0.150000006 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 72.0%

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

   4. Taylor expanded in v around -inf 70.6%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + 2 \cdot u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(2 \cdot u - \frac{u \cdot -2 + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    -1.0
    (- (* 2.0 u) (/ (+ (* u -2.0) (* (/ u v) -1.3333333333333333)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((2.0f * u) - (((u * -2.0f) + ((u / v) * -1.3333333333333333f)) / 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 <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((2.0e0 * u) - (((u * (-2.0e0)) + ((u / v) * (-1.3333333333333333e0))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(Float32(2.0) * u) - Float32(Float32(Float32(u * Float32(-2.0)) + Float32(Float32(u / v) * Float32(-1.3333333333333333))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((single(2.0) * u) - (((u * single(-2.0)) + ((u / v) * single(-1.3333333333333333))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   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 93.3%

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

   if 0.150000006 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 72.0%

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

   4. Taylor expanded in v around -inf 70.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(2 \cdot u - \frac{u \cdot -2 + \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.5% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(\frac{-1}{u} - -2\right) - \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (* u (- (- (/ -1.0 u) -2.0) (/ (+ -2.0 (/ -1.3333333333333333 v)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = u * (((-1.0f / u) - -2.0f) - ((-2.0f + (-1.3333333333333333f / v)) / 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 <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = u * ((((-1.0e0) / u) - (-2.0e0)) - (((-2.0e0) + ((-1.3333333333333333e0) / v)) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(u * Float32(Float32(Float32(Float32(-1.0) / u) - Float32(-2.0)) - Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / v)) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = u * (((single(-1.0) / u) - single(-2.0)) - ((single(-2.0) + (single(-1.3333333333333333) / v)) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   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 93.3%

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

   if 0.150000006 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 72.0%

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

   4. Taylor expanded in v around -inf 70.0%

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

   5. Taylor expanded in u around -inf 69.9%

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

      1. mul-1-neg 69.9%

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

      2. *-commutative 69.9%

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

      3. distribute-rgt-neg-in 69.9%

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

      4. associate--l+ 69.9%

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

      5. associate-*r/ 69.9%

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

      6. associate-*r/ 69.9%

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

      7. metadata-eval 69.9%

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

      8. distribute-lft-in 69.9%

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

      9. metadata-eval 69.9%

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

      10. neg-mul-1 69.9%

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

      11. distribute-neg-frac 69.9%

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

      12. metadata-eval 69.9%

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

      13. sub-neg 69.9%

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

      14. metadata-eval 69.9%

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

   7. Simplified 69.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(\frac{-1}{u} - -2\right) - \frac{-2 + \frac{-1.3333333333333333}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.3% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \frac{2 + \frac{2}{v}}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+ 1.0 (- (* u (* v (/ (+ 2.0 (/ 2.0 v)) v))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (v * ((2.0f + (2.0f / v)) / v))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (v * ((2.0e0 + (2.0e0 / v)) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(v * Float32(Float32(Float32(2.0) + Float32(Float32(2.0) / v)) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (v * ((single(2.0) + (single(2.0) / v)) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   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 93.3%

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

   if 0.150000006 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 72.0%

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

   4. Taylor expanded in v around inf 65.6%

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

      1. associate-*r/ 65.6%

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

      2. metadata-eval 65.6%

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

   6. Simplified 65.6%

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

Alternative 10: 90.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448) 1.0 (+ 1.0 (- (* u (+ 2.0 (/ 2.0 v))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f + (2.0f / v))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 + (2.0e0 / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(2.0) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) + (single(2.0) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   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 93.3%

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

   if 0.150000006 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 72.0%

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

   4. Taylor expanded in v around -inf 70.0%

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

   5. Taylor expanded in v around inf 65.6%

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

      1. associate-*r/ 65.6%

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

      2. metadata-eval 65.6%

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

   7. Simplified 65.6%

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

Alternative 11: 90.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(2 \cdot \left(u + \frac{u}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448) 1.0 (+ 1.0 (- (* 2.0 (+ u (/ u v))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((2.0f * (u + (u / v))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((2.0e0 * (u + (u / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(2.0) * Float32(u + Float32(u / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(2.0) * (u + (u / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   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 93.3%

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

   if 0.150000006 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 72.0%

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

   4. Taylor expanded in v around inf 65.6%

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

      1. distribute-lft-out 65.6%

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

   6. Simplified 65.6%

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

Alternative 12: 90.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448) 1.0 (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (2.0f * (u + (u / 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 <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (2.0e0 * (u + (u / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(2.0) * Float32(u + Float32(u / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (single(2.0) * (u + (u / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   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 93.3%

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

   if 0.150000006 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 72.0%

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

   4. Taylor expanded in v around inf 65.5%

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

      1. sub-neg 65.5%

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

      2. distribute-lft-out 65.5%

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

      3. metadata-eval 65.5%

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

   6. Simplified 65.5%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.7% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448) 1.0 (+ -1.0 (* 2.0 u))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		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 <= 0.15000000596046448e0) 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(0.15000000596046448))
		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(0.15000000596046448))
		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 < 0.150000006

   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 93.3%

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

   if 0.150000006 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 72.0%

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

   4. Taylor expanded in v around inf 54.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 86.9% 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 99.5%

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

   1. add-sqr-sqrt 99.5%

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

   2. log-prod 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-undefine 99.5%

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

   5. +-commutative 99.5%

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

   6. fma-undefine 99.5%

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

4. Applied egg-rr 99.5%

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

   1. count-2 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in v around 0 87.7%

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

Alternative 15: 5.7% 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 99.5%

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

3. Taylor expanded in u around 0 5.9%

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

Reproduce

herbie shell --seed 2024102 
(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))))))))