HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 13.2s

Alternatives: 17

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, 0.7× 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 99.4%

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

   1. +-commutative 99.4%

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

   2. +-commutative 99.4%

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

   3. fma-undefine 99.4%

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

4. Applied egg-rr 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 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 99.4%

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

Alternative 3: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return 1.0f + (v * logf((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 + exp(((-2.0e0) / v)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-define 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-define 99.4%

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

3. Simplified 99.4%

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

   1. fma-undefine 99.4%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in u around 0 96.0%

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

   1. fma-undefine 96.0%

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

9. Applied egg-rr 96.0%

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

10. Final simplification 96.0%

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

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in u around 0 99.5%

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

   8. Taylor expanded in u around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. log-rec 99.1%

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

      3. remove-double-neg 99.1%

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

   10. Simplified 99.1%

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

   if 0.5 < v

   1. Initial program 92.2%

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

   3. Taylor expanded in u around 0 76.8%

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

4. Final simplification 97.7%

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

Alternative 5: 97.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 99.9%

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

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in u around 0 99.5%

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

   8. Taylor expanded in u around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. distribute-rgt-neg-in 99.1%

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

      3. log-rec 99.1%

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

      4. remove-double-neg 99.1%

         \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

   10. Simplified 99.1%

       \[\leadsto \color{blue}{1 + v \cdot \log u} \]

   if 0.5 < v

   1. Initial program 92.2%

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

   3. Taylor expanded in u around 0 76.8%

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

4. Final simplification 97.6%

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

Alternative 6: 97.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(u * Float32(v * 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 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%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in u around 0 99.5%

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

   8. Taylor expanded in u around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. distribute-rgt-neg-in 99.1%

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

      3. log-rec 99.1%

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

      4. remove-double-neg 99.1%

         \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

   10. Simplified 99.1%

       \[\leadsto \color{blue}{1 + v \cdot \log u} \]

   if 0.5 < v

   1. Initial program 92.2%

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

   3. Taylor expanded in u around 0 75.7%

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

   4. Taylor expanded in u around -inf 76.3%

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

      1. mul-1-neg 76.3%

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

      2. distribute-lft-in 76.6%

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

      3. rgt-mult-inverse 76.8%

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

      4. distribute-neg-in 76.8%

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

      5. mul-1-neg 76.8%

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

      6. *-commutative 76.8%

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

      7. distribute-rgt-neg-in 76.8%

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

      8. rec-exp 76.8%

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

      9. expm1-define 76.8%

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

      10. distribute-neg-frac 76.8%

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

      11. metadata-eval 76.8%

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

      12. metadata-eval 76.8%

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

   6. Simplified 76.8%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = 1.0f + ((((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / v) - (2.0f + (u * -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 <= 0.5e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = 1.0e0 + ((((((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v) - (2.0e0 + (u * (-2.0e0))))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   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%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in u around 0 99.5%

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

   8. Taylor expanded in u around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. distribute-rgt-neg-in 99.1%

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

      3. log-rec 99.1%

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

      4. remove-double-neg 99.1%

         \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

   10. Simplified 99.1%

       \[\leadsto \color{blue}{1 + v \cdot \log u} \]

   if 0.5 < v

   1. Initial program 92.2%

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

   3. Taylor expanded in u around 0 75.7%

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

   4. Taylor expanded in v around -inf 68.3%

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

4. Final simplification 97.1%

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

Alternative 8: 91.1% accurate, 7.1× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / v) - (2.0f + (u * -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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((((((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v) - (2.0e0 + (u * (-2.0e0))))
    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(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) + Float32(u * Float32(-2.0)))));
	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(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) - (single(2.0) + (u * single(-2.0))));
	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. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 92.9%

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

   3. Taylor expanded in u around 0 65.6%

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

   4. Taylor expanded in v around -inf 62.6%

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

4. Final simplification 90.5%

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

Alternative 9: 91.0% accurate, 9.7× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * ((2.0f + (1.3333333333333333f / v)) / v)) - (2.0f + (u * -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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * ((2.0e0 + (1.3333333333333333e0 / v)) / v)) - (2.0e0 + (u * (-2.0e0))))
    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(u * Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v)) - Float32(Float32(2.0) + Float32(u * Float32(-2.0)))));
	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(1.3333333333333333) / v)) / v)) - (single(2.0) + (u * single(-2.0))));
	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. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 92.9%

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

   3. Taylor expanded in u around 0 65.6%

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

   4. Taylor expanded in v around -inf 60.7%

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

   5. Taylor expanded in u around 0 60.7%

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

      1. mul-1-neg 60.7%

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

      2. associate-/l* 60.7%

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

      3. distribute-rgt-neg-in 60.7%

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

      4. distribute-neg-frac2 60.7%

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

      5. associate-*r/ 60.7%

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

      6. metadata-eval 60.7%

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

   7. Simplified 60.7%

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

4. Final simplification 90.4%

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

Alternative 10: 91.0% accurate, 9.7× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (-
    1.0
    (* u (+ -2.0 (- (/ 2.0 u) (/ (+ 2.0 (/ 1.3333333333333333 v)) 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) - ((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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 - (u * ((-2.0e0) + ((2.0e0 / u) - ((2.0e0 + (1.3333333333333333e0 / v)) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) - Float32(u * Float32(Float32(-2.0) + Float32(Float32(Float32(2.0) / u) - 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(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) - (u * (single(-2.0) + ((single(2.0) / u) - ((single(2.0) + (single(1.3333333333333333) / v)) / 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. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 92.9%

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

   3. Taylor expanded in u around 0 65.6%

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

   4. Taylor expanded in v around -inf 60.7%

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

   5. Taylor expanded in u around -inf 60.7%

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

      1. mul-1-neg 60.7%

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

      2. distribute-rgt-neg-in 60.7%

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

      3. sub-neg 60.7%

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

      4. +-commutative 60.7%

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

      5. mul-1-neg 60.7%

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

      6. unsub-neg 60.7%

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

      7. associate-*r/ 60.7%

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

      8. metadata-eval 60.7%

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

      9. associate-*r/ 60.7%

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

      10. metadata-eval 60.7%

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

      11. metadata-eval 60.7%

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

   7. Simplified 60.7%

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

4. Final simplification 90.4%

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

Alternative 11: 91.0% accurate, 10.6× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = u * ((((2.0f + (1.3333333333333333f / v)) / v) + (-1.0f / u)) - -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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = u * ((((2.0e0 + (1.3333333333333333e0 / v)) / v) + ((-1.0e0) / u)) - (-2.0e0))
    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(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v) + Float32(Float32(-1.0) / u)) - Float32(-2.0)));
	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(1.3333333333333333) / v)) / v) + (single(-1.0) / u)) - single(-2.0));
	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. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 92.9%

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

   3. Taylor expanded in u around 0 65.6%

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

   4. Taylor expanded in v around -inf 60.7%

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

   5. Taylor expanded in u around -inf 60.6%

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

      1. mul-1-neg 60.6%

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

      2. *-commutative 60.6%

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

      3. distribute-rgt-neg-in 60.6%

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

      4. sub-neg 60.6%

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

      5. +-commutative 60.6%

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

      6. mul-1-neg 60.6%

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

      7. unsub-neg 60.6%

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

      8. associate-*r/ 60.6%

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

      9. metadata-eval 60.6%

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

      10. metadata-eval 60.6%

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

   7. Simplified 60.6%

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

4. Final simplification 90.4%

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

Alternative 12: 90.8% accurate, 11.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (- 1.0 (* u (- -2.0 (- (/ 2.0 v) (/ 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 - ((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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 - (u * ((-2.0e0) - ((2.0e0 / v) - (2.0e0 / u))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) - Float32(u * Float32(Float32(-2.0) - Float32(Float32(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(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) - (u * (single(-2.0) - ((single(2.0) / v) - (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. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 92.9%

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

   3. Taylor expanded in u around 0 65.6%

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

   4. Taylor expanded in v around -inf 60.7%

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

   5. Taylor expanded in u around -inf 60.7%

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

      1. mul-1-neg 60.7%

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

      2. distribute-rgt-neg-in 60.7%

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

      3. sub-neg 60.7%

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

      4. +-commutative 60.7%

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

      5. mul-1-neg 60.7%

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

      6. unsub-neg 60.7%

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

      7. associate-*r/ 60.7%

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

      8. metadata-eval 60.7%

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

      9. associate-*r/ 60.7%

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

      10. metadata-eval 60.7%

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

      11. metadata-eval 60.7%

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

   7. Simplified 60.7%

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

   8. Taylor expanded in v around inf 58.5%

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

4. Final simplification 90.2%

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

Alternative 13: 90.8% accurate, 13.3× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ -1.0 (+ (* u 2.0) (* 2.0 (/ u 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)));
	}
	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)))
    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(u * Float32(2.0)) + Float32(Float32(2.0) * Float32(u / 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)));
	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. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 92.9%

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

   3. Taylor expanded in u around 0 65.6%

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

   4. Taylor expanded in v around inf 58.5%

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

4. Final simplification 90.2%

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

Alternative 14: 90.2% accurate, 15.2× speedup

Math

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

FPCore

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

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)));
	}
	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)))
    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) / 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) - (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. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 92.9%

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

   3. Taylor expanded in u around 0 65.6%

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

   4. Taylor expanded in v around -inf 60.7%

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

   5. Taylor expanded in u around -inf 60.7%

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

      1. mul-1-neg 60.7%

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

      2. distribute-rgt-neg-in 60.7%

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

      3. sub-neg 60.7%

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

      4. +-commutative 60.7%

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

      5. mul-1-neg 60.7%

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

      6. unsub-neg 60.7%

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

      7. associate-*r/ 60.7%

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

      8. metadata-eval 60.7%

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

      9. associate-*r/ 60.7%

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

      10. metadata-eval 60.7%

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

      11. metadata-eval 60.7%

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

   7. Simplified 60.7%

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

   8. Taylor expanded in v around inf 52.8%

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

      1. associate-*r/ 52.8%

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

      2. metadata-eval 52.8%

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

   10. Simplified 52.8%

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

Alternative 15: 90.2% accurate, 21.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * 2.0e0)
    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(2.0)));
	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));
	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. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 92.9%

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

   3. Taylor expanded in u around 0 65.6%

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

   4. Taylor expanded in v around inf 52.7%

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

4. Final simplification 89.8%

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

Alternative 16: 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 99.4%

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

   1. +-commutative 99.4%

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

   2. +-commutative 99.4%

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

   3. fma-undefine 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in v around 0 86.1%

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

Alternative 17: 5.9% 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 99.4%

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

3. Taylor expanded in u around 0 6.4%

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

Reproduce

herbie shell --seed 2024096 
(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))))))))