HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 4.9s

Alternatives: 14

Speedup: 0.9×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(u, v)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(u, v)
use fmin_fmax_functions
    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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(fma(Float32(Float32(1.0) - u), (Float32(exp(1)) ^ Float32(Float32(-2.0) / v)), u)), Float32(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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 99.5

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

   5. lift-+.f32 N/A

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

   6. +-commutative N/A

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

   7. add-flip N/A

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

   8. sub-flip N/A

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

   9. lift-*.f32 N/A

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

   10. remove-double-neg N/A

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

   11. lower-fma.f32 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. Applied rewrites 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. Step-by-step derivation

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. mult-flip N/A

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

   11. *-commutative N/A

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

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

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

   13. distribute-rgt-neg-out N/A

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

   14. remove-double-neg N/A

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

   15. mult-flip N/A

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

   16. associate-*l* N/A

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

   17. *-commutative N/A

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

   18. mult-flip N/A

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

   19. lift-/.f32 N/A

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

   20. exp-prod N/A

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

   21. lower-pow.f32 N/A

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

   22. lower-exp.f32 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-exp.f32 N/A

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

   2. exp-1-e N/A

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

   3. lower-E.f32 99.5

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

7. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(fma(Float32(Float32(1.0) - u), exp(Float32(Float32(-2.0) / v)), u)), Float32(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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 99.5

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

   5. lift-+.f32 N/A

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

   6. +-commutative N/A

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

   7. add-flip N/A

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

   8. sub-flip N/A

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

   9. lift-*.f32 N/A

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

   10. remove-double-neg N/A

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

   11. lower-fma.f32 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. Applied rewrites 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

Alternative 3: 97.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lower-fma.f32 99.5

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

      5. lift-+.f32 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. sub-flip N/A

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

      9. lift-*.f32 N/A

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

      10. remove-double-neg N/A

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

      11. lower-fma.f32 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. Applied rewrites 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. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f32 N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-/.f32 N/A

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

      10. mult-flip N/A

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

      11. *-commutative N/A

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

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

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

      13. distribute-rgt-neg-out N/A

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

      14. remove-double-neg N/A

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

      15. mult-flip N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. mult-flip N/A

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

      19. lift-/.f32 N/A

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

      20. exp-prod N/A

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

      21. lower-pow.f32 N/A

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

      22. lower-exp.f32 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in u around 0

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

      1. lower--.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-exp.f32 N/A

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

      7. lower-/.f32 10.3

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

   8. Applied rewrites 10.3%

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

      1. lift--.f32 N/A

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

      2. metadata-eval N/A

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

      3. add-flip N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f32 N/A

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

   10. Applied rewrites 10.3%

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

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

   1. Initial program 99.5%

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

   2. Taylor expanded in u around -inf

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-expm1.f32 N/A

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

      4. lower-/.f32 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f32 94.6

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

   6. Applied rewrites 94.6%

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

Alternative 4: 97.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lower-fma.f32 99.5

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

      5. lift-+.f32 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. sub-flip N/A

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

      9. lift-*.f32 N/A

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

      10. remove-double-neg N/A

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

      11. lower-fma.f32 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. Applied rewrites 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. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f32 N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-/.f32 N/A

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

      10. mult-flip N/A

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

      11. *-commutative N/A

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

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

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

      13. distribute-rgt-neg-out N/A

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

      14. remove-double-neg N/A

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

      15. mult-flip N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. mult-flip N/A

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

      19. lift-/.f32 N/A

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

      20. exp-prod N/A

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

      21. lower-pow.f32 N/A

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

      22. lower-exp.f32 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in u around 0

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

      1. lower--.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-exp.f32 N/A

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

      7. lower-/.f32 10.3

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

   8. Applied rewrites 10.3%

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

      1. lift--.f32 N/A

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

      2. metadata-eval N/A

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

      3. add-flip N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f32 N/A

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

   10. Applied rewrites 10.3%

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

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

   1. Initial program 99.5%

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

   2. Taylor expanded in u around -inf

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-expm1.f32 N/A

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

      4. lower-/.f32 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 94.6

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

      6. lift-*.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f32 N/A

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

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

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 94.6

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

   6. Applied rewrites 94.6%

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

Alternative 5: 96.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return Float32(Float32(log(fma(Float32(1.0), exp(Float32(Float32(-2.0) / v)), u)) * v) - Float32(-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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. lower--.f32 99.5

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 99.5

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. add-flip N/A

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

   12. sub-flip N/A

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

   13. lift-*.f32 N/A

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

   14. remove-double-neg N/A

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

   15. lower-fma.f32 99.5

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in u around 0

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

6. Applied rewrites 96.1%

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

Alternative 6: 96.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(fma(Float32(1.0), exp(Float32(Float32(-2.0) / v)), u)), Float32(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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 99.5

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

   5. lift-+.f32 N/A

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

   6. +-commutative N/A

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

   7. add-flip N/A

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

   8. sub-flip N/A

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

   9. lift-*.f32 N/A

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

   10. remove-double-neg N/A

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

   11. lower-fma.f32 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. Applied rewrites 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. Taylor expanded in u around 0

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

6. Applied rewrites 96.1%

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

Alternative 7: 90.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lower-fma.f32 99.5

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

      5. lift-+.f32 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. sub-flip N/A

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

      9. lift-*.f32 N/A

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

      10. remove-double-neg N/A

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

      11. lower-fma.f32 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. Applied rewrites 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. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f32 N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-/.f32 N/A

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

      10. mult-flip N/A

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

      11. *-commutative N/A

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

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

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

      13. distribute-rgt-neg-out N/A

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

      14. remove-double-neg N/A

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

      15. mult-flip N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. mult-flip N/A

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

      19. lift-/.f32 N/A

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

      20. exp-prod N/A

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

      21. lower-pow.f32 N/A

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

      22. lower-exp.f32 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in u around 0

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

      1. lower--.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-exp.f32 N/A

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

      7. lower-/.f32 10.3

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

   8. Applied rewrites 10.3%

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

      1. lift--.f32 N/A

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

      2. metadata-eval N/A

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

      3. add-flip N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f32 N/A

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

   10. Applied rewrites 10.3%

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

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

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f32 87.1

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

   4. Applied rewrites 87.1%

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

Alternative 8: 90.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{fma}\left(2, u, 2 \cdot \frac{u}{v}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \log \left(u + \left(1 - u\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<=
      (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))))
      -0.10000000149011612)
   (- (fma 2.0 u (* 2.0 (/ u v))) 1.0)
   (+ 1.0 (* v (log (+ u (- 1.0 u)))))))

C

float code(float u, float v) {
	float tmp;
	if ((1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))))) <= -0.10000000149011612f) {
		tmp = fmaf(2.0f, u, (2.0f * (u / v))) - 1.0f;
	} else {
		tmp = 1.0f + (v * logf((u + (1.0f - u))));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v))))))) <= Float32(-0.10000000149011612))
		tmp = Float32(fma(Float32(2.0), u, Float32(Float32(2.0) * Float32(u / v))) - Float32(1.0));
	else
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(1.0) - u)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))) < -0.100000001

   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. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lower-fma.f32 99.5

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

      5. lift-+.f32 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. sub-flip N/A

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

      9. lift-*.f32 N/A

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

      10. remove-double-neg N/A

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

      11. lower-fma.f32 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. Applied rewrites 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. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f32 N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-/.f32 N/A

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

      10. mult-flip N/A

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

      11. *-commutative N/A

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

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

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

      13. distribute-rgt-neg-out N/A

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

      14. remove-double-neg N/A

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

      15. mult-flip N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. mult-flip N/A

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

      19. lift-/.f32 N/A

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

      20. exp-prod N/A

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

      21. lower-pow.f32 N/A

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

      22. lower-exp.f32 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in u around 0

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

      1. lower--.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-exp.f32 N/A

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

      7. lower-/.f32 10.3

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

   8. Applied rewrites 10.3%

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

   9. Taylor expanded in v around inf

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

       1. lower--.f32 N/A

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

       2. lower-fma.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-/.f32 14.0

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

   11. Applied rewrites 14.0%

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

   if -0.100000001 < (+.f32 #s(literal 1 binary32) (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))))

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

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

      1. lower--.f32 87.1

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

   4. Applied rewrites 87.1%

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

Alternative 9: 50.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lower-fma.f32 99.5

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

      5. lift-+.f32 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. sub-flip N/A

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

      9. lift-*.f32 N/A

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

      10. remove-double-neg N/A

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

      11. lower-fma.f32 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. Applied rewrites 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. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f32 N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-/.f32 N/A

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

      10. mult-flip N/A

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

      11. *-commutative N/A

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

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

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

      13. distribute-rgt-neg-out N/A

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

      14. remove-double-neg N/A

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

      15. mult-flip N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. mult-flip N/A

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

      19. lift-/.f32 N/A

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

      20. exp-prod N/A

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

      21. lower-pow.f32 N/A

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

      22. lower-exp.f32 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in u around 0

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

      1. lower--.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-exp.f32 N/A

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

      7. lower-/.f32 10.3

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

   8. Applied rewrites 10.3%

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

   9. Taylor expanded in v around inf

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

       1. lower--.f32 N/A

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

       2. lower-fma.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-/.f32 14.0

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

   11. Applied rewrites 14.0%

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

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

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 7.8

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

   4. Applied rewrites 7.8%

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

   5. Taylor expanded in u around inf

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

      1. lower-*.f32 46.7

         \[\leadsto 1 + 2 \cdot u \]

   7. Applied rewrites 46.7%

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

   9. Applied rewrites 46.7%

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

Alternative 10: 50.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   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. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lower-fma.f32 99.5

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

      5. lift-+.f32 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. sub-flip N/A

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

      9. lift-*.f32 N/A

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

      10. remove-double-neg N/A

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

      11. lower-fma.f32 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. Applied rewrites 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. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f32 N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-/.f32 N/A

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

      10. mult-flip N/A

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

      11. *-commutative N/A

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

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

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

      13. distribute-rgt-neg-out N/A

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

      14. remove-double-neg N/A

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

      15. mult-flip N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. mult-flip N/A

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

      19. lift-/.f32 N/A

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

      20. exp-prod N/A

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

      21. lower-pow.f32 N/A

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

      22. lower-exp.f32 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in u around 0

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

      1. lower--.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-exp.f32 N/A

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

      7. lower-/.f32 10.3

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

   8. Applied rewrites 10.3%

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

   9. Taylor expanded in v around inf

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

       1. lower--.f32 N/A

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

       2. lower-fma.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-/.f32 14.0

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

   11. Applied rewrites 14.0%

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

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

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 7.8

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

   4. Applied rewrites 7.8%

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

   5. Taylor expanded in u around inf

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

      1. lower-*.f32 46.7

         \[\leadsto 1 + 2 \cdot u \]

   7. Applied rewrites 46.7%

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

   9. Applied rewrites 46.7%

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

Alternative 11: 49.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.10000000149011612:\\ \;\;\;\;\left(\left(u + u\right) + -2\right) - -1\\ \mathbf{else}:\\ \;\;\;\;\left(u + u\right) - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<=
      (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))))
      -0.10000000149011612)
   (- (+ (+ u u) -2.0) -1.0)
   (- (+ u u) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if ((1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))))) <= -0.10000000149011612f) {
		tmp = ((u + u) + -2.0f) - -1.0f;
	} else {
		tmp = (u + u) - -1.0f;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(u, v)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if ((1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))) <= (-0.10000000149011612e0)) then
        tmp = ((u + u) + (-2.0e0)) - (-1.0e0)
    else
        tmp = (u + u) - (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v))))))) <= Float32(-0.10000000149011612))
		tmp = Float32(Float32(Float32(u + u) + Float32(-2.0)) - Float32(-1.0));
	else
		tmp = Float32(Float32(u + u) - Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))))) <= single(-0.10000000149011612))
		tmp = ((u + u) + single(-2.0)) - single(-1.0);
	else
		tmp = (u + u) - single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))) < -0.100000001

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 7.8

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

   4. Applied rewrites 7.8%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f32 7.8

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

   6. Applied rewrites 7.8%

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

   if -0.100000001 < (+.f32 #s(literal 1 binary32) (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))))

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 7.8

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

   4. Applied rewrites 7.8%

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

   5. Taylor expanded in u around inf

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

      1. lower-*.f32 46.7

         \[\leadsto 1 + 2 \cdot u \]

   7. Applied rewrites 46.7%

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

   9. Applied rewrites 46.7%

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

Alternative 12: 49.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \leq -0.10000000149011612:\\ \;\;\;\;u \cdot 2 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(u + u\right) - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<=
      (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))))
      -0.10000000149011612)
   (- (* u 2.0) 1.0)
   (- (+ u u) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if ((1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))))) <= -0.10000000149011612f) {
		tmp = (u * 2.0f) - 1.0f;
	} else {
		tmp = (u + u) - -1.0f;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(u, v)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if ((1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))) <= (-0.10000000149011612e0)) then
        tmp = (u * 2.0e0) - 1.0e0
    else
        tmp = (u + u) - (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v))))))) <= Float32(-0.10000000149011612))
		tmp = Float32(Float32(u * Float32(2.0)) - Float32(1.0));
	else
		tmp = Float32(Float32(u + u) - Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if ((single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))))) <= single(-0.10000000149011612))
		tmp = (u * single(2.0)) - single(1.0);
	else
		tmp = (u + u) - single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))) < -0.100000001

   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. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lower-fma.f32 99.5

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

      5. lift-+.f32 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. sub-flip N/A

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

      9. lift-*.f32 N/A

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

      10. remove-double-neg N/A

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

      11. lower-fma.f32 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. Applied rewrites 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. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f32 N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-/.f32 N/A

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

      10. mult-flip N/A

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

      11. *-commutative N/A

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

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

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

      13. distribute-rgt-neg-out N/A

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

      14. remove-double-neg N/A

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

      15. mult-flip N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. mult-flip N/A

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

      19. lift-/.f32 N/A

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

      20. exp-prod N/A

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

      21. lower-pow.f32 N/A

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

      22. lower-exp.f32 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in u around 0

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

      1. lower--.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-exp.f32 N/A

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

      7. lower-/.f32 10.3

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

   8. Applied rewrites 10.3%

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

   9. Taylor expanded in v around inf

      \[\leadsto u \cdot 2 - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 7.8%

       \[\leadsto u \cdot 2 - 1 \]

   if -0.100000001 < (+.f32 #s(literal 1 binary32) (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))))

   1. Initial program 99.5%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 7.8

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

   4. Applied rewrites 7.8%

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

   5. Taylor expanded in u around inf

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

      1. lower-*.f32 46.7

         \[\leadsto 1 + 2 \cdot u \]

   7. Applied rewrites 46.7%

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

   9. Applied rewrites 46.7%

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

Alternative 13: 46.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \left(u + u\right) - -1 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (- (+ u u) -1.0))

C

float code(float u, float v) {
	return (u + u) - -1.0f;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(u, v)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (u + u) - (-1.0e0)
end function

Julia

function code(u, v)
	return Float32(Float32(u + u) - Float32(-1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (u + u) - 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. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 7.8

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

4. Applied rewrites 7.8%

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

5. Taylor expanded in u around inf

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

   1. lower-*.f32 46.7

      \[\leadsto 1 + 2 \cdot u \]

7. Applied rewrites 46.7%

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

9. Applied rewrites 46.7%

   \[\leadsto \color{blue}{\left(u + u\right) - -1} \]
10. Add Preprocessing

Alternative 14: 5.7% accurate, 34.9× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(u, v)
use fmin_fmax_functions
    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. Taylor expanded in u around 0

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

4. Applied rewrites 5.7%

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

Reproduce

herbie shell --seed 2025151 
(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))))))))