Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 60.7% → 97.8%

Time: 9.1s

Alternatives: 5

Speedup: 11.4×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Initial Program: 60.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Alternative 1: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.009999999776482582:\\ \;\;\;\;s \cdot \left(\left(\left(\frac{\left(\frac{4}{u \cdot u} + \frac{8}{u}\right) + 21.333333333333332}{u} + 64\right) \cdot {u}^{3}\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.009999999776482582)
   (*
    s
    (*
     (*
      (+ (/ (+ (+ (/ 4.0 (* u u)) (/ 8.0 u)) 21.333333333333332) u) 64.0)
      (pow u 3.0))
     u))
   (* s (log (/ 1.0 (- 1.0 (* 4.0 u)))))))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.009999999776482582f) {
		tmp = s * (((((((4.0f / (u * u)) + (8.0f / u)) + 21.333333333333332f) / u) + 64.0f) * powf(u, 3.0f)) * u);
	} else {
		tmp = s * logf((1.0f / (1.0f - (4.0f * u))));
	}
	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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if (u <= 0.009999999776482582e0) then
        tmp = s * (((((((4.0e0 / (u * u)) + (8.0e0 / u)) + 21.333333333333332e0) / u) + 64.0e0) * (u ** 3.0e0)) * u)
    else
        tmp = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.009999999776482582))
		tmp = Float32(s * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(4.0) / Float32(u * u)) + Float32(Float32(8.0) / u)) + Float32(21.333333333333332)) / u) + Float32(64.0)) * (u ^ Float32(3.0))) * u));
	else
		tmp = Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if (u <= single(0.009999999776482582))
		tmp = s * (((((((single(4.0) / (u * u)) + (single(8.0) / u)) + single(21.333333333333332)) / u) + single(64.0)) * (u ^ single(3.0))) * u);
	else
		tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00999999978

   1. Initial program 56.4%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around 0

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

      1. lower-*.f32 81.4

         \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]

   5. Applied rewrites 81.4%

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

   7. Applied rewrites 81.4%

      \[\leadsto s \cdot \left(\sqrt{4 \cdot u} \cdot \color{blue}{\sqrt{4 \cdot u}}\right) \]

   8. Taylor expanded in u around 0

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      3. +-commutative N/A

         \[\leadsto s \cdot \left(\color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)} \cdot u\right) \]

      4. *-commutative N/A

         \[\leadsto s \cdot \left(\left(\color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) \cdot u} + 4\right) \cdot u\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), u, 4\right)} \cdot u\right) \]

      6. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, u, 4\right) \cdot u\right) \]

      7. *-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{64}{3} + 64 \cdot u\right) \cdot u} + 8, u, 4\right) \cdot u\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64}{3} + 64 \cdot u, u, 8\right)}, u, 4\right) \cdot u\right) \]

      9. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{64 \cdot u + \frac{64}{3}}, u, 8\right), u, 4\right) \cdot u\right) \]

      10. lower-fma.f32 81.4

          \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(64, u, 21.333333333333332\right)}, u, 8\right), u, 4\right) \cdot u\right) \]

   10. Applied rewrites 81.2%

       \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]

   11. Taylor expanded in u around -inf

       \[\leadsto s \cdot \left(\left(-1 \cdot \left({u}^{3} \cdot \left(-1 \cdot \frac{\frac{64}{3} + \left(8 \cdot \frac{1}{u} + \frac{4}{{u}^{2}}\right)}{u} - 64\right)\right)\right) \cdot u\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 98.3%

       \[\leadsto s \cdot \left(\left(\left(-\left(\frac{\left(\frac{4}{u \cdot u} + \frac{8}{u}\right) + 21.333333333333332}{-u} - 64\right)\right) \cdot {u}^{3}\right) \cdot u\right) \]

   if 0.00999999978 < u

   1. Initial program 96.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 0.009999999776482582:\\ \;\;\;\;s \cdot \left(\left(\left(\frac{\left(\frac{4}{u \cdot u} + \frac{8}{u}\right) + 21.333333333333332}{u} + 64\right) \cdot {u}^{3}\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.00800000037997961:\\ \;\;\;\;s \cdot \left(\left(\left(\left(\frac{4}{{u}^{3}} + \frac{8}{u \cdot u}\right) + \frac{21.333333333333332}{u}\right) + 64\right) \cdot {u}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\log \left(\mathsf{fma}\left(-16, u \cdot u, 1\right)\right) - \log \left(\left(-4\right) \cdot u - -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.00800000037997961)
   (*
    s
    (*
     (+
      (+ (+ (/ 4.0 (pow u 3.0)) (/ 8.0 (* u u))) (/ 21.333333333333332 u))
      64.0)
     (pow u 4.0)))
   (* s (- (log (fma -16.0 (* u u) 1.0)) (log (- (* (- 4.0) u) -1.0))))))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.00800000037997961f) {
		tmp = s * (((((4.0f / powf(u, 3.0f)) + (8.0f / (u * u))) + (21.333333333333332f / u)) + 64.0f) * powf(u, 4.0f));
	} else {
		tmp = s * (logf(fmaf(-16.0f, (u * u), 1.0f)) - logf(((-4.0f * u) - -1.0f)));
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.00800000037997961))
		tmp = Float32(s * Float32(Float32(Float32(Float32(Float32(Float32(4.0) / (u ^ Float32(3.0))) + Float32(Float32(8.0) / Float32(u * u))) + Float32(Float32(21.333333333333332) / u)) + Float32(64.0)) * (u ^ Float32(4.0))));
	else
		tmp = Float32(s * Float32(log(fma(Float32(-16.0), Float32(u * u), Float32(1.0))) - log(Float32(Float32(Float32(-Float32(4.0)) * u) - Float32(-1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00800000038

   1. Initial program 56.2%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around 0

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

      1. lower-*.f32 81.6

         \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 81.5%

      \[\leadsto s \cdot \left(\sqrt{4 \cdot u} \cdot \color{blue}{\sqrt{4 \cdot u}}\right) \]

   8. Taylor expanded in u around 0

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      3. +-commutative N/A

         \[\leadsto s \cdot \left(\color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)} \cdot u\right) \]

      4. *-commutative N/A

         \[\leadsto s \cdot \left(\left(\color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) \cdot u} + 4\right) \cdot u\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), u, 4\right)} \cdot u\right) \]

      6. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, u, 4\right) \cdot u\right) \]

      7. *-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{64}{3} + 64 \cdot u\right) \cdot u} + 8, u, 4\right) \cdot u\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64}{3} + 64 \cdot u, u, 8\right)}, u, 4\right) \cdot u\right) \]

      9. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{64 \cdot u + \frac{64}{3}}, u, 8\right), u, 4\right) \cdot u\right) \]

      10. lower-fma.f32 81.6

          \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(64, u, 21.333333333333332\right)}, u, 8\right), u, 4\right) \cdot u\right) \]

   10. Applied rewrites 81.4%

       \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]

   11. Taylor expanded in u around inf

       \[\leadsto s \cdot \left({u}^{4} \cdot \color{blue}{\left(64 + \left(\frac{64}{3} \cdot \frac{1}{u} + \left(4 \cdot \frac{1}{{u}^{3}} + \frac{8}{{u}^{2}}\right)\right)\right)}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 98.6%

       \[\leadsto s \cdot \left(\left(\left(\left(\frac{4}{{u}^{3}} + \frac{8}{u \cdot u}\right) + \frac{21.333333333333332}{u}\right) + 64\right) \cdot \color{blue}{{u}^{4}}\right) \]

   if 0.00800000038 < u

   1. Initial program 96.3%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 39.8%

      \[\leadsto s \cdot \color{blue}{\left(\log \left(1 \cdot \mathsf{fma}\left(16 \cdot u, u, 1\right)\right) - \left(\mathsf{log1p}\left(-16 \cdot \left(u \cdot u\right)\right) + \mathsf{log1p}\left(-4 \cdot u\right)\right)\right)} \]

   6. Applied rewrites 33.5%

      \[\leadsto s \cdot \color{blue}{\left(\log \left(-\left(\left(u \cdot u\right) \cdot 16 - 1\right)\right) - \log \left(-\left(4 \cdot u - 1\right)\right)\right)} \]

   7. Taylor expanded in u around 0

      \[\leadsto s \cdot \left(\log \color{blue}{\left(1 + -16 \cdot {u}^{2}\right)} - \log \left(-\left(4 \cdot u - 1\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto s \cdot \left(\log \color{blue}{\left(-16 \cdot {u}^{2} + 1\right)} - \log \left(-\left(4 \cdot u - 1\right)\right)\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(-16, {u}^{2}, 1\right)\right)} - \log \left(-\left(4 \cdot u - 1\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto s \cdot \left(\log \left(\mathsf{fma}\left(-16, \color{blue}{u \cdot u}, 1\right)\right) - \log \left(-\left(4 \cdot u - 1\right)\right)\right) \]

      4. lower-*.f32 14.2

         \[\leadsto s \cdot \left(\log \left(\mathsf{fma}\left(-16, \color{blue}{u \cdot u}, 1\right)\right) - \log \left(-\left(4 \cdot u - 1\right)\right)\right) \]

   9. Applied rewrites 12.2%

      \[\leadsto s \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(-16, u \cdot u, 1\right)\right)} - \log \left(-\left(4 \cdot u - 1\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 0.00800000037997961:\\ \;\;\;\;s \cdot \left(\left(\left(\left(\frac{4}{{u}^{3}} + \frac{8}{u \cdot u}\right) + \frac{21.333333333333332}{u}\right) + 64\right) \cdot {u}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\log \left(\mathsf{fma}\left(-16, u \cdot u, 1\right)\right) - \log \left(\left(-4\right) \cdot u - -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.00800000037997961:\\ \;\;\;\;s \cdot \left(\left(\left(\frac{\left(\frac{4}{u \cdot u} + \frac{8}{u}\right) + 21.333333333333332}{u} + 64\right) \cdot {u}^{3}\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\log \left(\mathsf{fma}\left(-16, u \cdot u, 1\right)\right) - \log \left(\left(-4\right) \cdot u - -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.00800000037997961)
   (*
    s
    (*
     (*
      (+ (/ (+ (+ (/ 4.0 (* u u)) (/ 8.0 u)) 21.333333333333332) u) 64.0)
      (pow u 3.0))
     u))
   (* s (- (log (fma -16.0 (* u u) 1.0)) (log (- (* (- 4.0) u) -1.0))))))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.00800000037997961f) {
		tmp = s * (((((((4.0f / (u * u)) + (8.0f / u)) + 21.333333333333332f) / u) + 64.0f) * powf(u, 3.0f)) * u);
	} else {
		tmp = s * (logf(fmaf(-16.0f, (u * u), 1.0f)) - logf(((-4.0f * u) - -1.0f)));
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.00800000037997961))
		tmp = Float32(s * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(4.0) / Float32(u * u)) + Float32(Float32(8.0) / u)) + Float32(21.333333333333332)) / u) + Float32(64.0)) * (u ^ Float32(3.0))) * u));
	else
		tmp = Float32(s * Float32(log(fma(Float32(-16.0), Float32(u * u), Float32(1.0))) - log(Float32(Float32(Float32(-Float32(4.0)) * u) - Float32(-1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00800000038

   1. Initial program 56.2%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around 0

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

      1. lower-*.f32 81.6

         \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 81.5%

      \[\leadsto s \cdot \left(\sqrt{4 \cdot u} \cdot \color{blue}{\sqrt{4 \cdot u}}\right) \]

   8. Taylor expanded in u around 0

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      3. +-commutative N/A

         \[\leadsto s \cdot \left(\color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)} \cdot u\right) \]

      4. *-commutative N/A

         \[\leadsto s \cdot \left(\left(\color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) \cdot u} + 4\right) \cdot u\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), u, 4\right)} \cdot u\right) \]

      6. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, u, 4\right) \cdot u\right) \]

      7. *-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{64}{3} + 64 \cdot u\right) \cdot u} + 8, u, 4\right) \cdot u\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64}{3} + 64 \cdot u, u, 8\right)}, u, 4\right) \cdot u\right) \]

      9. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{64 \cdot u + \frac{64}{3}}, u, 8\right), u, 4\right) \cdot u\right) \]

      10. lower-fma.f32 81.6

          \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(64, u, 21.333333333333332\right)}, u, 8\right), u, 4\right) \cdot u\right) \]

   10. Applied rewrites 81.4%

       \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]

   11. Taylor expanded in u around -inf

       \[\leadsto s \cdot \left(\left(-1 \cdot \left({u}^{3} \cdot \left(-1 \cdot \frac{\frac{64}{3} + \left(8 \cdot \frac{1}{u} + \frac{4}{{u}^{2}}\right)}{u} - 64\right)\right)\right) \cdot u\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 98.3%

       \[\leadsto s \cdot \left(\left(\left(-\left(\frac{\left(\frac{4}{u \cdot u} + \frac{8}{u}\right) + 21.333333333333332}{-u} - 64\right)\right) \cdot {u}^{3}\right) \cdot u\right) \]

   if 0.00800000038 < u

   1. Initial program 96.3%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 38.4%

      \[\leadsto s \cdot \color{blue}{\left(\log \left(1 \cdot \mathsf{fma}\left(16 \cdot u, u, 1\right)\right) - \left(\mathsf{log1p}\left(-16 \cdot \left(u \cdot u\right)\right) + \mathsf{log1p}\left(-4 \cdot u\right)\right)\right)} \]

   6. Applied rewrites 33.5%

      \[\leadsto s \cdot \color{blue}{\left(\log \left(-\left(\left(u \cdot u\right) \cdot 16 - 1\right)\right) - \log \left(-\left(4 \cdot u - 1\right)\right)\right)} \]

   7. Taylor expanded in u around 0

      \[\leadsto s \cdot \left(\log \color{blue}{\left(1 + -16 \cdot {u}^{2}\right)} - \log \left(-\left(4 \cdot u - 1\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto s \cdot \left(\log \color{blue}{\left(-16 \cdot {u}^{2} + 1\right)} - \log \left(-\left(4 \cdot u - 1\right)\right)\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(-16, {u}^{2}, 1\right)\right)} - \log \left(-\left(4 \cdot u - 1\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto s \cdot \left(\log \left(\mathsf{fma}\left(-16, \color{blue}{u \cdot u}, 1\right)\right) - \log \left(-\left(4 \cdot u - 1\right)\right)\right) \]

      4. lower-*.f32 12.4

         \[\leadsto s \cdot \left(\log \left(\mathsf{fma}\left(-16, \color{blue}{u \cdot u}, 1\right)\right) - \log \left(-\left(4 \cdot u - 1\right)\right)\right) \]

   9. Applied rewrites 12.8%

      \[\leadsto s \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(-16, u \cdot u, 1\right)\right)} - \log \left(-\left(4 \cdot u - 1\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 0.00800000037997961:\\ \;\;\;\;s \cdot \left(\left(\left(\frac{\left(\frac{4}{u \cdot u} + \frac{8}{u}\right) + 21.333333333333332}{u} + 64\right) \cdot {u}^{3}\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\log \left(\mathsf{fma}\left(-16, u \cdot u, 1\right)\right) - \log \left(\left(-4\right) \cdot u - -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 7.000000186963007 \cdot 10^{-5}:\\ \;\;\;\;s \cdot \left(4 \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 7.000000186963007e-5)
   (* s (* 4.0 u))
   (* s (log (/ 1.0 (- 1.0 (* 4.0 u)))))))

C

float code(float s, float u) {
	float tmp;
	if (u <= 7.000000186963007e-5f) {
		tmp = s * (4.0f * u);
	} else {
		tmp = s * logf((1.0f / (1.0f - (4.0f * u))));
	}
	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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if (u <= 7.000000186963007e-5) then
        tmp = s * (4.0e0 * u)
    else
        tmp = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(7.000000186963007e-5))
		tmp = Float32(s * Float32(Float32(4.0) * u));
	else
		tmp = Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if (u <= single(7.000000186963007e-5))
		tmp = s * (single(4.0) * u);
	else
		tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u < 7.00000019e-5

   1. Initial program 46.1%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u around 0

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

      1. lower-*.f32 91.5

         \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]

   5. Applied rewrites 91.5%

      \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]

   if 7.00000019e-5 < u

   1. Initial program 86.5%

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

Alternative 5: 74.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(4 \cdot u\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (* 4.0 u)))

C

float code(float s, float u) {
	return s * (4.0f * u);
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (4.0e0 * u)
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(4.0) * u))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (single(4.0) * u);
end

Derivation

1. Initial program 62.0%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

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

   1. lower-*.f32 75.0

      \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]

5. Applied rewrites 75.0%

   \[\leadsto s \cdot \color{blue}{\left(4 \cdot u\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024346 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))