Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.0% → 98.5%

Time: 9.0s

Alternatives: 7

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: 61.0% 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: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.010499999858438969:\\ \;\;\;\;\left(\left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left|\log \left(-\left(\left(u \cdot u\right) \cdot 16 - 1\right)\right) - \log \left(\left(--4\right) \cdot u - -1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.010499999858438969)
   (*
    (* (- (* (- (* (- (* -64.0 u) 21.333333333333332) u) 8.0) u) 4.0) u)
    (- s))
   (*
    s
    (fabs
     (- (log (- (- (* (* u u) 16.0) 1.0))) (log (- (* (- -4.0) u) -1.0)))))))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.010499999858438969f) {
		tmp = (((((((-64.0f * u) - 21.333333333333332f) * u) - 8.0f) * u) - 4.0f) * u) * -s;
	} else {
		tmp = s * fabsf((logf(-(((u * u) * 16.0f) - 1.0f)) - logf(((-(-4.0f) * 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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if (u <= 0.010499999858438969e0) then
        tmp = ((((((((-64.0e0) * u) - 21.333333333333332e0) * u) - 8.0e0) * u) - 4.0e0) * u) * -s
    else
        tmp = s * abs((log(-(((u * u) * 16.0e0) - 1.0e0)) - log(((-(-4.0e0) * u) - (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.010499999858438969))
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-64.0) * u) - Float32(21.333333333333332)) * u) - Float32(8.0)) * u) - Float32(4.0)) * u) * Float32(-s));
	else
		tmp = Float32(s * abs(Float32(log(Float32(-Float32(Float32(Float32(u * u) * Float32(16.0)) - Float32(1.0)))) - log(Float32(Float32(Float32(-Float32(-4.0)) * u) - Float32(-1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if (u <= single(0.010499999858438969))
		tmp = (((((((single(-64.0) * u) - single(21.333333333333332)) * u) - single(8.0)) * u) - single(4.0)) * u) * -s;
	else
		tmp = s * abs((log(-(((u * u) * single(16.0)) - single(1.0))) - log(((-single(-4.0) * u) - single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u < 0.0104999999

   1. Initial program 55.6%

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

   4. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-4 \cdot u\right)\right) \cdot \log \left(e^{s}\right)} \]

   5. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 32.9

         \[\leadsto \left(-\left(\left(\color{blue}{-21.333333333333332 \cdot u} - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \log \left(e^{s}\right) \]

   7. Applied rewrites 32.9%

      \[\leadsto \left(-\color{blue}{\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u}\right) \cdot \log \left(e^{s}\right) \]
   8. Step-by-step derivation

      1. lift-log.f32 N/A

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

      2. lift-exp.f32 N/A

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

      3. rem-log-exp 97.4

         \[\leadsto \left(-\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \color{blue}{s} \]

   9. Applied rewrites 97.4%

      \[\leadsto \left(-\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \color{blue}{s} \]

   10. Taylor expanded in u around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f32 N/A

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

       6. lower--.f32 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f32 N/A

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

       9. lower--.f32 N/A

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

       10. lower-*.f32 98.6

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

   12. Applied rewrites 98.6%

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

   if 0.0104999999 < u

   1. Initial program 96.4%

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

   4. Applied rewrites 38.9%

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

   6. Applied rewrites 96.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 0.010499999858438969:\\ \;\;\;\;\left(\left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left|\log \left(-\left(\left(u \cdot u\right) \cdot 16 - 1\right)\right) - \log \left(\left(--4\right) \cdot u - -1\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t\_0 \leq 0.9599999785423279:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* 4.0 u))))
   (if (<= t_0 0.9599999785423279)
     (* s (log (/ 1.0 t_0)))
     (*
      (* (- (* (- (* (- (* -64.0 u) 21.333333333333332) u) 8.0) u) 4.0) u)
      (- s)))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (4.0f * u);
	float tmp;
	if (t_0 <= 0.9599999785423279f) {
		tmp = s * logf((1.0f / t_0));
	} else {
		tmp = (((((((-64.0f * u) - 21.333333333333332f) * u) - 8.0f) * u) - 4.0f) * u) * -s;
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = 1.0e0 - (4.0e0 * u)
    if (t_0 <= 0.9599999785423279e0) then
        tmp = s * log((1.0e0 / t_0))
    else
        tmp = ((((((((-64.0e0) * u) - 21.333333333333332e0) * u) - 8.0e0) * u) - 4.0e0) * u) * -s
    end if
    code = tmp
end function

Julia

function code(s, u)
	t_0 = Float32(Float32(1.0) - Float32(Float32(4.0) * u))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9599999785423279))
		tmp = Float32(s * log(Float32(Float32(1.0) / t_0)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-64.0) * u) - Float32(21.333333333333332)) * u) - Float32(8.0)) * u) - Float32(4.0)) * u) * Float32(-s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	t_0 = single(1.0) - (single(4.0) * u);
	tmp = single(0.0);
	if (t_0 <= single(0.9599999785423279))
		tmp = s * log((single(1.0) / t_0));
	else
		tmp = (((((((single(-64.0) * u) - single(21.333333333333332)) * u) - single(8.0)) * u) - single(4.0)) * u) * -s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.959999979

   1. Initial program 96.3%

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

   if 0.959999979 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 55.4%

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

   4. Applied rewrites 28.2%

      \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-4 \cdot u\right)\right) \cdot \log \left(e^{s}\right)} \]

   5. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 33.1

         \[\leadsto \left(-\left(\left(\color{blue}{-21.333333333333332 \cdot u} - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \log \left(e^{s}\right) \]

   7. Applied rewrites 33.1%

      \[\leadsto \left(-\color{blue}{\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u}\right) \cdot \log \left(e^{s}\right) \]
   8. Step-by-step derivation

      1. lift-log.f32 N/A

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

      2. lift-exp.f32 N/A

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

      3. rem-log-exp 97.5

         \[\leadsto \left(-\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \color{blue}{s} \]

   9. Applied rewrites 97.5%

      \[\leadsto \left(-\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \color{blue}{s} \]

   10. Taylor expanded in u around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f32 N/A

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

       6. lower--.f32 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f32 N/A

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

       9. lower--.f32 N/A

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

       10. lower-*.f32 98.7

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

   12. Applied rewrites 98.7%

       \[\leadsto \left(-\color{blue}{\left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right) \cdot u}\right) \cdot s \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9599999785423279:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* (- (* (- (* (- (* -64.0 u) 21.333333333333332) u) 8.0) u) 4.0) u)
  (- s)))

C

float code(float s, float u) {
	return (((((((-64.0f * u) - 21.333333333333332f) * u) - 8.0f) * u) - 4.0f) * u) * -s;
}

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 = ((((((((-64.0e0) * u) - 21.333333333333332e0) * u) - 8.0e0) * u) - 4.0e0) * u) * -s
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-64.0) * u) - Float32(21.333333333333332)) * u) - Float32(8.0)) * u) - Float32(4.0)) * u) * Float32(-s))
end

MATLAB

function tmp = code(s, u)
	tmp = (((((((single(-64.0) * u) - single(21.333333333333332)) * u) - single(8.0)) * u) - single(4.0)) * u) * -s;
end

Derivation

1. Initial program 62.8%

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

4. Applied rewrites 25.4%

   \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-4 \cdot u\right)\right) \cdot \log \left(e^{s}\right)} \]

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-*.f32 30.1

      \[\leadsto \left(-\left(\left(\color{blue}{-21.333333333333332 \cdot u} - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \log \left(e^{s}\right) \]

7. Applied rewrites 30.1%

   \[\leadsto \left(-\color{blue}{\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u}\right) \cdot \log \left(e^{s}\right) \]
8. Step-by-step derivation

   1. lift-log.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. rem-log-exp 89.6

      \[\leadsto \left(-\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \color{blue}{s} \]

9. Applied rewrites 89.6%

   \[\leadsto \left(-\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \color{blue}{s} \]

10. Taylor expanded in u around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. lower--.f32 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. lower--.f32 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f32 N/A

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

    9. lower--.f32 N/A

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

    10. lower-*.f32 91.9

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

12. Applied rewrites 91.9%

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

13. Final simplification 91.9%

    \[\leadsto \left(\left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \left(-s\right) \]
14. Add Preprocessing

Alternative 4: 91.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* (- (* (- (* -21.333333333333332 u) 8.0) u) 4.0) u) (- s)))

C

float code(float s, float u) {
	return (((((-21.333333333333332f * u) - 8.0f) * u) - 4.0f) * u) * -s;
}

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 = ((((((-21.333333333333332e0) * u) - 8.0e0) * u) - 4.0e0) * u) * -s
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(Float32(Float32(Float32(-21.333333333333332) * u) - Float32(8.0)) * u) - Float32(4.0)) * u) * Float32(-s))
end

MATLAB

function tmp = code(s, u)
	tmp = (((((single(-21.333333333333332) * u) - single(8.0)) * u) - single(4.0)) * u) * -s;
end

Derivation

1. Initial program 62.8%

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

4. Applied rewrites 25.2%

   \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-4 \cdot u\right)\right) \cdot \log \left(e^{s}\right)} \]

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-*.f32 30.1

      \[\leadsto \left(-\left(\left(\color{blue}{-21.333333333333332 \cdot u} - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \log \left(e^{s}\right) \]

7. Applied rewrites 30.1%

   \[\leadsto \left(-\color{blue}{\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u}\right) \cdot \log \left(e^{s}\right) \]
8. Step-by-step derivation

   1. lift-log.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. rem-log-exp 89.6

      \[\leadsto \left(-\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \color{blue}{s} \]

9. Applied rewrites 89.6%

   \[\leadsto \left(-\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \color{blue}{s} \]

10. Final simplification 89.6%

    \[\leadsto \left(\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u\right) \cdot \left(-s\right) \]
11. Add Preprocessing

Alternative 5: 87.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(8 \cdot u\right) \cdot u - -4 \cdot u\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (- (* (* 8.0 u) u) (* -4.0 u))))

C

float code(float s, float u) {
	return s * (((8.0f * u) * u) - (-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 * (((8.0e0 * u) * u) - ((-4.0e0) * u))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(Float32(8.0) * u) * u) - Float32(Float32(-4.0) * u)))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (((single(8.0) * u) * u) - (single(-4.0) * u));
end

Derivation

1. Initial program 62.8%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 73.5

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

5. Applied rewrites 73.5%

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

6. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 73.5

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

8. Applied rewrites 73.3%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(8, u, 4\right) \cdot u\right)} \]
9. Step-by-step derivation

10. Applied rewrites 85.6%

    \[\leadsto s \cdot \left(\left(8 \cdot u\right) \cdot u - \color{blue}{-4 \cdot u}\right) \]
11. Add Preprocessing

Alternative 6: 86.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(4 - -8 \cdot u\right) \cdot u\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (* (- 4.0 (* -8.0 u)) u)))

C

float code(float s, float u) {
	return s * ((4.0f - (-8.0f * u)) * 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 - ((-8.0e0) * u)) * u)
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(4.0) - Float32(Float32(-8.0) * u)) * u))
end

MATLAB

function tmp = code(s, u)
	tmp = s * ((single(4.0) - (single(-8.0) * u)) * u);
end

Derivation

1. Initial program 62.8%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 73.5

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

5. Applied rewrites 73.5%

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

6. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 73.5

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

8. Applied rewrites 73.3%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(8, u, 4\right) \cdot u\right)} \]
9. Step-by-step derivation

10. Applied rewrites 85.3%

    \[\leadsto s \cdot \left(\left(4 - -8 \cdot u\right) \cdot u\right) \]
11. Add Preprocessing

Alternative 7: 74.1% 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 62.8%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 73.5

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

5. Applied rewrites 73.5%

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

Reproduce

herbie shell --seed 2024347 
(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))))))