Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.6% → 98.9%

Time: 3.1s

Alternatives: 15

Speedup: 2.7×

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: 61.6% 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: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\\ \mathbf{if}\;u \leq 0.008200000040233135:\\ \;\;\;\;s \cdot \mathsf{fma}\left(u, 4, \left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right) \cdot u\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{0 - {t\_0}^{3}}{0 + \mathsf{fma}\left(t\_0, t\_0, 0 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (log (fma -4.0 u 1.0))))
   (if (<= u 0.008200000040233135)
     (* s (fma u 4.0 (* (* (fma (fma 64.0 u 21.333333333333332) u 8.0) u) u)))
     (* s (/ (- 0.0 (pow t_0 3.0)) (+ 0.0 (fma t_0 t_0 (* 0.0 t_0))))))))

C

float code(float s, float u) {
	float t_0 = logf(fmaf(-4.0f, u, 1.0f));
	float tmp;
	if (u <= 0.008200000040233135f) {
		tmp = s * fmaf(u, 4.0f, ((fmaf(fmaf(64.0f, u, 21.333333333333332f), u, 8.0f) * u) * u));
	} else {
		tmp = s * ((0.0f - powf(t_0, 3.0f)) / (0.0f + fmaf(t_0, t_0, (0.0f * t_0))));
	}
	return tmp;
}

Julia

function code(s, u)
	t_0 = log(fma(Float32(-4.0), u, Float32(1.0)))
	tmp = Float32(0.0)
	if (u <= Float32(0.008200000040233135))
		tmp = Float32(s * fma(u, Float32(4.0), Float32(Float32(fma(fma(Float32(64.0), u, Float32(21.333333333333332)), u, Float32(8.0)) * u) * u)));
	else
		tmp = Float32(s * Float32(Float32(Float32(0.0) - (t_0 ^ Float32(3.0))) / Float32(Float32(0.0) + fma(t_0, t_0, Float32(Float32(0.0) * t_0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00820000004

   1. Initial program 55.0%

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

   2. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 99.0

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

   4. Applied rewrites 99.0%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

   6. Applied rewrites 99.3%

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

   if 0.00820000004 < u

   1. Initial program 95.7%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Step-by-step derivation

      1. lift-log.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift--.f32 N/A

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

      5. log-div N/A

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

      6. metadata-eval N/A

         \[\leadsto s \cdot \left(\color{blue}{0} - \log \left(1 - 4 \cdot u\right)\right) \]

      7. flip3-- N/A

         \[\leadsto s \cdot \color{blue}{\frac{{0}^{3} - {\log \left(1 - 4 \cdot u\right)}^{3}}{0 \cdot 0 + \left(\log \left(1 - 4 \cdot u\right) \cdot \log \left(1 - 4 \cdot u\right) + 0 \cdot \log \left(1 - 4 \cdot u\right)\right)}} \]

      8. lower-/.f32 N/A

         \[\leadsto s \cdot \color{blue}{\frac{{0}^{3} - {\log \left(1 - 4 \cdot u\right)}^{3}}{0 \cdot 0 + \left(\log \left(1 - 4 \cdot u\right) \cdot \log \left(1 - 4 \cdot u\right) + 0 \cdot \log \left(1 - 4 \cdot u\right)\right)}} \]

   3. Applied rewrites 96.5%

      \[\leadsto s \cdot \color{blue}{\frac{0 - {\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\mathsf{fma}\left(-4, u, 1\right)\right), \log \left(\mathsf{fma}\left(-4, u, 1\right)\right), 0 \cdot \log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.008200000040233135:\\ \;\;\;\;s \cdot \mathsf{fma}\left(u, 4, \left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right) \cdot u\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.008200000040233135)
   (* s (fma u 4.0 (* (* (fma (fma 64.0 u 21.333333333333332) u 8.0) u) u)))
   (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.008200000040233135f) {
		tmp = s * fmaf(u, 4.0f, ((fmaf(fmaf(64.0f, u, 21.333333333333332f), u, 8.0f) * u) * u));
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.008200000040233135))
		tmp = Float32(s * fma(u, Float32(4.0), Float32(Float32(fma(fma(Float32(64.0), u, Float32(21.333333333333332)), u, Float32(8.0)) * u) * u)));
	else
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00820000004

   1. Initial program 55.0%

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

   2. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 99.0

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

   4. Applied rewrites 99.0%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

   6. Applied rewrites 99.3%

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

   if 0.00820000004 < u

   1. Initial program 95.7%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift--.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. log-rec N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-log.f32 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f32 96.7

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

   3. Applied rewrites 96.7%

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

Alternative 3: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.008200000040233135:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right) \cdot s, u, 4 \cdot s\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.008200000040233135)
   (* (fma (* (fma (fma 64.0 u 21.333333333333332) u 8.0) s) u (* 4.0 s)) u)
   (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.008200000040233135f) {
		tmp = fmaf((fmaf(fmaf(64.0f, u, 21.333333333333332f), u, 8.0f) * s), u, (4.0f * s)) * u;
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.008200000040233135))
		tmp = Float32(fma(Float32(fma(fma(Float32(64.0), u, Float32(21.333333333333332)), u, Float32(8.0)) * s), u, Float32(Float32(4.0) * s)) * u);
	else
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00820000004

   1. Initial program 55.0%

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

   2. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in s around 0

      \[\leadsto \mathsf{fma}\left(s \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right), u, 4 \cdot s\right) \cdot u \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift-fma.f32 N/A

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

      7. lift-fma.f32 99.3

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

   7. Applied rewrites 99.3%

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

   if 0.00820000004 < u

   1. Initial program 95.7%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift--.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. log-rec N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-log.f32 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f32 96.7

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

   3. Applied rewrites 96.7%

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

Alternative 4: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9670000076293945:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (* 4.0 u)) 0.9670000076293945)
   (* (- (log (fma -4.0 u 1.0))) s)
   (* s (* (fma (fma (fma 64.0 u 21.333333333333332) u 8.0) u 4.0) u))))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.9670000076293945f) {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	} else {
		tmp = s * (fmaf(fmaf(fmaf(64.0f, u, 21.333333333333332f), u, 8.0f), u, 4.0f) * u);
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.9670000076293945))
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	else
		tmp = Float32(s * Float32(fma(fma(fma(Float32(64.0), u, Float32(21.333333333333332)), u, Float32(8.0)), u, Float32(4.0)) * u));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.96700001

   1. Initial program 95.7%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift--.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. log-rec N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-log.f32 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f32 96.7

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

   3. Applied rewrites 96.7%

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

   if 0.96700001 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 55.0%

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

   2. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 99.0

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

   4. Applied rewrites 99.0%

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

Alternative 5: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9670000076293945:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot s\right) \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (* 4.0 u)) 0.9670000076293945)
   (* (- (log (fma -4.0 u 1.0))) s)
   (* (* (fma (fma (fma 64.0 u 21.333333333333332) u 8.0) u 4.0) s) u)))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.9670000076293945f) {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	} else {
		tmp = (fmaf(fmaf(fmaf(64.0f, u, 21.333333333333332f), u, 8.0f), u, 4.0f) * s) * u;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.9670000076293945))
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	else
		tmp = Float32(Float32(fma(fma(fma(Float32(64.0), u, Float32(21.333333333333332)), u, Float32(8.0)), u, Float32(4.0)) * s) * u);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.96700001

   1. Initial program 95.7%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift--.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. log-rec N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-log.f32 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f32 96.7

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

   3. Applied rewrites 96.7%

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

   if 0.96700001 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 55.0%

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

   2. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in s around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lift-fma.f32 N/A

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

      9. lift-fma.f32 N/A

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

      10. lift-fma.f32 99.0

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

   7. Applied rewrites 99.0%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot s\right) \cdot u \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9847999811172485:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;s \cdot \mathsf{fma}\left(u, 4, \left(\mathsf{fma}\left(21.333333333333332, u, 8\right) \cdot u\right) \cdot u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (* 4.0 u)) 0.9847999811172485)
   (* (- (log (fma -4.0 u 1.0))) s)
   (* s (fma u 4.0 (* (* (fma 21.333333333333332 u 8.0) u) u)))))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.9847999811172485f) {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	} else {
		tmp = s * fmaf(u, 4.0f, ((fmaf(21.333333333333332f, u, 8.0f) * u) * u));
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.9847999811172485))
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	else
		tmp = Float32(s * fma(u, Float32(4.0), Float32(Float32(fma(Float32(21.333333333333332), u, Float32(8.0)) * u) * u)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.984799981

   1. Initial program 94.5%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift--.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. log-rec N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-log.f32 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f32 95.7

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

   3. Applied rewrites 95.7%

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

   if 0.984799981 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 53.4%

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

   2. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 98.8

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

   4. Applied rewrites 98.8%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u, 4\right) \cdot u\right)} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lift-fma.f32 99.1

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

   6. Applied rewrites 99.1%

      \[\leadsto s \cdot \mathsf{fma}\left(u, \color{blue}{4}, \left(\mathsf{fma}\left(21.333333333333332, u, 8\right) \cdot u\right) \cdot u\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.003800000064074993:\\ \;\;\;\;s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u, 4\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.003800000064074993)
   (* s (* (fma (fma 21.333333333333332 u 8.0) u 4.0) u))
   (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.003800000064074993f) {
		tmp = s * (fmaf(fmaf(21.333333333333332f, u, 8.0f), u, 4.0f) * u);
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.003800000064074993))
		tmp = Float32(s * Float32(fma(fma(Float32(21.333333333333332), u, Float32(8.0)), u, Float32(4.0)) * u));
	else
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00380000006

   1. Initial program 53.4%

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

   2. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 98.8

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

   4. Applied rewrites 98.8%

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

   if 0.00380000006 < u

   1. Initial program 94.5%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift--.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. log-rec N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-log.f32 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f32 95.7

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

   3. Applied rewrites 95.7%

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

Alternative 8: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.003800000064074993:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u, 4\right) \cdot s\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.003800000064074993)
   (* (* (fma (fma 21.333333333333332 u 8.0) u 4.0) s) u)
   (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.003800000064074993f) {
		tmp = (fmaf(fmaf(21.333333333333332f, u, 8.0f), u, 4.0f) * s) * u;
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.003800000064074993))
		tmp = Float32(Float32(fma(fma(Float32(21.333333333333332), u, Float32(8.0)), u, Float32(4.0)) * s) * u);
	else
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00380000006

   1. Initial program 53.4%

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

   2. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-*.f32 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in s around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 98.8

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

   10. Applied rewrites 98.8%

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

   if 0.00380000006 < u

   1. Initial program 94.5%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift--.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. log-rec N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-log.f32 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f32 95.7

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

   3. Applied rewrites 95.7%

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

Alternative 9: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9973000288009644:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;s \cdot \mathsf{fma}\left(u, 4, \left(8 \cdot u\right) \cdot u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (* 4.0 u)) 0.9973000288009644)
   (* (- (log (fma -4.0 u 1.0))) s)
   (* s (fma u 4.0 (* (* 8.0 u) u)))))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.9973000288009644f) {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	} else {
		tmp = s * fmaf(u, 4.0f, ((8.0f * u) * u));
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.9973000288009644))
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	else
		tmp = Float32(s * fma(u, Float32(4.0), Float32(Float32(Float32(8.0) * u) * u)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.997300029

   1. Initial program 91.1%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift--.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. log-rec N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-log.f32 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f32 92.8

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

   3. Applied rewrites 92.8%

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

   if 0.997300029 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 49.8%

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

   2. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 99.1

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

   4. Applied rewrites 99.1%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in u around 0

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

   9. Applied rewrites 98.5%

      \[\leadsto s \cdot \mathsf{fma}\left(u, 4, \left(8 \cdot u\right) \cdot u\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 87.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.6%

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

2. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 93.2

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

4. Applied rewrites 93.2%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-in N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-*.f32 N/A

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

6. Applied rewrites 93.5%

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

7. Taylor expanded in u around 0

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

9. Applied rewrites 87.1%

   \[\leadsto s \cdot \mathsf{fma}\left(u, 4, \left(8 \cdot u\right) \cdot u\right) \]
10. Add Preprocessing

Alternative 11: 87.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.6%

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

2. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

      \[\leadsto \left(8 \cdot \left(s \cdot u\right) + 4 \cdot s\right) \cdot u \]

   4. lower-fma.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 87.1

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

4. Applied rewrites 87.1%

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

Alternative 12: 86.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.6%

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

2. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

      \[\leadsto s \cdot \left(\left(8 \cdot u + 4\right) \cdot u\right) \]

   4. lower-fma.f32 86.9

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

4. Applied rewrites 86.9%

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

Alternative 13: 86.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.6%

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

2. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

      \[\leadsto \left(8 \cdot \left(s \cdot u\right) + 4 \cdot s\right) \cdot u \]

   4. lower-fma.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 87.1

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

4. Applied rewrites 87.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

      \[\leadsto \left(8 \cdot \left(u \cdot s\right) + 4 \cdot s\right) \cdot u \]

   4. +-commutative N/A

      \[\leadsto \left(4 \cdot s + 8 \cdot \left(u \cdot s\right)\right) \cdot u \]

   5. associate-*r* N/A

      \[\leadsto \left(4 \cdot s + \left(8 \cdot u\right) \cdot s\right) \cdot u \]

   6. distribute-rgt-in N/A

      \[\leadsto \left(s \cdot \left(4 + 8 \cdot u\right)\right) \cdot u \]

   7. *-commutative N/A

      \[\leadsto \left(\left(4 + 8 \cdot u\right) \cdot s\right) \cdot u \]

   8. lower-*.f32 N/A

      \[\leadsto \left(\left(4 + 8 \cdot u\right) \cdot s\right) \cdot u \]

   9. +-commutative N/A

      \[\leadsto \left(\left(8 \cdot u + 4\right) \cdot s\right) \cdot u \]

   10. lower-fma.f32 86.9

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

6. Applied rewrites 86.9%

   \[\leadsto \left(\mathsf{fma}\left(8, u, 4\right) \cdot s\right) \cdot u \]
7. Add Preprocessing

Alternative 14: 73.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.6%

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

2. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 73.8

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

4. Applied rewrites 73.8%

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

Alternative 15: 73.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.6%

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

2. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 93.5%

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

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

      \[\leadsto \left(u \cdot s\right) \cdot 4 \]

   4. lift-*.f32 73.6

      \[\leadsto \left(u \cdot s\right) \cdot 4 \]

7. Applied rewrites 73.6%

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

Reproduce

herbie shell --seed 2025122 
(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))))))