Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.9% → 98.9%

Time: 3.6s

Alternatives: 12

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

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

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: 61.9% accurate, 1.0× speedup

Math

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

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, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u \leq 0.008700000122189522:\\ \;\;\;\;u \cdot \mathsf{fma}\left(4, s, u \cdot \mathsf{fma}\left(8, s, u \cdot \mathsf{fma}\left(21.333333333333332, s, 64 \cdot \left(s \cdot u\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.008700000122189522)
   (*
    u
    (fma
     4.0
     s
     (* u (fma 8.0 s (* u (fma 21.333333333333332 s (* 64.0 (* s u))))))))
   (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.008700000122189522f) {
		tmp = u * fmaf(4.0f, s, (u * fmaf(8.0f, s, (u * fmaf(21.333333333333332f, s, (64.0f * (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(0.008700000122189522))
		tmp = Float32(u * fma(Float32(4.0), s, Float32(u * fma(Float32(8.0), s, Float32(u * fma(Float32(21.333333333333332), s, Float32(Float32(64.0) * Float32(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 < 0.00870000012

   1. Initial program 61.9%

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

   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. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 93.2%

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

   4. Applied rewrites 93.2%

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

   if 0.00870000012 < u

   1. Initial program 61.9%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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. *-commutative N/A

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

      3. lower-*.f32 61.9%

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-log.f32 N/A

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

      8. lower-unsound-log.f32 N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-unsound-log.f32 64.3%

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

      11. lift--.f32 N/A

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

      12. lift-*.f32 N/A

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

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

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. metadata-eval 64.3%

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

   3. Applied rewrites 64.3%

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

Alternative 2: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u \leq 0.008700000122189522:\\ \;\;\;\;s \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u \cdot u, u \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.008700000122189522)
   (* s (fma (fma (fma 64.0 u 21.333333333333332) u 8.0) (* u u) (* u 4.0)))
   (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.008700000122189522f) {
		tmp = s * fmaf(fmaf(fmaf(64.0f, u, 21.333333333333332f), u, 8.0f), (u * u), (u * 4.0f));
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.008700000122189522))
		tmp = Float32(s * fma(fma(fma(Float32(64.0), u, Float32(21.333333333333332)), u, Float32(8.0)), Float32(u * u), Float32(u * Float32(4.0))));
	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 < 0.00870000012

   1. Initial program 61.9%

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

   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. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 92.9%

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

   4. Applied rewrites 92.9%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lift-+.f32 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lower-fma.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f32 93.2%

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

   6. Applied rewrites 93.2%

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

   if 0.00870000012 < u

   1. Initial program 61.9%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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. *-commutative N/A

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

      3. lower-*.f32 61.9%

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-log.f32 N/A

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

      8. lower-unsound-log.f32 N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-unsound-log.f32 64.3%

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

      11. lift--.f32 N/A

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

      12. lift-*.f32 N/A

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

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

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. metadata-eval 64.3%

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

   3. Applied rewrites 64.3%

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

Alternative 3: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9652000069618225:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot s\right) \cdot u\\ \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (* 4.0 u)) 0.9652000069618225)
   (* (- (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)) <= 0.9652000069618225f) {
		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(0.9652000069618225))
		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)) < 0.965200007

   1. Initial program 61.9%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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. *-commutative N/A

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

      3. lower-*.f32 61.9%

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-log.f32 N/A

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

      8. lower-unsound-log.f32 N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-unsound-log.f32 64.3%

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

      11. lift--.f32 N/A

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

      12. lift-*.f32 N/A

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

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

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. metadata-eval 64.3%

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

   3. Applied rewrites 64.3%

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

   if 0.965200007 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 61.9%

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

   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. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 93.2%

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in s around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 92.9%

         \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + 64 \cdot u\right)\right)\right)\right) \]

   7. Applied rewrites 92.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 92.9%

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

   9. Applied rewrites 92.9%

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

Alternative 4: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9860000014305115:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;u \cdot \mathsf{fma}\left(21.333333333333332 \cdot u - -8, u \cdot s, 4 \cdot s\right)\\ \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (* 4.0 u)) 0.9860000014305115)
   (* (- (log (fma -4.0 u 1.0))) s)
   (* u (fma (- (* 21.333333333333332 u) -8.0) (* u s) (* 4.0 s)))))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.9860000014305115f) {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	} else {
		tmp = u * fmaf(((21.333333333333332f * u) - -8.0f), (u * s), (4.0f * s));
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.9860000014305115))
		tmp = Float32(Float32(-log(fma(Float32(-4.0), u, Float32(1.0)))) * s);
	else
		tmp = Float32(u * fma(Float32(Float32(Float32(21.333333333333332) * u) - Float32(-8.0)), Float32(u * s), Float32(Float32(4.0) * s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.986000001

   1. Initial program 61.9%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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. *-commutative N/A

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

      3. lower-*.f32 61.9%

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-log.f32 N/A

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

      8. lower-unsound-log.f32 N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-unsound-log.f32 64.3%

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

      11. lift--.f32 N/A

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

      12. lift-*.f32 N/A

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

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

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. metadata-eval 64.3%

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

   3. Applied rewrites 64.3%

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

   if 0.986000001 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 61.9%

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

   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. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 93.2%

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in s around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 92.9%

         \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + 64 \cdot u\right)\right)\right)\right) \]

   7. Applied rewrites 92.9%

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

   8. Taylor expanded in u around 0

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

      1. lower-*.f32 90.8%

         \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot \left(8 + 21.333333333333332 \cdot u\right)\right)\right) \]

   10. Applied rewrites 90.8%

       \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot \left(8 + 21.333333333333332 \cdot u\right)\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f32 N/A

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

       2. lift-+.f32 N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. lift-*.f32 N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f32 N/A

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

   12. Applied rewrites 91.0%

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

Alternative 5: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u \leq 0.0035000001080334187:\\ \;\;\;\;s \cdot \mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u \cdot u, u \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.0035000001080334187)
   (* s (fma (fma 21.333333333333332 u 8.0) (* u u) (* u 4.0)))
   (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.0035000001080334187f) {
		tmp = s * fmaf(fmaf(21.333333333333332f, u, 8.0f), (u * u), (u * 4.0f));
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.0035000001080334187))
		tmp = Float32(s * fma(fma(Float32(21.333333333333332), u, Float32(8.0)), Float32(u * u), Float32(u * Float32(4.0))));
	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 < 0.00350000011

   1. Initial program 61.9%

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

   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. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 92.9%

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

   4. Applied rewrites 92.9%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lift-+.f32 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lower-fma.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f32 93.2%

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

   6. Applied rewrites 93.2%

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

   7. Taylor expanded in u around 0

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

   9. Applied rewrites 91.0%

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

   if 0.00350000011 < u

   1. Initial program 61.9%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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. *-commutative N/A

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

      3. lower-*.f32 61.9%

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-log.f32 N/A

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

      8. lower-unsound-log.f32 N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-unsound-log.f32 64.3%

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

      11. lift--.f32 N/A

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

      12. lift-*.f32 N/A

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

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

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. metadata-eval 64.3%

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

   3. Applied rewrites 64.3%

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

Alternative 6: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u \leq 0.0035000001080334187:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(21.333333333333332, u, 8\right), u, 4\right) \cdot s\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.0035000001080334187)
   (* (* (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 <= 0.0035000001080334187f) {
		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(0.0035000001080334187))
		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 < 0.00350000011

   1. Initial program 61.9%

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

   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. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 93.2%

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in s around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 92.9%

         \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + 64 \cdot u\right)\right)\right)\right) \]

   7. Applied rewrites 92.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 92.9%

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

   9. Applied rewrites 92.9%

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

   10. Taylor expanded in u around 0

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

   12. Applied rewrites 90.8%

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

   if 0.00350000011 < u

   1. Initial program 61.9%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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. *-commutative N/A

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

      3. lower-*.f32 61.9%

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-log.f32 N/A

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

      8. lower-unsound-log.f32 N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-unsound-log.f32 64.3%

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

      11. lift--.f32 N/A

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

      12. lift-*.f32 N/A

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

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

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. metadata-eval 64.3%

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

   3. Applied rewrites 64.3%

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

Alternative 7: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u \leq 0.0010000000474974513:\\ \;\;\;\;s \cdot \mathsf{fma}\left(8, u \cdot u, u \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\log \left(\mathsf{fma}\left(-4, u, 1\right)\right)\right) \cdot s\\ \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.0010000000474974513)
   (* s (fma 8.0 (* u u) (* u 4.0)))
   (* (- (log (fma -4.0 u 1.0))) s)))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.0010000000474974513f) {
		tmp = s * fmaf(8.0f, (u * u), (u * 4.0f));
	} else {
		tmp = -logf(fmaf(-4.0f, u, 1.0f)) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.0010000000474974513))
		tmp = Float32(s * fma(Float32(8.0), Float32(u * u), Float32(u * Float32(4.0))));
	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 < 0.00100000005

   1. Initial program 61.9%

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

   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. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 92.9%

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

   4. Applied rewrites 92.9%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lift-+.f32 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lower-fma.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f32 93.2%

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

   6. Applied rewrites 93.2%

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

   7. Taylor expanded in u around 0

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

   9. Applied rewrites 86.4%

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

   if 0.00100000005 < u

   1. Initial program 61.9%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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. *-commutative N/A

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

      3. lower-*.f32 61.9%

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

      4. lift-log.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. log-rec N/A

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

      7. lower-log.f32 N/A

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

      8. lower-unsound-log.f32 N/A

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

      9. lower-neg.f32 N/A

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

      10. lower-unsound-log.f32 64.3%

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

      11. lift--.f32 N/A

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

      12. lift-*.f32 N/A

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

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

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. metadata-eval 64.3%

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

   3. Applied rewrites 64.3%

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

Alternative 8: 86.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

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. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lower-*.f32 92.9%

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

4. Applied rewrites 92.9%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. lower-fma.f32 N/A

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

   18. lower-*.f32 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f32 93.2%

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

6. Applied rewrites 93.2%

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

7. Taylor expanded in u around 0

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

9. Applied rewrites 86.4%

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

Alternative 9: 86.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

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. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 86.4%

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

4. Applied rewrites 86.4%

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

Alternative 10: 86.3% accurate, 1.6× speedup

Math

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

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 61.9%

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

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. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 86.4%

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

4. Applied rewrites 86.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 86.4%

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

   4. lift-fma.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. associate-*l* N/A

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

   11. distribute-lft-out N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 86.3%

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

6. Applied rewrites 86.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 86.3%

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

8. Applied rewrites 86.3%

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

Alternative 11: 73.4% accurate, 2.7× speedup

Math

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

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 61.9%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f32 73.4%

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

4. Applied rewrites 73.4%

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

Alternative 12: 73.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.9%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 73.1%

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

4. Applied rewrites 73.1%

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

Reproduce

herbie shell --seed 2025179 
(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))))))