Rust f32::acosh

Percentage Accurate: 14.3% → 99.6%

Time: 1.8s

Alternatives: 4

Speedup: 4.3×

• Could not converge on a ground truth

  1. x = 4.2758711267766454e-38

Specification

\[x \geq 1\]

Math

\[\cosh^{-1} x \]

FPCore

(FPCore (x)
  :precision binary32
  (acosh x))

C

float code(float x) {
	return acoshf(x);
}

Julia

function code(x)
	return acosh(x)
end

MATLAB

function tmp = code(x)
	tmp = acosh(x);
end

Initial Program: 14.3% accurate, 1.0× speedup

Math

\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]

FPCore

(FPCore (x)
  :precision binary32
  (log (+ x (sqrt (- (* x x) 1.0)))))

C

float code(float x) {
	return logf((x + sqrtf(((x * x) - 1.0f))));
}

Fortran

real(4) function code(x)
use fmin_fmax_functions
    real(4), intent (in) :: x
    code = log((x + sqrt(((x * x) - 1.0e0))))
end function

Julia

function code(x)
	return log(Float32(x + sqrt(Float32(Float32(x * x) - Float32(1.0)))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) - single(1.0)))));
end

Alternative 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\sqrt{\left|1 - x\right|}, \sqrt{\left|-1 - x\right|}, x\right)\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary32
  (if (<= x 0.5)
  (* 0.0 1.0)
  (log (fma (sqrt (fabs (- 1.0 x))) (sqrt (fabs (- -1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= 0.5f) {
		tmp = 0.0f * 1.0f;
	} else {
		tmp = logf(fmaf(sqrtf(fabsf((1.0f - x))), sqrtf(fabsf((-1.0f - x))), x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.5))
		tmp = Float32(Float32(0.0) * Float32(1.0));
	else
		tmp = log(fma(sqrt(abs(Float32(Float32(1.0) - x))), sqrt(abs(Float32(Float32(-1.0) - x))), x));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 14.3%

      \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
   2. Step-by-step derivation

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. fabs-sqr N/A

         \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|\sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}\right|}}\right) \]

      5. lift-sqrt.f32 N/A

         \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{\sqrt{x \cdot x - 1}} \cdot \sqrt{x \cdot x - 1}\right|}\right) \]

      6. lift-sqrt.f32 N/A

         \[\leadsto \log \left(x + \sqrt{\left|\sqrt{x \cdot x - 1} \cdot \color{blue}{\sqrt{x \cdot x - 1}}\right|}\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{x \cdot x - 1}\right|}\right) \]

      8. lift--.f32 N/A

         \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{x \cdot x - 1}\right|}\right) \]

      9. lift-*.f32 N/A

         \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{x \cdot x} - 1\right|}\right) \]

      10. difference-of-sqr-1 N/A

          \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}\right|}\right) \]

      11. *-commutative N/A

          \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{\left(x - 1\right) \cdot \left(x + 1\right)}\right|}\right) \]

      12. fabs-mul N/A

          \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|x - 1\right| \cdot \left|x + 1\right|}}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|x - 1\right| \cdot \left|x + 1\right|}}\right) \]

      14. fabs-sub N/A

          \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|1 - x\right|} \cdot \left|x + 1\right|}\right) \]

      15. lower-fabs.f32 N/A

          \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|1 - x\right|} \cdot \left|x + 1\right|}\right) \]

      16. lower--.f32 N/A

          \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{1 - x}\right| \cdot \left|x + 1\right|}\right) \]

      17. add-flip N/A

          \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \left|\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right|}\right) \]

      18. fabs-sub N/A

          \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \color{blue}{\left|\left(\mathsf{neg}\left(1\right)\right) - x\right|}}\right) \]

      19. lower-fabs.f32 N/A

          \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \color{blue}{\left|\left(\mathsf{neg}\left(1\right)\right) - x\right|}}\right) \]

      20. lower--.f32 N/A

          \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \left|\color{blue}{\left(\mathsf{neg}\left(1\right)\right) - x}\right|}\right) \]

      21. metadata-eval 55.1%

          \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \left|\color{blue}{-1} - x\right|}\right) \]

   3. Applied rewrites 55.1%

      \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|1 - x\right| \cdot \left|-1 - x\right|}}\right) \]

   4. Taylor expanded in x around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 6.7%

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

   6. Applied rewrites 6.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto x \cdot 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 9.4%

      \[\leadsto x \cdot 1 \]

   10. Taylor expanded in undef-var around zero

       \[\leadsto 0 \cdot 1 \]
   11. Step-by-step derivation

   12. Applied rewrites 76.1%

       \[\leadsto 0 \cdot 1 \]

   if 0.5 < x

   1. Initial program 14.3%

      \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
   2. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. difference-of-sqr-1 N/A

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

      7. sqrt-prod N/A

         \[\leadsto \log \left(\color{blue}{\sqrt{\left|x + 1\right|} \cdot \sqrt{\left|x - 1\right|}} + x\right) \]

      8. *-commutative N/A

         \[\leadsto \log \left(\color{blue}{\sqrt{\left|x - 1\right|} \cdot \sqrt{\left|x + 1\right|}} + x\right) \]

      9. lower-fma.f32 N/A

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\sqrt{\left|x - 1\right|}, \sqrt{\left|x + 1\right|}, x\right)\right)} \]

      10. lower-sqrt.f32 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\sqrt{\left|x - 1\right|}}, \sqrt{\left|x + 1\right|}, x\right)\right) \]

      11. fabs-sub N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{\color{blue}{\left|1 - x\right|}}, \sqrt{\left|x + 1\right|}, x\right)\right) \]

      12. lower-fabs.f32 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{\color{blue}{\left|1 - x\right|}}, \sqrt{\left|x + 1\right|}, x\right)\right) \]

      13. lower--.f32 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{\left|\color{blue}{1 - x}\right|}, \sqrt{\left|x + 1\right|}, x\right)\right) \]

      14. lower-sqrt.f32 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{\left|1 - x\right|}, \color{blue}{\sqrt{\left|x + 1\right|}}, x\right)\right) \]

      15. add-flip N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{\left|1 - x\right|}, \sqrt{\left|\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right|}, x\right)\right) \]

      16. fabs-sub N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{\left|1 - x\right|}, \sqrt{\color{blue}{\left|\left(\mathsf{neg}\left(1\right)\right) - x\right|}}, x\right)\right) \]

      17. lower-fabs.f32 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{\left|1 - x\right|}, \sqrt{\color{blue}{\left|\left(\mathsf{neg}\left(1\right)\right) - x\right|}}, x\right)\right) \]

      18. lower--.f32 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{\left|1 - x\right|}, \sqrt{\left|\color{blue}{\left(\mathsf{neg}\left(1\right)\right) - x}\right|}, x\right)\right) \]

      19. metadata-eval 67.1%

          \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{\left|1 - x\right|}, \sqrt{\left|\color{blue}{-1} - x\right|}, x\right)\right) \]

   3. Applied rewrites 67.1%

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

Alternative 2: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cosh^{-1} x\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary32
  (if (<= x 0.5) (* 0.0 1.0) (acosh x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.5f) {
		tmp = 0.0f * 1.0f;
	} else {
		tmp = acoshf(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.5))
		tmp = Float32(Float32(0.0) * Float32(1.0));
	else
		tmp = acosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(0.5))
		tmp = single(0.0) * single(1.0);
	else
		tmp = acosh(x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 14.3%

      \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
   2. Step-by-step derivation

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. fabs-sqr N/A

         \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|\sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}\right|}}\right) \]

      5. lift-sqrt.f32 N/A

         \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{\sqrt{x \cdot x - 1}} \cdot \sqrt{x \cdot x - 1}\right|}\right) \]

      6. lift-sqrt.f32 N/A

         \[\leadsto \log \left(x + \sqrt{\left|\sqrt{x \cdot x - 1} \cdot \color{blue}{\sqrt{x \cdot x - 1}}\right|}\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{x \cdot x - 1}\right|}\right) \]

      8. lift--.f32 N/A

         \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{x \cdot x - 1}\right|}\right) \]

      9. lift-*.f32 N/A

         \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{x \cdot x} - 1\right|}\right) \]

      10. difference-of-sqr-1 N/A

          \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}\right|}\right) \]

      11. *-commutative N/A

          \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{\left(x - 1\right) \cdot \left(x + 1\right)}\right|}\right) \]

      12. fabs-mul N/A

          \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|x - 1\right| \cdot \left|x + 1\right|}}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|x - 1\right| \cdot \left|x + 1\right|}}\right) \]

      14. fabs-sub N/A

          \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|1 - x\right|} \cdot \left|x + 1\right|}\right) \]

      15. lower-fabs.f32 N/A

          \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|1 - x\right|} \cdot \left|x + 1\right|}\right) \]

      16. lower--.f32 N/A

          \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{1 - x}\right| \cdot \left|x + 1\right|}\right) \]

      17. add-flip N/A

          \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \left|\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right|}\right) \]

      18. fabs-sub N/A

          \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \color{blue}{\left|\left(\mathsf{neg}\left(1\right)\right) - x\right|}}\right) \]

      19. lower-fabs.f32 N/A

          \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \color{blue}{\left|\left(\mathsf{neg}\left(1\right)\right) - x\right|}}\right) \]

      20. lower--.f32 N/A

          \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \left|\color{blue}{\left(\mathsf{neg}\left(1\right)\right) - x}\right|}\right) \]

      21. metadata-eval 55.1%

          \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \left|\color{blue}{-1} - x\right|}\right) \]

   3. Applied rewrites 55.1%

      \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|1 - x\right| \cdot \left|-1 - x\right|}}\right) \]

   4. Taylor expanded in x around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 6.7%

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

   6. Applied rewrites 6.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto x \cdot 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 9.4%

      \[\leadsto x \cdot 1 \]

   10. Taylor expanded in undef-var around zero

       \[\leadsto 0 \cdot 1 \]
   11. Step-by-step derivation

   12. Applied rewrites 76.1%

       \[\leadsto 0 \cdot 1 \]

   if 0.5 < x

   1. Initial program 14.3%

      \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
   2. Step-by-step derivation

      1. lift-log.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. acosh-def-rev N/A

         \[\leadsto \color{blue}{\cosh^{-1} x} \]

      7. lower-acosh.f32 25.3%

         \[\leadsto \color{blue}{\cosh^{-1} x} \]

   3. Applied rewrites 25.3%

      \[\leadsto \color{blue}{\cosh^{-1} x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 76.1% accurate, 4.3× speedup

Math

\[0 \cdot 1 \]

FPCore

(FPCore (x)
  :precision binary32
  (* 0.0 1.0))

C

float code(float x) {
	return 0.0f * 1.0f;
}

Fortran

real(4) function code(x)
use fmin_fmax_functions
    real(4), intent (in) :: x
    code = 0.0e0 * 1.0e0
end function

Julia

function code(x)
	return Float32(Float32(0.0) * Float32(1.0))
end

MATLAB

function tmp = code(x)
	tmp = single(0.0) * single(1.0);
end

Derivation

1. Initial program 14.3%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Step-by-step derivation

   1. rem-square-sqrt N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. fabs-sqr N/A

      \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|\sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}\right|}}\right) \]

   5. lift-sqrt.f32 N/A

      \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{\sqrt{x \cdot x - 1}} \cdot \sqrt{x \cdot x - 1}\right|}\right) \]

   6. lift-sqrt.f32 N/A

      \[\leadsto \log \left(x + \sqrt{\left|\sqrt{x \cdot x - 1} \cdot \color{blue}{\sqrt{x \cdot x - 1}}\right|}\right) \]

   7. rem-square-sqrt N/A

      \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{x \cdot x - 1}\right|}\right) \]

   8. lift--.f32 N/A

      \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{x \cdot x - 1}\right|}\right) \]

   9. lift-*.f32 N/A

      \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{x \cdot x} - 1\right|}\right) \]

   10. difference-of-sqr-1 N/A

       \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}\right|}\right) \]

   11. *-commutative N/A

       \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{\left(x - 1\right) \cdot \left(x + 1\right)}\right|}\right) \]

   12. fabs-mul N/A

       \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|x - 1\right| \cdot \left|x + 1\right|}}\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|x - 1\right| \cdot \left|x + 1\right|}}\right) \]

   14. fabs-sub N/A

       \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|1 - x\right|} \cdot \left|x + 1\right|}\right) \]

   15. lower-fabs.f32 N/A

       \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|1 - x\right|} \cdot \left|x + 1\right|}\right) \]

   16. lower--.f32 N/A

       \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{1 - x}\right| \cdot \left|x + 1\right|}\right) \]

   17. add-flip N/A

       \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \left|\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right|}\right) \]

   18. fabs-sub N/A

       \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \color{blue}{\left|\left(\mathsf{neg}\left(1\right)\right) - x\right|}}\right) \]

   19. lower-fabs.f32 N/A

       \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \color{blue}{\left|\left(\mathsf{neg}\left(1\right)\right) - x\right|}}\right) \]

   20. lower--.f32 N/A

       \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \left|\color{blue}{\left(\mathsf{neg}\left(1\right)\right) - x}\right|}\right) \]

   21. metadata-eval 55.1%

       \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \left|\color{blue}{-1} - x\right|}\right) \]

3. Applied rewrites 55.1%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|1 - x\right| \cdot \left|-1 - x\right|}}\right) \]

4. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 6.7%

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

6. Applied rewrites 6.7%

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

7. Taylor expanded in x around 0

   \[\leadsto x \cdot 1 \]
8. Step-by-step derivation

9. Applied rewrites 9.4%

   \[\leadsto x \cdot 1 \]

10. Taylor expanded in undef-var around zero

    \[\leadsto 0 \cdot 1 \]
11. Step-by-step derivation

12. Applied rewrites 76.1%

    \[\leadsto 0 \cdot 1 \]
13. Add Preprocessing

Alternative 4: 9.4% accurate, 4.3× speedup

Math

\[x \cdot 1 \]

FPCore

(FPCore (x)
  :precision binary32
  (* x 1.0))

C

float code(float x) {
	return x * 1.0f;
}

Fortran

real(4) function code(x)
use fmin_fmax_functions
    real(4), intent (in) :: x
    code = x * 1.0e0
end function

Julia

function code(x)
	return Float32(x * Float32(1.0))
end

MATLAB

function tmp = code(x)
	tmp = x * single(1.0);
end

Derivation

1. Initial program 14.3%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Step-by-step derivation

   1. rem-square-sqrt N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. fabs-sqr N/A

      \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|\sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}\right|}}\right) \]

   5. lift-sqrt.f32 N/A

      \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{\sqrt{x \cdot x - 1}} \cdot \sqrt{x \cdot x - 1}\right|}\right) \]

   6. lift-sqrt.f32 N/A

      \[\leadsto \log \left(x + \sqrt{\left|\sqrt{x \cdot x - 1} \cdot \color{blue}{\sqrt{x \cdot x - 1}}\right|}\right) \]

   7. rem-square-sqrt N/A

      \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{x \cdot x - 1}\right|}\right) \]

   8. lift--.f32 N/A

      \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{x \cdot x - 1}\right|}\right) \]

   9. lift-*.f32 N/A

      \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{x \cdot x} - 1\right|}\right) \]

   10. difference-of-sqr-1 N/A

       \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}\right|}\right) \]

   11. *-commutative N/A

       \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{\left(x - 1\right) \cdot \left(x + 1\right)}\right|}\right) \]

   12. fabs-mul N/A

       \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|x - 1\right| \cdot \left|x + 1\right|}}\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|x - 1\right| \cdot \left|x + 1\right|}}\right) \]

   14. fabs-sub N/A

       \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|1 - x\right|} \cdot \left|x + 1\right|}\right) \]

   15. lower-fabs.f32 N/A

       \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|1 - x\right|} \cdot \left|x + 1\right|}\right) \]

   16. lower--.f32 N/A

       \[\leadsto \log \left(x + \sqrt{\left|\color{blue}{1 - x}\right| \cdot \left|x + 1\right|}\right) \]

   17. add-flip N/A

       \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \left|\color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right|}\right) \]

   18. fabs-sub N/A

       \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \color{blue}{\left|\left(\mathsf{neg}\left(1\right)\right) - x\right|}}\right) \]

   19. lower-fabs.f32 N/A

       \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \color{blue}{\left|\left(\mathsf{neg}\left(1\right)\right) - x\right|}}\right) \]

   20. lower--.f32 N/A

       \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \left|\color{blue}{\left(\mathsf{neg}\left(1\right)\right) - x}\right|}\right) \]

   21. metadata-eval 55.1%

       \[\leadsto \log \left(x + \sqrt{\left|1 - x\right| \cdot \left|\color{blue}{-1} - x\right|}\right) \]

3. Applied rewrites 55.1%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{\left|1 - x\right| \cdot \left|-1 - x\right|}}\right) \]

4. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 6.7%

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

6. Applied rewrites 6.7%

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

7. Taylor expanded in x around 0

   \[\leadsto x \cdot 1 \]
8. Step-by-step derivation

9. Applied rewrites 9.4%

   \[\leadsto x \cdot 1 \]
10. Add Preprocessing

Developer Target 1: 25.2% accurate, 0.8× speedup

Math

\[\log \left(x + \sqrt{x - 1} \cdot \sqrt{x + 1}\right) \]

FPCore

(FPCore (x)
  :precision binary32
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0))))))

C

float code(float x) {
	return logf((x + (sqrtf((x - 1.0f)) * sqrtf((x + 1.0f)))));
}

Fortran

real(4) function code(x)
use fmin_fmax_functions
    real(4), intent (in) :: x
    code = log((x + (sqrt((x - 1.0e0)) * sqrt((x + 1.0e0)))))
end function

Julia

function code(x)
	return log(Float32(x + Float32(sqrt(Float32(x - Float32(1.0))) * sqrt(Float32(x + Float32(1.0))))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + (sqrt((x - single(1.0))) * sqrt((x + single(1.0))))));
end

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :alt
  (! :herbie-platform c (log (+ x (* (sqrt (- x 1)) (sqrt (+ x 1))))))

  (log (+ x (sqrt (- (* x x) 1.0)))))