Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 2.8s

Alternatives: 5

Speedup: 0.7×

Specification

Math

\[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]

FPCore

(FPCore (x) :precision binary32 (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))

C

float code(float x) {
	return 0.5f * log1pf(((2.0f * x) / (1.0f - x)));
}

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(2.0) * x) / Float32(Float32(1.0) - x))))
end

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]

FPCore

(FPCore (x) :precision binary32 (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))

C

float code(float x) {
	return 0.5f * log1pf(((2.0f * x) / (1.0f - x)));
}

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(2.0) * x) / Float32(Float32(1.0) - x))))
end

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \left(0.5 \cdot \mathsf{log1p}\left(\frac{\left|x\right| + \left|x\right|}{1 - \left|x\right|}\right)\right) \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  (copysign 1.0 x)
  (* 0.5 (log1p (/ (+ (fabs x) (fabs x)) (- 1.0 (fabs x)))))))

C

float code(float x) {
	return copysignf(1.0f, x) * (0.5f * log1pf(((fabsf(x) + fabsf(x)) / (1.0f - fabsf(x)))));
}

Julia

function code(x)
	return Float32(copysign(Float32(1.0), x) * Float32(Float32(0.5) * log1p(Float32(Float32(abs(x) + abs(x)) / Float32(Float32(1.0) - abs(x))))))
end

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{2 \cdot x}}{1 - x}\right) \]

   2. count-2-rev N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{x + x}}{1 - x}\right) \]

   3. lower-+.f32 99.8%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{x + x}}{1 - x}\right) \]

3. Applied rewrites 99.8%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{x + x}}{1 - x}\right) \]
4. Add Preprocessing

Alternative 2: 99.3% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004000000189989805:\\ \;\;\;\;\left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{-2}{\left|x\right| - 1}, \left|x\right|, 1\right)\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  (copysign 1.0 x)
  (if (<= (fabs x) 0.004000000189989805)
    (fabs x)
    (* (log (fma (/ -2.0 (- (fabs x) 1.0)) (fabs x) 1.0)) 0.5))))

C

float code(float x) {
	float tmp;
	if (fabsf(x) <= 0.004000000189989805f) {
		tmp = fabsf(x);
	} else {
		tmp = logf(fmaf((-2.0f / (fabsf(x) - 1.0f)), fabsf(x), 1.0f)) * 0.5f;
	}
	return copysignf(1.0f, x) * tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.004000000189989805))
		tmp = abs(x);
	else
		tmp = Float32(log(fma(Float32(Float32(-2.0) / Float32(abs(x) - Float32(1.0))), abs(x), Float32(1.0))) * Float32(0.5));
	end
	return Float32(copysign(Float32(1.0), x) * tmp)
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00400000019

   1. Initial program 99.8%

      \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 97.0%

      \[\leadsto \color{blue}{x} \]

   if 0.00400000019 < x

   1. Initial program 99.8%

      \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \cdot \frac{1}{2}} \]

      3. lower-*.f32 99.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \cdot 0.5} \]

      4. lift-log1p.f32 N/A

         \[\leadsto \color{blue}{\log \left(1 + \frac{2 \cdot x}{1 - x}\right)} \cdot \frac{1}{2} \]

      5. lower-log.f32 N/A

         \[\leadsto \color{blue}{\log \left(1 + \frac{2 \cdot x}{1 - x}\right)} \cdot \frac{1}{2} \]

      6. +-commutative N/A

         \[\leadsto \log \color{blue}{\left(\frac{2 \cdot x}{1 - x} + 1\right)} \cdot \frac{1}{2} \]

      7. metadata-eval N/A

         \[\leadsto \log \left(\frac{2 \cdot x}{1 - x} + \color{blue}{\left(1 + 0\right)}\right) \cdot \frac{1}{2} \]

      8. lift-/.f32 N/A

         \[\leadsto \log \left(\color{blue}{\frac{2 \cdot x}{1 - x}} + \left(1 + 0\right)\right) \cdot \frac{1}{2} \]

      9. lift-*.f32 N/A

         \[\leadsto \log \left(\frac{\color{blue}{2 \cdot x}}{1 - x} + \left(1 + 0\right)\right) \cdot \frac{1}{2} \]

      10. *-commutative N/A

          \[\leadsto \log \left(\frac{\color{blue}{x \cdot 2}}{1 - x} + \left(1 + 0\right)\right) \cdot \frac{1}{2} \]

      11. associate-/l* N/A

          \[\leadsto \log \left(\color{blue}{x \cdot \frac{2}{1 - x}} + \left(1 + 0\right)\right) \cdot \frac{1}{2} \]

      12. *-commutative N/A

          \[\leadsto \log \left(\color{blue}{\frac{2}{1 - x} \cdot x} + \left(1 + 0\right)\right) \cdot \frac{1}{2} \]

      13. metadata-eval N/A

          \[\leadsto \log \left(\frac{2}{1 - x} \cdot x + \color{blue}{1}\right) \cdot \frac{1}{2} \]

      14. lower-fma.f32 N/A

          \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\frac{2}{1 - x}, x, 1\right)\right)} \cdot \frac{1}{2} \]

      15. frac-2neg N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(1 - x\right)\right)}}, x, 1\right)\right) \cdot \frac{1}{2} \]

      16. lower-/.f32 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(1 - x\right)\right)}}, x, 1\right)\right) \cdot \frac{1}{2} \]

      17. metadata-eval N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\frac{\color{blue}{-2}}{\mathsf{neg}\left(\left(1 - x\right)\right)}, x, 1\right)\right) \cdot \frac{1}{2} \]

      18. lift--.f32 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\frac{-2}{\mathsf{neg}\left(\color{blue}{\left(1 - x\right)}\right)}, x, 1\right)\right) \cdot \frac{1}{2} \]

      19. sub-negate-rev N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\frac{-2}{\color{blue}{x - 1}}, x, 1\right)\right) \cdot \frac{1}{2} \]

      20. lower--.f32 23.6%

          \[\leadsto \log \left(\mathsf{fma}\left(\frac{-2}{\color{blue}{x - 1}}, x, 1\right)\right) \cdot 0.5 \]

   3. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\log \left(\mathsf{fma}\left(\frac{-2}{x - 1}, x, 1\right)\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.3% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004000000189989805:\\ \;\;\;\;\left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(-2, \frac{\left|x\right|}{\left|x\right| - 1}, 1\right)\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  (copysign 1.0 x)
  (if (<= (fabs x) 0.004000000189989805)
    (fabs x)
    (* (log (fma -2.0 (/ (fabs x) (- (fabs x) 1.0)) 1.0)) 0.5))))

C

float code(float x) {
	float tmp;
	if (fabsf(x) <= 0.004000000189989805f) {
		tmp = fabsf(x);
	} else {
		tmp = logf(fmaf(-2.0f, (fabsf(x) / (fabsf(x) - 1.0f)), 1.0f)) * 0.5f;
	}
	return copysignf(1.0f, x) * tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.004000000189989805))
		tmp = abs(x);
	else
		tmp = Float32(log(fma(Float32(-2.0), Float32(abs(x) / Float32(abs(x) - Float32(1.0))), Float32(1.0))) * Float32(0.5));
	end
	return Float32(copysign(Float32(1.0), x) * tmp)
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00400000019

   1. Initial program 99.8%

      \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 97.0%

      \[\leadsto \color{blue}{x} \]

   if 0.00400000019 < x

   1. Initial program 99.8%

      \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \cdot \frac{1}{2}} \]

      3. lower-*.f32 99.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \cdot 0.5} \]

      4. lift-log1p.f32 N/A

         \[\leadsto \color{blue}{\log \left(1 + \frac{2 \cdot x}{1 - x}\right)} \cdot \frac{1}{2} \]

      5. lower-log.f32 N/A

         \[\leadsto \color{blue}{\log \left(1 + \frac{2 \cdot x}{1 - x}\right)} \cdot \frac{1}{2} \]

      6. +-commutative N/A

         \[\leadsto \log \color{blue}{\left(\frac{2 \cdot x}{1 - x} + 1\right)} \cdot \frac{1}{2} \]

      7. metadata-eval N/A

         \[\leadsto \log \left(\frac{2 \cdot x}{1 - x} + \color{blue}{\left(1 + 0\right)}\right) \cdot \frac{1}{2} \]

      8. lift-/.f32 N/A

         \[\leadsto \log \left(\color{blue}{\frac{2 \cdot x}{1 - x}} + \left(1 + 0\right)\right) \cdot \frac{1}{2} \]

      9. lift-*.f32 N/A

         \[\leadsto \log \left(\frac{\color{blue}{2 \cdot x}}{1 - x} + \left(1 + 0\right)\right) \cdot \frac{1}{2} \]

      10. *-commutative N/A

          \[\leadsto \log \left(\frac{\color{blue}{x \cdot 2}}{1 - x} + \left(1 + 0\right)\right) \cdot \frac{1}{2} \]

      11. associate-/l* N/A

          \[\leadsto \log \left(\color{blue}{x \cdot \frac{2}{1 - x}} + \left(1 + 0\right)\right) \cdot \frac{1}{2} \]

      12. *-commutative N/A

          \[\leadsto \log \left(\color{blue}{\frac{2}{1 - x} \cdot x} + \left(1 + 0\right)\right) \cdot \frac{1}{2} \]

      13. metadata-eval N/A

          \[\leadsto \log \left(\frac{2}{1 - x} \cdot x + \color{blue}{1}\right) \cdot \frac{1}{2} \]

      14. lower-fma.f32 N/A

          \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\frac{2}{1 - x}, x, 1\right)\right)} \cdot \frac{1}{2} \]

      15. frac-2neg N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(1 - x\right)\right)}}, x, 1\right)\right) \cdot \frac{1}{2} \]

      16. lower-/.f32 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(1 - x\right)\right)}}, x, 1\right)\right) \cdot \frac{1}{2} \]

      17. metadata-eval N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\frac{\color{blue}{-2}}{\mathsf{neg}\left(\left(1 - x\right)\right)}, x, 1\right)\right) \cdot \frac{1}{2} \]

      18. lift--.f32 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\frac{-2}{\mathsf{neg}\left(\color{blue}{\left(1 - x\right)}\right)}, x, 1\right)\right) \cdot \frac{1}{2} \]

      19. sub-negate-rev N/A

          \[\leadsto \log \left(\mathsf{fma}\left(\frac{-2}{\color{blue}{x - 1}}, x, 1\right)\right) \cdot \frac{1}{2} \]

      20. lower--.f32 23.6%

          \[\leadsto \log \left(\mathsf{fma}\left(\frac{-2}{\color{blue}{x - 1}}, x, 1\right)\right) \cdot 0.5 \]

   3. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\log \left(\mathsf{fma}\left(\frac{-2}{x - 1}, x, 1\right)\right) \cdot 0.5} \]
   4. Step-by-step derivation

      1. lift-fma.f32 N/A

         \[\leadsto \log \color{blue}{\left(\frac{-2}{x - 1} \cdot x + 1\right)} \cdot \frac{1}{2} \]

      2. lift-/.f32 N/A

         \[\leadsto \log \left(\color{blue}{\frac{-2}{x - 1}} \cdot x + 1\right) \cdot \frac{1}{2} \]

      3. lift--.f32 N/A

         \[\leadsto \log \left(\frac{-2}{\color{blue}{x - 1}} \cdot x + 1\right) \cdot \frac{1}{2} \]

      4. lift--.f32 N/A

         \[\leadsto \log \left(\frac{-2}{\color{blue}{x - 1}} \cdot x + 1\right) \cdot \frac{1}{2} \]

      5. associate-*l/ N/A

         \[\leadsto \log \left(\color{blue}{\frac{-2 \cdot x}{x - 1}} + 1\right) \cdot \frac{1}{2} \]

      6. associate-/l* N/A

         \[\leadsto \log \left(\color{blue}{-2 \cdot \frac{x}{x - 1}} + 1\right) \cdot \frac{1}{2} \]

      7. lower-fma.f32 N/A

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(-2, \frac{x}{x - 1}, 1\right)\right)} \cdot \frac{1}{2} \]

      8. lower-/.f32 23.6%

         \[\leadsto \log \left(\mathsf{fma}\left(-2, \color{blue}{\frac{x}{x - 1}}, 1\right)\right) \cdot 0.5 \]

   5. Applied rewrites 23.6%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(-2, \frac{x}{x - 1}, 1\right)\right)} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.2% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004999999888241291:\\ \;\;\;\;\left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\mathsf{fma}\left(\frac{-2}{\left|x\right| - 1}, \left|x\right|, 1\right)}\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  (copysign 1.0 x)
  (if (<= (fabs x) 0.004999999888241291)
    (fabs x)
    (log (sqrt (fma (/ -2.0 (- (fabs x) 1.0)) (fabs x) 1.0))))))

C

float code(float x) {
	float tmp;
	if (fabsf(x) <= 0.004999999888241291f) {
		tmp = fabsf(x);
	} else {
		tmp = logf(sqrtf(fmaf((-2.0f / (fabsf(x) - 1.0f)), fabsf(x), 1.0f)));
	}
	return copysignf(1.0f, x) * tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.004999999888241291))
		tmp = abs(x);
	else
		tmp = log(sqrt(fma(Float32(Float32(-2.0) / Float32(abs(x) - Float32(1.0))), abs(x), Float32(1.0))));
	end
	return Float32(copysign(Float32(1.0), x) * tmp)
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00499999989

   1. Initial program 99.8%

      \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 97.0%

      \[\leadsto \color{blue}{x} \]

   if 0.00499999989 < x

   1. Initial program 99.8%

      \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
   2. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)} \]

      2. lift-log1p.f32 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\log \left(1 + \frac{2 \cdot x}{1 - x}\right)} \]

      3. log-pow-rev N/A

         \[\leadsto \color{blue}{\log \left({\left(1 + \frac{2 \cdot x}{1 - x}\right)}^{\frac{1}{2}}\right)} \]

      4. lower-log.f32 N/A

         \[\leadsto \color{blue}{\log \left({\left(1 + \frac{2 \cdot x}{1 - x}\right)}^{\frac{1}{2}}\right)} \]

      5. unpow1/2 N/A

         \[\leadsto \log \color{blue}{\left(\sqrt{1 + \frac{2 \cdot x}{1 - x}}\right)} \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \log \color{blue}{\left(\sqrt{1 + \frac{2 \cdot x}{1 - x}}\right)} \]

      7. +-commutative N/A

         \[\leadsto \log \left(\sqrt{\color{blue}{\frac{2 \cdot x}{1 - x} + 1}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \log \left(\sqrt{\frac{2 \cdot x}{1 - x} + \color{blue}{\left(1 + 0\right)}}\right) \]

      9. lift-/.f32 N/A

         \[\leadsto \log \left(\sqrt{\color{blue}{\frac{2 \cdot x}{1 - x}} + \left(1 + 0\right)}\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \log \left(\sqrt{\frac{\color{blue}{2 \cdot x}}{1 - x} + \left(1 + 0\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log \left(\sqrt{\frac{\color{blue}{x \cdot 2}}{1 - x} + \left(1 + 0\right)}\right) \]

      12. associate-/l* N/A

          \[\leadsto \log \left(\sqrt{\color{blue}{x \cdot \frac{2}{1 - x}} + \left(1 + 0\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log \left(\sqrt{\color{blue}{\frac{2}{1 - x} \cdot x} + \left(1 + 0\right)}\right) \]

      14. metadata-eval N/A

          \[\leadsto \log \left(\sqrt{\frac{2}{1 - x} \cdot x + \color{blue}{1}}\right) \]

      15. lower-fma.f32 N/A

          \[\leadsto \log \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{2}{1 - x}, x, 1\right)}}\right) \]

      16. frac-2neg N/A

          \[\leadsto \log \left(\sqrt{\mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(1 - x\right)\right)}}, x, 1\right)}\right) \]

      17. lower-/.f32 N/A

          \[\leadsto \log \left(\sqrt{\mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(\left(1 - x\right)\right)}}, x, 1\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \log \left(\sqrt{\mathsf{fma}\left(\frac{\color{blue}{-2}}{\mathsf{neg}\left(\left(1 - x\right)\right)}, x, 1\right)}\right) \]

      19. lift--.f32 N/A

          \[\leadsto \log \left(\sqrt{\mathsf{fma}\left(\frac{-2}{\mathsf{neg}\left(\color{blue}{\left(1 - x\right)}\right)}, x, 1\right)}\right) \]

      20. sub-negate-rev N/A

          \[\leadsto \log \left(\sqrt{\mathsf{fma}\left(\frac{-2}{\color{blue}{x - 1}}, x, 1\right)}\right) \]

      21. lower--.f32 22.8%

          \[\leadsto \log \left(\sqrt{\mathsf{fma}\left(\frac{-2}{\color{blue}{x - 1}}, x, 1\right)}\right) \]

   3. Applied rewrites 22.8%

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

Alternative 5: 97.0% accurate, 23.2× speedup

Math

\[x \]

FPCore

(FPCore (x) :precision binary32 x)

C

float code(float x) {
	return x;
}

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

Julia

function code(x)
	return x
end

MATLAB

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

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 97.0%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))