Rust f32::atanh

Percentage Accurate: 99.8% → 99.8%

Time: 1.8s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

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

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

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, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(0.5 \cdot \mathsf{log1p}\left(\frac{x\_m + x\_m}{1 - x\_m}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f32 x)
x\_s = (copysign.f32 #s(literal 1 binary32) x)
(FPCore (x_s x_m)
 :precision binary32
 (* x_s (* 0.5 (log1p (/ (+ x_m x_m) (- 1.0 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
float code(float x_s, float x_m) {
	return x_s * (0.5f * log1pf(((x_m + x_m) / (1.0f - x_m))));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float32(x_s * Float32(Float32(0.5) * log1p(Float32(Float32(x_m + x_m) / Float32(Float32(1.0) - x_m)))))
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%

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

Alternative 2: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.2, 0.3333333333333333\right) \cdot x\_m\right) \cdot x\_m, x\_m, x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f32 x)
x\_s = (copysign.f32 #s(literal 1 binary32) x)
(FPCore (x_s x_m)
 :precision binary32
 (*
  x_s
  (fma (* (* (fma (* x_m x_m) 0.2 0.3333333333333333) x_m) x_m) x_m x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
float code(float x_s, float x_m) {
	return x_s * fmaf(((fmaf((x_m * x_m), 0.2f, 0.3333333333333333f) * x_m) * x_m), x_m, x_m);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float32(x_s * fma(Float32(Float32(fma(Float32(x_m * x_m), Float32(0.2), Float32(0.3333333333333333)) * x_m) * x_m), x_m, x_m))
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

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot \color{blue}{x} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot \color{blue}{x} \]

   3. +-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) + 1\right) \cdot x \]

   4. *-commutative N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot x \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}, {x}^{2}, 1\right) \cdot x \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{5} \cdot {x}^{2} + \frac{1}{3}, {x}^{2}, 1\right) \cdot x \]

   7. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 1\right) \cdot x \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), {x}^{2}, 1\right) \cdot x \]

   9. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), {x}^{2}, 1\right) \cdot x \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), x \cdot x, 1\right) \cdot x \]

   11. lower-*.f32 98.9

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x \]

4. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), x \cdot x, 1\right) \cdot \color{blue}{x} \]

   2. lift-fma.f32 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

   3. lift-fma.f32 N/A

      \[\leadsto \left(\left(\frac{1}{5} \cdot \left(x \cdot x\right) + \frac{1}{3}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

   5. lift-*.f32 N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

   6. pow2 N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

   7. lift-*.f32 N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

   8. pow2 N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot x \]

   9. *-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) + 1\right) \cdot x \]

   10. +-commutative N/A

       \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot x \]

   11. *-commutative N/A

       \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]

   12. distribute-lft-in N/A

       \[\leadsto x \cdot 1 + \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]

   13. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{1}, x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(x, 1, x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, 1, x \cdot \left(\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right) \]

6. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{1}, x \cdot \left(\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right) \cdot x\right) \cdot x\right)\right) \]
7. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto x \cdot 1 + \color{blue}{x \cdot \left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x\right)} \]

   2. *-rgt-identity N/A

      \[\leadsto x + \color{blue}{x} \cdot \left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x\right) \]

   3. lift-*.f32 N/A

      \[\leadsto x + x \cdot \color{blue}{\left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x\right)} \]

   4. lift-*.f32 N/A

      \[\leadsto x + x \cdot \left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot \color{blue}{x}\right) \]

   5. lift-*.f32 N/A

      \[\leadsto x + x \cdot \left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x\right) \]

   6. lift-*.f32 N/A

      \[\leadsto x + x \cdot \left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x\right) \]

   7. lift-fma.f32 N/A

      \[\leadsto x + x \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}\right) \cdot x\right) \cdot x\right) \]

   8. +-commutative N/A

      \[\leadsto x \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}\right) \cdot x\right) \cdot x\right) + \color{blue}{x} \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}\right) \cdot x\right) \cdot x\right) \cdot x + x \]

   10. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}\right) \cdot x\right) \cdot x, \color{blue}{x}, x\right) \]

   11. lift-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x, x, x\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x, x, x\right) \]

   13. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x, x, x\right) \]

   14. lift-*.f32 99.0

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right) \cdot x\right) \cdot x, x, x\right) \]

8. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right) \cdot x\right) \cdot x, \color{blue}{x}, x\right) \]
9. Add Preprocessing

Alternative 3: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.2, x\_m \cdot x\_m, 0.3333333333333333\right), x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f32 x)
x\_s = (copysign.f32 #s(literal 1 binary32) x)
(FPCore (x_s x_m)
 :precision binary32
 (*
  x_s
  (* (fma (fma 0.2 (* x_m x_m) 0.3333333333333333) (* x_m x_m) 1.0) x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
float code(float x_s, float x_m) {
	return x_s * (fmaf(fmaf(0.2f, (x_m * x_m), 0.3333333333333333f), (x_m * x_m), 1.0f) * x_m);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float32(x_s * Float32(fma(fma(Float32(0.2), Float32(x_m * x_m), Float32(0.3333333333333333)), Float32(x_m * x_m), Float32(1.0)) * x_m))
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

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot \color{blue}{x} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot \color{blue}{x} \]

   3. +-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) + 1\right) \cdot x \]

   4. *-commutative N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot x \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}, {x}^{2}, 1\right) \cdot x \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{5} \cdot {x}^{2} + \frac{1}{3}, {x}^{2}, 1\right) \cdot x \]

   7. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 1\right) \cdot x \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), {x}^{2}, 1\right) \cdot x \]

   9. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), {x}^{2}, 1\right) \cdot x \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), x \cdot x, 1\right) \cdot x \]

   11. lower-*.f32 98.9

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x \]

4. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \mathsf{fma}\left(\left(0.3333333333333333 \cdot x\_m\right) \cdot x\_m, x\_m, x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f32 x)
x\_s = (copysign.f32 #s(literal 1 binary32) x)
(FPCore (x_s x_m)
 :precision binary32
 (* x_s (fma (* (* 0.3333333333333333 x_m) x_m) x_m x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
float code(float x_s, float x_m) {
	return x_s * fmaf(((0.3333333333333333f * x_m) * x_m), x_m, x_m);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float32(x_s * fma(Float32(Float32(Float32(0.3333333333333333) * x_m) * x_m), x_m, x_m))
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

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot \color{blue}{x} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot \color{blue}{x} \]

   3. +-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) + 1\right) \cdot x \]

   4. *-commutative N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot x \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}, {x}^{2}, 1\right) \cdot x \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{5} \cdot {x}^{2} + \frac{1}{3}, {x}^{2}, 1\right) \cdot x \]

   7. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 1\right) \cdot x \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), {x}^{2}, 1\right) \cdot x \]

   9. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), {x}^{2}, 1\right) \cdot x \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), x \cdot x, 1\right) \cdot x \]

   11. lower-*.f32 98.9

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x \]

4. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), x \cdot x, 1\right) \cdot \color{blue}{x} \]

   2. lift-fma.f32 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

   3. lift-fma.f32 N/A

      \[\leadsto \left(\left(\frac{1}{5} \cdot \left(x \cdot x\right) + \frac{1}{3}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

   5. lift-*.f32 N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

   6. pow2 N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

   7. lift-*.f32 N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

   8. pow2 N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot x \]

   9. *-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) + 1\right) \cdot x \]

   10. +-commutative N/A

       \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot x \]

   11. *-commutative N/A

       \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]

   12. distribute-lft-in N/A

       \[\leadsto x \cdot 1 + \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]

   13. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{1}, x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(x, 1, x \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, 1, x \cdot \left(\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right) \]

6. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{1}, x \cdot \left(\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right) \cdot x\right) \cdot x\right)\right) \]
7. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto x \cdot 1 + \color{blue}{x \cdot \left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x\right)} \]

   2. *-rgt-identity N/A

      \[\leadsto x + \color{blue}{x} \cdot \left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x\right) \]

   3. lift-*.f32 N/A

      \[\leadsto x + x \cdot \color{blue}{\left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x\right)} \]

   4. lift-*.f32 N/A

      \[\leadsto x + x \cdot \left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot \color{blue}{x}\right) \]

   5. lift-*.f32 N/A

      \[\leadsto x + x \cdot \left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x\right) \]

   6. lift-*.f32 N/A

      \[\leadsto x + x \cdot \left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x\right) \]

   7. lift-fma.f32 N/A

      \[\leadsto x + x \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}\right) \cdot x\right) \cdot x\right) \]

   8. +-commutative N/A

      \[\leadsto x \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}\right) \cdot x\right) \cdot x\right) + \color{blue}{x} \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}\right) \cdot x\right) \cdot x\right) \cdot x + x \]

   10. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}\right) \cdot x\right) \cdot x, \color{blue}{x}, x\right) \]

   11. lift-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x, x, x\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x, x, x\right) \]

   13. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{1}{3}\right) \cdot x\right) \cdot x, x, x\right) \]

   14. lift-*.f32 99.0

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right) \cdot x\right) \cdot x, x, x\right) \]

8. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right) \cdot x\right) \cdot x, \color{blue}{x}, x\right) \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x\right) \cdot x, x, x\right) \]
10. Step-by-step derivation

11. Applied rewrites 98.5%

    \[\leadsto \mathsf{fma}\left(\left(0.3333333333333333 \cdot x\right) \cdot x, x, x\right) \]
12. Add Preprocessing

Alternative 5: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.3333333333333333, 1\right) \cdot x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f32 x)
x\_s = (copysign.f32 #s(literal 1 binary32) x)
(FPCore (x_s x_m)
 :precision binary32
 (* x_s (* (fma (* x_m x_m) 0.3333333333333333 1.0) x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
float code(float x_s, float x_m) {
	return x_s * (fmaf((x_m * x_m), 0.3333333333333333f, 1.0f) * x_m);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float32(x_s * Float32(fma(Float32(x_m * x_m), Float32(0.3333333333333333), Float32(1.0)) * x_m))
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

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{3} \cdot {x}^{2}\right) \cdot \color{blue}{x} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(1 + \frac{1}{3} \cdot {x}^{2}\right) \cdot \color{blue}{x} \]

   3. +-commutative N/A

      \[\leadsto \left(\frac{1}{3} \cdot {x}^{2} + 1\right) \cdot x \]

   4. *-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \frac{1}{3} + 1\right) \cdot x \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left({x}^{2}, \frac{1}{3}, 1\right) \cdot x \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{3}, 1\right) \cdot x \]

   7. lower-*.f32 98.5

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right) \cdot x \]

4. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right) \cdot x} \]
5. Add Preprocessing

Alternative 6: 96.8% accurate, 23.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f32 x)
x\_s = (copysign.f32 #s(literal 1 binary32) x)
(FPCore (x_s x_m) :precision binary32 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
float code(float x_s, float x_m) {
	return x_s * x_m;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m)
use fmin_fmax_functions
    real(4), intent (in) :: x_s
    real(4), intent (in) :: x_m
    code = x_s * x_m
end function

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float32(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
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 96.8%

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

Reproduce

herbie shell --seed 2025139 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))