Average Error: 21.9 → 0.1

Time: 8.7s

Precision: binary64

Cost: 15426

\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1175788890.492152:\\ \;\;\;\;\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + {\left(\frac{1}{y}\right)}^{3}\right)\right)\right) - \left(\frac{x}{y} + \left(1 + \frac{x}{y}\right) \cdot \frac{1}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 11013.798141579568:\\ \;\;\;\;\left(1 + \left(y \cdot y\right) \cdot \frac{y \cdot x - y}{1 + {y}^{3}}\right) + \frac{y \cdot x - y}{1 + {y}^{3}} \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + {\left(\frac{1}{y}\right)}^{3}\right)\right)\right) - \left(\frac{x}{y} + \left(1 + \frac{x}{y}\right) \cdot \frac{1}{y \cdot y}\right)\\ \end{array}\]

1 - \frac{\left(1 - x\right) \cdot y}{y + 1}

↓

\begin{array}{l}
\mathbf{if}\;y \leq -1175788890.492152:\\
\;\;\;\;\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + {\left(\frac{1}{y}\right)}^{3}\right)\right)\right) - \left(\frac{x}{y} + \left(1 + \frac{x}{y}\right) \cdot \frac{1}{y \cdot y}\right)\\

\mathbf{elif}\;y \leq 11013.798141579568:\\
\;\;\;\;\left(1 + \left(y \cdot y\right) \cdot \frac{y \cdot x - y}{1 + {y}^{3}}\right) + \frac{y \cdot x - y}{1 + {y}^{3}} \cdot \left(1 - y\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + {\left(\frac{1}{y}\right)}^{3}\right)\right)\right) - \left(\frac{x}{y} + \left(1 + \frac{x}{y}\right) \cdot \frac{1}{y \cdot y}\right)\\

\end{array}

(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -1175788890.492152)
   (-
    (+ (/ x (* y y)) (+ x (+ (/ 1.0 y) (pow (/ 1.0 y) 3.0))))
    (+ (/ x y) (* (+ 1.0 (/ x y)) (/ 1.0 (* y y)))))
   (if (<= y 11013.798141579568)
     (+
      (+ 1.0 (* (* y y) (/ (- (* y x) y) (+ 1.0 (pow y 3.0)))))
      (* (/ (- (* y x) y) (+ 1.0 (pow y 3.0))) (- 1.0 y)))
     (-
      (+ (/ x (* y y)) (+ x (+ (/ 1.0 y) (pow (/ 1.0 y) 3.0))))
      (+ (/ x y) (* (+ 1.0 (/ x y)) (/ 1.0 (* y y))))))))

double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}

↓

double code(double x, double y) {
	double tmp;
	if (y <= -1175788890.492152) {
		tmp = ((x / (y * y)) + (x + ((1.0 / y) + pow((1.0 / y), 3.0)))) - ((x / y) + ((1.0 + (x / y)) * (1.0 / (y * y))));
	} else if (y <= 11013.798141579568) {
		tmp = (1.0 + ((y * y) * (((y * x) - y) / (1.0 + pow(y, 3.0))))) + ((((y * x) - y) / (1.0 + pow(y, 3.0))) * (1.0 - y));
	} else {
		tmp = ((x / (y * y)) + (x + ((1.0 / y) + pow((1.0 / y), 3.0)))) - ((x / y) + ((1.0 + (x / y)) * (1.0 / (y * y))));
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	21.9
Target	0.2
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Alternatives

Alternative 1
Error	30.0
Cost	27648

\[\left(1 + \left(y \cdot y\right) \cdot \frac{x \cdot y - y}{1 + {y}^{3}}\right) + \sqrt[3]{{\left(\left(1 - y\right) \cdot \frac{x \cdot y - y}{1 + {y}^{3}}\right)}^{3}}\]

Alternative 2
Error	22.3
Cost	21440

\[\sqrt[3]{1 - \frac{\left(1 - x\right) \cdot y}{1 + y}} \cdot \left(\sqrt[3]{1 - \frac{\left(1 - x\right) \cdot y}{1 + y}} \cdot \sqrt[3]{1 - \frac{\left(1 - x\right) \cdot y}{1 + y}}\right)\]

Alternative 3
Error	22.3
Cost	21184

\[1 - \sqrt[3]{\frac{\left(1 - x\right) \cdot y}{1 + y}} \cdot \left(\sqrt[3]{\frac{\left(1 - x\right) \cdot y}{1 + y}} \cdot \sqrt[3]{\frac{\left(1 - x\right) \cdot y}{1 + y}}\right)\]

Alternative 4
Error	22.3
Cost	20672

\[1 - \frac{\sqrt[3]{\left(1 - x\right) \cdot y} \cdot \left(\sqrt[3]{\left(1 - x\right) \cdot y} \cdot \sqrt[3]{\left(1 - x\right) \cdot y}\right)}{1 + y}\]

Alternative 5
Error	16.9
Cost	20416

\[1 - \frac{y \cdot \frac{1 - x}{\sqrt[3]{1 + y} \cdot \sqrt[3]{1 + y}}}{\sqrt[3]{1 + y}}\]

Alternative 6
Error	14.4
Cost	20416

\[1 - \frac{1 - x}{\sqrt[3]{1 + y} \cdot \sqrt[3]{1 + y}} \cdot \frac{y}{\sqrt[3]{1 + y}}\]

Alternative 7
Error	22.3
Cost	20160

\[1 - \frac{\sqrt[3]{y} \cdot \left(\left(1 - x\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)}{1 + y}\]

Alternative 8
Error	29.8
Cost	15424

\[\left(1 + \left(y \cdot y\right) \cdot \frac{x \cdot y - y}{1 + {y}^{3}}\right) + \left(1 - y\right) \cdot \frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) - y \cdot y}{\left(1 + x\right) \cdot \left(y + {y}^{4}\right)}\]

Alternative 9
Error	24.4
Cost	14784

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

Alternative 10
Error	33.1
Cost	14528

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

Alternative 11
Error	42.2
Cost	14144

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

Alternative 12
Error	42.9
Cost	13888

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

Alternative 13
Error	23.3
Cost	13760

\[1 - \frac{1 - x}{\sqrt{1 + y}} \cdot \frac{y}{\sqrt{1 + y}}\]

Alternative 14
Error	25.1
Cost	13760

\[1 - \frac{y \cdot \frac{1 - x}{\sqrt{1 + y}}}{\sqrt{1 + y}}\]

Alternative 15
Error	42.8
Cost	13632

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

Alternative 16
Error	33.9
Cost	13568

\[1 - \frac{\sqrt[3]{{\left(\left(1 - x\right) \cdot y\right)}^{3}}}{1 + y}\]

Alternative 17
Error	43.6
Cost	13504

\[1 - \frac{e^{\log \left(\left(1 - x\right) \cdot y\right)}}{1 + y}\]

Alternative 18
Error	31.3
Cost	13504

\[e^{\log \left(1 - \frac{\left(1 - x\right) \cdot y}{1 + y}\right)}\]

Alternative 19
Error	38.2
Cost	13504

\[\log \left(e^{1 - \frac{\left(1 - x\right) \cdot y}{1 + y}}\right)\]

Alternative 20
Error	29.3
Cost	8704

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

Alternative 21
Error	32.2
Cost	8448

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

Alternative 22
Error	26.5
Cost	7680

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

Alternative 23
Error	26.5
Cost	7680

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

Alternative 24
Error	26.1
Cost	2112

\[\frac{1 - \frac{\left(1 - x\right) \cdot y}{1 + y} \cdot \frac{\left(1 - x\right) \cdot y}{1 + y}}{1 + \frac{\left(1 - x\right) \cdot y}{1 + y}}\]

Alternative 25
Error	32.3
Cost	1344

\[\left(x + \left(\frac{x}{y \cdot y} + \frac{1}{y}\right)\right) - \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)\]

Alternative 26
Error	32.3
Cost	1216

\[\left(\frac{1 - x}{y} + \left(x + \frac{x}{y \cdot y}\right)\right) - \frac{1}{y \cdot y}\]

Alternative 27
Error	24.4
Cost	1088

\[1 - \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1} \cdot \left(y - 1\right)\]

Alternative 28
Error	34.4
Cost	1088

\[1 - \frac{y \cdot \left(1 - x \cdot x\right)}{\left(1 + y\right) \cdot \left(1 + x\right)}\]

Alternative 29
Error	22.0
Cost	832

\[1 - \left(\left(1 - x\right) \cdot y\right) \cdot \frac{1}{1 + y}\]

Alternative 30
Error	31.8
Cost	832

\[1 - \left(y + y \cdot \left(x \cdot y - \left(x + y\right)\right)\right)\]

Alternative 31
Error	14.0
Cost	704

\[1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\]

Alternative 32
Error	21.9
Cost	704

\[1 - \frac{\left(1 - x\right) \cdot y}{1 + y}\]

Alternative 33
Error	15.7
Cost	576

\[1 + \frac{x}{\frac{1 + y}{y}}\]

Alternative 34
Error	23.6
Cost	576

\[1 + \frac{x \cdot y}{1 + y}\]

Alternative 35
Error	31.9
Cost	576

\[1 - x \cdot \left(y \cdot y - y\right)\]

Alternative 36
Error	32.1
Cost	576

\[\left(x + \frac{1}{y}\right) - \frac{x}{y}\]

Alternative 37
Error	23.6
Cost	576

\[1 - \frac{x \cdot y}{-1 - y}\]

Alternative 38
Error	32.1
Cost	448

\[x + \frac{1 - x}{y}\]

Alternative 39
Error	30.8
Cost	448

\[\frac{x}{\frac{1 + y}{y}}\]

Alternative 40
Error	31.6
Cost	448

\[1 - \left(1 - x\right) \cdot y\]

Alternative 41
Error	38.8
Cost	448

\[1 - \frac{y}{1 + y}\]

Alternative 42
Error	45.8
Cost	320

\[1 - \left(1 - x\right)\]

Alternative 43
Error	31.5
Cost	320

\[1 + x \cdot y\]

Alternative 44
Error	38.8
Cost	64

\[x\]

Alternative 45
Error	39.4
Cost	64

\[1\]

Alternative 46
Error	62.0
Cost	64

\[0\]

Alternative 47
Error	62.1
Cost	64

\[-1\]

Derivation

Split input into 2 regimes
if y < -1175788890.49215198 or 11013.7981415795675 < y
1. Initial program 45.0
  \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
2. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(\frac{x}{{y}^{2}} + \left(\frac{1}{{y}^{3}} + \left(\frac{1}{y} + x\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{1}{{y}^{2}} + \frac{x}{y}\right)\right)}\]
3. Simplified0.0
  \[\leadsto \color{blue}{\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + {\left(\frac{1}{y}\right)}^{3}\right)\right)\right) - \left(\frac{x}{y} + \left(1 + \frac{x}{y}\right) \cdot \frac{1}{y \cdot y}\right)}\]
4. Simplified0.0
  \[\leadsto \color{blue}{\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + {\left(\frac{1}{y}\right)}^{3}\right)\right)\right) - \left(\frac{x}{y} + \left(1 + \frac{x}{y}\right) \cdot \frac{1}{y \cdot y}\right)}\]
if -1175788890.49215198 < y < 11013.7981415795675
1. Initial program 0.1
  \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
2. Using strategy rm
3. Applied flip3-+_binary64_213860.1
  \[\leadsto 1 - \frac{\left(1 - x\right) \cdot y}{\color{blue}{\frac{{y}^{3} + {1}^{3}}{y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)}}}\]
4. Applied associate-/r/_binary64_213290.1
  \[\leadsto 1 - \color{blue}{\frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right)}\]
5. Applied cancel-sign-sub-inv_binary64_213490.1
  \[\leadsto \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\right) \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right)}\]
6. Simplified0.1
  \[\leadsto 1 + \color{blue}{\frac{x \cdot y - y}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(1 - y\right)\right)}\]
7. Using strategy rm
8. Applied distribute-rgt-in_binary64_213330.1
  \[\leadsto 1 + \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{x \cdot y - y}{1 + {y}^{3}} + \left(1 - y\right) \cdot \frac{x \cdot y - y}{1 + {y}^{3}}\right)}\]
9. Applied associate-+r+_binary64_213150.1
  \[\leadsto \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \frac{x \cdot y - y}{1 + {y}^{3}}\right) + \left(1 - y\right) \cdot \frac{x \cdot y - y}{1 + {y}^{3}}}\]
10. Simplified0.1
  \[\leadsto \color{blue}{\left(1 + \frac{x \cdot y - y}{1 + {y}^{3}} \cdot \left(y \cdot y\right)\right)} + \left(1 - y\right) \cdot \frac{x \cdot y - y}{1 + {y}^{3}}\]
11. Simplified0.1
  \[\leadsto \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \frac{x \cdot y - y}{1 + {y}^{3}}\right) + \left(1 - y\right) \cdot \frac{x \cdot y - y}{1 + {y}^{3}}}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1175788890.492152:\\ \;\;\;\;\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + {\left(\frac{1}{y}\right)}^{3}\right)\right)\right) - \left(\frac{x}{y} + \left(1 + \frac{x}{y}\right) \cdot \frac{1}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 11013.798141579568:\\ \;\;\;\;\left(1 + \left(y \cdot y\right) \cdot \frac{y \cdot x - y}{1 + {y}^{3}}\right) + \frac{y \cdot x - y}{1 + {y}^{3}} \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + {\left(\frac{1}{y}\right)}^{3}\right)\right)\right) - \left(\frac{x}{y} + \left(1 + \frac{x}{y}\right) \cdot \frac{1}{y \cdot y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))

Error

Try it out

Target

Alternatives

Error

Derivation

`if y < -1175788890.49215198 or 11013.7981415795675 < y`

`if -1175788890.49215198 < y < 11013.7981415795675`

Reproduce