\[x + y \cdot \frac{z - t}{z - a} \]

↓

\[x + \frac{z - t}{\frac{z - a}{y}} \]

(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- z a)))))

↓

(FPCore (x y z t a) :precision binary64 (+ x (/ (- z t) (/ (- z a) y))))

double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (z - a)));
}

↓

double code(double x, double y, double z, double t, double a) {
	return x + ((z - t) / ((z - a) / y));
}

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (y * ((z - t) / (z - a)))
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((z - t) / ((z - a) / y))
end function

public static double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (z - a)));
}

↓

public static double code(double x, double y, double z, double t, double a) {
	return x + ((z - t) / ((z - a) / y));
}

def code(x, y, z, t, a):
	return x + (y * ((z - t) / (z - a)))

↓

def code(x, y, z, t, a):
	return x + ((z - t) / ((z - a) / y))

function code(x, y, z, t, a)
	return Float64(x + Float64(y * Float64(Float64(z - t) / Float64(z - a))))
end

↓

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(z - t) / Float64(Float64(z - a) / y)))
end

function tmp = code(x, y, z, t, a)
	tmp = x + (y * ((z - t) / (z - a)));
end

↓

function tmp = code(x, y, z, t, a)
	tmp = x + ((z - t) / ((z - a) / y));
end

code[x_, y_, z_, t_, a_] := N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(z - t), $MachinePrecision] / N[(N[(z - a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x + y \cdot \frac{z - t}{z - a}

↓

x + \frac{z - t}{\frac{z - a}{y}}

Original	1.2
Target	1.2
Herbie	3.1

Derivation

Initial program 1.2
\[x + y \cdot \frac{z - t}{z - a} \]
Taylor expanded in y around 0 10.5
\[\leadsto x + \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} \]
Simplified3.1
\[\leadsto x + \color{blue}{\frac{z - t}{\frac{z - a}{y}}} \]
Proof
(/.f64 (-.f64 z t) (/.f64 (-.f64 z a) y)): 0 points increase in error, 0 points decrease in error
(Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 z t) y) (-.f64 z a))): 77 points increase in error, 46 points decrease in error
(/.f64 (Rewrite=> *-commutative_binary64 (*.f64 y (-.f64 z t))) (-.f64 z a)): 0 points increase in error, 0 points decrease in error
Final simplification3.1
\[\leadsto x + \frac{z - t}{\frac{z - a}{y}} \]

Alternatives

Alternative 1
Error	13.0
Cost	1236

\[\begin{array}{l} t_1 := x + \frac{y}{a} \cdot \left(t - z\right)\\ t_2 := x + z \cdot \frac{y}{z - a}\\ \mathbf{if}\;a \leq -1.25 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{z - t}{\frac{z}{y}}\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	12.9
Cost	1236

\[\begin{array}{l} t_1 := x + z \cdot \frac{y}{z - a}\\ t_2 := x + \frac{t - z}{\frac{a}{y}}\\ \mathbf{if}\;a \leq -1.2 \cdot 10^{-50}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;a \leq 0.053:\\ \;\;\;\;x + \frac{z - t}{\frac{z}{y}}\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{+27}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	23.1
Cost	984

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-225}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-297}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-256}:\\ \;\;\;\;y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-236}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+18}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+133}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4
Error	24.2
Cost	852

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{+178}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.72 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-113}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-26}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5
Error	15.2
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+96}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-18}:\\ \;\;\;\;x - \frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 1.48 \cdot 10^{-84}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 6
Error	11.9
Cost	840

\[\begin{array}{l} t_1 := x + z \cdot \frac{y}{z - a}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-85}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	14.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00023:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.48 \cdot 10^{-84}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 8
Error	1.2
Cost	704

\[x + y \cdot \frac{z - t}{z - a} \]

Alternative 9
Error	19.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -0.03:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-79}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 10
Error	28.3
Cost	64

\[x \]

Error

Try it out

Target

Derivation

Alternatives

Error

Reproduce

In
Out