\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]

↓

\[4 \cdot \frac{x - y}{z} + -2 \]

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

↓

(FPCore (x y z) :precision binary64 (+ (* 4.0 (/ (- x y) z)) -2.0))

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

↓

double code(double x, double y, double z) {
	return (4.0 * ((x - y) / z)) + -2.0;
}

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) / z)) + (-2.0d0)
end function

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

↓

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) / z)) + -2.0;
}

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

↓

def code(x, y, z):
	return (4.0 * ((x - y) / z)) + -2.0

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

↓

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) / z)) + -2.0)
end

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

↓

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) / z)) + -2.0;
end

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]

\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}

↓

4 \cdot \frac{x - y}{z} + -2

Original	0.1
Target	0.0
Herbie	0.0

Derivation

Initial program 0.1
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
Simplified0.2
\[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]
Proof
(*.f64 (/.f64 4 z) (+.f64 (-.f64 x y) (*.f64 z -1/2))): 0 points increase in error, 0 points decrease in error
(*.f64 (/.f64 4 z) (+.f64 (-.f64 x y) (*.f64 z (Rewrite<= metadata-eval (neg.f64 1/2))))): 0 points increase in error, 0 points decrease in error
(*.f64 (/.f64 4 z) (+.f64 (-.f64 x y) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 z 1/2))))): 0 points increase in error, 0 points decrease in error
(*.f64 (/.f64 4 z) (+.f64 (-.f64 x y) (neg.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (*.f64 z 1/2) 1))))): 0 points increase in error, 0 points decrease in error
(*.f64 (/.f64 4 z) (+.f64 (-.f64 x y) (neg.f64 (/.f64 (*.f64 z 1/2) (Rewrite<= metadata-eval (neg.f64 -1)))))): 0 points increase in error, 0 points decrease in error
(*.f64 (/.f64 4 z) (+.f64 (-.f64 x y) (Rewrite<= distribute-frac-neg_binary64 (/.f64 (neg.f64 (*.f64 z 1/2)) (neg.f64 -1))))): 0 points increase in error, 0 points decrease in error
(*.f64 (/.f64 4 z) (+.f64 (-.f64 x y) (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 z 1/2))) (neg.f64 -1)))): 0 points increase in error, 0 points decrease in error
(*.f64 (/.f64 4 z) (+.f64 (-.f64 x y) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 z 1/2) -1)) (neg.f64 -1)))): 0 points increase in error, 0 points decrease in error
(*.f64 (/.f64 4 z) (+.f64 (-.f64 x y) (Rewrite=> associate-/l*_binary64 (/.f64 (*.f64 z 1/2) (/.f64 (neg.f64 -1) -1))))): 0 points increase in error, 0 points decrease in error
(*.f64 (/.f64 4 z) (+.f64 (-.f64 x y) (/.f64 (*.f64 z 1/2) (/.f64 (Rewrite=> metadata-eval 1) -1)))): 0 points increase in error, 0 points decrease in error
(*.f64 (/.f64 4 z) (+.f64 (-.f64 x y) (/.f64 (*.f64 z 1/2) (Rewrite=> metadata-eval -1)))): 0 points increase in error, 0 points decrease in error
(Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 4 z) (-.f64 x y)) (*.f64 (/.f64 4 z) (/.f64 (*.f64 z 1/2) -1)))): 2 points increase in error, 4 points decrease in error
(+.f64 (*.f64 (/.f64 4 z) (-.f64 x y)) (*.f64 (/.f64 4 z) (/.f64 (*.f64 z 1/2) (Rewrite<= metadata-eval (/.f64 1 -1))))): 0 points increase in error, 0 points decrease in error
(+.f64 (*.f64 (/.f64 4 z) (-.f64 x y)) (*.f64 (/.f64 4 z) (/.f64 (*.f64 z 1/2) (/.f64 (Rewrite<= metadata-eval (neg.f64 -1)) -1)))): 0 points increase in error, 0 points decrease in error
(+.f64 (*.f64 (/.f64 4 z) (-.f64 x y)) (*.f64 (/.f64 4 z) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 z 1/2) -1) (neg.f64 -1))))): 0 points increase in error, 0 points decrease in error
(+.f64 (*.f64 (/.f64 4 z) (-.f64 x y)) (*.f64 (/.f64 4 z) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (*.f64 z 1/2))) (neg.f64 -1)))): 0 points increase in error, 0 points decrease in error
(+.f64 (*.f64 (/.f64 4 z) (-.f64 x y)) (*.f64 (/.f64 4 z) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (*.f64 z 1/2))) (neg.f64 -1)))): 0 points increase in error, 0 points decrease in error
(+.f64 (*.f64 (/.f64 4 z) (-.f64 x y)) (*.f64 (/.f64 4 z) (/.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 z) 1/2)) (neg.f64 -1)))): 0 points increase in error, 0 points decrease in error
(+.f64 (*.f64 (/.f64 4 z) (-.f64 x y)) (*.f64 (/.f64 4 z) (/.f64 (*.f64 (neg.f64 z) 1/2) (Rewrite=> metadata-eval 1)))): 0 points increase in error, 0 points decrease in error
(+.f64 (*.f64 (/.f64 4 z) (-.f64 x y)) (*.f64 (/.f64 4 z) (Rewrite=> /-rgt-identity_binary64 (*.f64 (neg.f64 z) 1/2)))): 0 points increase in error, 0 points decrease in error
(Rewrite<= distribute-lft-in_binary64 (*.f64 (/.f64 4 z) (+.f64 (-.f64 x y) (*.f64 (neg.f64 z) 1/2)))): 4 points increase in error, 2 points decrease in error
(*.f64 (/.f64 4 z) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (-.f64 x y) (*.f64 z 1/2)))): 0 points increase in error, 0 points decrease in error
(Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 4 (-.f64 (-.f64 x y) (*.f64 z 1/2))) z)): 3 points increase in error, 59 points decrease in error
Taylor expanded in z around 0 0.0
\[\leadsto \color{blue}{4 \cdot \frac{x - y}{z} - 2} \]
Final simplification0.0
\[\leadsto 4 \cdot \frac{x - y}{z} + -2 \]

Alternatives

Alternative 1
Error	30.6
Cost	1376

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot -4\\ t_1 := 4 \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+47}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -680000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-39}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 2
Error	14.6
Cost	976

\[\begin{array}{l} t_0 := 4 \cdot \frac{x}{z}\\ t_1 := 4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	10.1
Cost	976

\[\begin{array}{l} t_0 := 4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ t_1 := 4 \cdot \frac{x}{z} + -2\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+51}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	30.1
Cost	848

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot -4\\ \mathbf{if}\;z \leq -200000000000:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-39}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 5
Error	11.8
Cost	712

\[\begin{array}{l} t_0 := 4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+80}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	36.4
Cost	64

\[-2 \]

Error

Try it out

Target

Derivation

Alternatives

Error

Reproduce

In
Out